Introduction and main results
The compressible Navier–Stokes–Poisson (NSP) system is a classical model that couples the Navier–Stokes equations with a self-consistent Poisson equation, and is commonly employed to describe the motion of charged particles such as electrons or holes. In this paper, we study the initial-boundary value problem for the compressible Navier–Stokes–Poisson system in a three-dimensional smooth bounded domain Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3
{ ρ t ε + div ( ρ ε u ε ) = 0 , ρ ε u t ε + ρ ε u ε ⋅ ∇ u ε + ∇ p ε = μ ε Δ u ε + ( λ + μ ) ε ∇ div u ε + ρ ε ∇ ϕ ε , Δ ϕ ε = ρ ε − 1 , (1) \begin{cases}
\rho^{\varepsilon}_{t}+\operatorname{div}(\rho^{\varepsilon}u^{\varepsilon})=0,\\
\rho^{\varepsilon}u^{\varepsilon}_{t}+\rho^{\varepsilon}u^{\varepsilon}\cdot\nabla u^{\varepsilon}+\nabla p^{\varepsilon}=\mu\varepsilon\Delta u^{\varepsilon}+(\lambda+\mu)\varepsilon\nabla\operatorname{div}u^{\varepsilon}+\rho^{\varepsilon}\nabla\phi^{\varepsilon},\\
\Delta\phi^{\varepsilon}=\rho^{\varepsilon}-1, \tag*{(1)}
\end{cases} ⎩ ⎨ ⎧ ρ t ε + div ( ρ ε u ε ) = 0 , ρ ε u t ε + ρ ε u ε ⋅ ∇ u ε + ∇ p ε = μ ε Δ u ε + ( λ + μ ) ε ∇ div u ε + ρ ε ∇ ϕ ε , Δ ϕ ε = ρ ε − 1 , (1) where ρ ε \rho^{\varepsilon} ρ ε u ε u^{\varepsilon} u ε ϕ ε \phi^{\varepsilon} ϕ ε
p ε = ( ρ ε ) γ , γ > 1 , p^{\varepsilon}=(\rho^{\varepsilon})^{\gamma},\qquad\gamma>1, p ε = ( ρ ε ) γ , γ > 1 , and the viscosity coefficients satisfy
μ > 0 , 2 μ + 3 λ > 0. \mu>0,\qquad2\mu+3\lambda>0. μ > 0 , 2 μ + 3 λ > 0. Here 0 < ε ≤ 1 0<\varepsilon\leq1 0 < ε ≤ 1
We impose a general Navier–slip boundary condition on the velocity. In its stress form, this condition is written as
u ε ⋅ n = 0 , ( S u ε ⋅ n ) τ = − ( A u ε ) τ on ∂ Ω , u^\varepsilon\cdot n = 0,\qquad\left(Su^\varepsilon\cdot n\right)_\tau= -\left(Au^\varepsilon\right)_\tau\quad\text{on } \partial\Omega, u ε ⋅ n = 0 , ( S u ε ⋅ n ) τ = − ( A u ε ) τ on ∂ Ω , where
S u ε = 1 2 ( ∇ u ε + ( ∇ u ε ) T ) . Su^\varepsilon= \frac{1}{2}\left(\nabla u^\varepsilon+ \left(\nabla u^\varepsilon\right)^{T}\right). S u ε = 2 1 ( ∇ u ε + ( ∇ u ε ) T ) . n n n ∂ Ω \partial\Omega ∂ Ω A A A ( ⋅ ) τ (\cdot)_\tau ( ⋅ ) τ
( 2 S ( v ) n − ( ∇ × v ) × n ) τ = − ( 2 S ( n ) v ) τ , \left(2S(v)n - (\nabla\times v)\times n\right)_\tau= -\left(2S(n)v\right)_\tau, ( 2 S ( v ) n − ( ∇ × v ) × n ) τ = − ( 2 S ( n ) v ) τ , this boundary condition can be equivalently rewritten in the vorticity form
u ε ⋅ n = 0 , n × ω ε = [ B u ε ] τ on ∂ Ω , (2) u^\varepsilon\cdot n = 0,\qquad n \times\omega^\varepsilon= \left[Bu^\varepsilon\right]_\tau\quad\text{on } \partial\Omega, \tag*{(2)} u ε ⋅ n = 0 , n × ω ε = [ B u ε ] τ on ∂ Ω , (2) where
ω ε = ∇ × u ε , B = 2 ( A − S ( n ) ) . \omega^\varepsilon= \nabla\times u^\varepsilon,\qquad B = 2(A-S(n)). ω ε = ∇ × u ε , B = 2 ( A − S ( n )) . This vorticity formulation is more convenient for the normal derivative and boundary vorticity estimates. For the electric potential, we impose the homogeneous Neumann boundary condition
∂ n ϕ ε = 0 on ∂ Ω . (3) \partial_n\phi^\varepsilon= 0 \quad\text{on } \partial\Omega. \tag*{(3)} ∂ n ϕ ε = 0 on ∂ Ω. (3) As ε → 0 \varepsilon\to0 ε → 0 (1) formally vanish, and the corresponding limiting system is the compressible Euler–Poisson system
{ ρ t + div ( ρ u ) = 0 , ρ u t + ρ u ⋅ ∇ u + ∇ p = ρ ∇ ϕ , x ∈ Ω , t > 0 , Δ ϕ = ρ − 1 , (4) \begin{cases}
\rho_t + \operatorname{div}(\rho u) = 0,\\
\rho u_t + \rho u\cdot\nabla u + \nabla p = \rho\nabla\phi,\qquad x\in\Omega,\quad t>0,\\
\Delta\phi= \rho- 1, \tag*{(4)}
\end{cases} ⎩ ⎨ ⎧ ρ t + div ( ρ u ) = 0 , ρ u t + ρ u ⋅ ∇ u + ∇ p = ρ ∇ ϕ , x ∈ Ω , t > 0 , Δ ϕ = ρ − 1 , (4) where p = ρ γ p=\rho^\gamma p = ρ γ
u ⋅ n = 0 , ∂ n ϕ = 0 on ∂ Ω . (5) u\cdot n = 0,\qquad\partial_n\phi= 0 \quad\text{on } \partial\Omega. \tag*{(5)} u ⋅ n = 0 , ∂ n ϕ = 0 on ∂ Ω. (5) Thus, our aim is to establish uniform estimates for (1) –(3) in a conormal Sobolev framework and to justify the inviscid limit from (1) to (4) .
The vanishing viscosity limit for the Navier–Stokes equations is a classical problem in mathematical fluid mechanics. In the absence of boundaries, the limit can be justified in the standard Sobolev framework as long as the corresponding Euler solution remains smooth [15 ] . In domains with physical boundaries, however, the problem becomes much more delicate because viscous and inviscid flows usually satisfy different boundary conditions. For the no-slip boundary condition, the inviscid limit is closely related to the Prandtl boundary layer theory and remains largely open, except in some special settings such as analytic data, vorticity initially separated from the boundary, or certain steady regimes [24 , 25 , 21 , 23 ] . In contrast, Navier–slip type boundary conditions are more compatible with the inviscid boundary condition and therefore make it possible to establish uniform regularity estimates. For incompressible flows, uniform regularity and the inviscid limit under slip or Navier-type boundary conditions were studied in [35 , 2 , 10 , 36 ] . A decisive progress was made by Masmoudi and Rousset [22 ] , who developed conormal Sobolev estimates in general domains and justified the vanishing viscosity limit by compactness. This conormal approach was later adapted to the compressible Navier–Stokes equations with general Navier–slip boundary conditions by Wang, Xin and Yong [34 ] , and to the full compressible Navier–Stokes system by Wang [33 ] . These works provide the main analytical background for the present paper, where conormal derivatives are used to capture tangential regularity while the missing normal regularity is recovered from the equations and the boundary structure.
The main object of this paper is the compressible Navier–Stokes–Poisson system. This model describes the motion of a compressible charged fluid coupled with a self-consistent electrostatic field and appears in plasma physics, semiconductor theory, ionized gas dynamics and related hydrodynamic models of charged particles. A systematic introduction to the compressible Navier–Stokes–Poisson equations can be found in Hsiao and Li [9 ] . Compared with the usual compressible Navier–Stokes equations, the Poisson equation introduces a nonlocal elliptic coupling into the momentum equation, and the electric force has to be estimated together with the fluid variables. This changes both the energy structure and the regularity analysis, especially in the presence of physical boundaries.
There is a large literature on the well-posedness and large-time behavior of the Navier–Stokes–Poisson system. Weak solutions were studied, for example, in [5 , 16 , 37 ] , where the main difficulty is to combine the compactness of the fluid variables with the elliptic structure of the self-consistent potential. For strong solutions in the Cauchy problem, global existence and decay around equilibrium states have been obtained in various perturbative frameworks [8 , 17 , 26 , 32 , 28 , 3 ] . These works show that the electric field may produce additional dissipation or dispersive effects, but at the same time creates new coupling terms in the high-order energy estimates. For problems with boundaries or nontrivial background profiles, stability of steady states, global well-posedness and decay have also been investigated in bounded or exterior domains [27 , 19 , 20 , 4 ] . In particular, the treatment of boundary conditions for the electric potential is essential, since different choices of boundary conditions may lead to different boundary mechanisms. There are also related models with additional physical effects, such as damping term, heat conductivity, capillarity or quantum/Korteweg-type corrections [29 , 30 , 14 , 1 ] . These variants further illustrate that the Poisson coupling interacts strongly with the dissipative and dispersive structures of the fluid equations.
Singular limits for the Navier–Stokes–Poisson system form another important direction. The quasi-neutral limit, the vanishing Debye length limit, the relaxation limit and the vanishing viscosity limit have been studied by compactness methods, relative entropy methods, dispersive estimates, boundary layer analysis and energy estimates. In the whole space or periodic domains, the convergence from Navier–Stokes–Poisson type systems to incompressible Euler, Euler–Poisson or related limiting systems has been considered in [31 , 6 , 11 , 18 ] . For half-space or boundary problems, the interaction between the Poisson equation and the boundary conditions becomes more subtle. Ju and Xu [12 ] studied quasi-neutral and zero-viscosity limits for the Navier–Stokes–Poisson equations in the half-space. Ju, Luo and Xu [13 ] justified singular limits for a viscous plasma under Navier–slip boundary condition for the velocity and Dirichlet boundary condition for the electric potential; in that problem, the main difficulty is the strong density boundary layer generated by the prescribed boundary potential and its interaction with the weak velocity boundary layer. In contrast, when the electric potential satisfies a homogeneous Neumann boundary condition, the electrostatic energy is naturally compatible with integration by parts, and the strong boundary layer mechanism caused by the prescribed potential is absent. Nevertheless, uniform high-order estimates remain nontrivial because one still has to control the nonlocal electric field together with the compressible fluid variables.
Recently, uniform regularity and vanishing viscosity limits for Navier–Stokes–Poisson type systems have attracted further attention. For the compressible Navier–Stokes–Poisson system with damping term under the slip boundary condition, Gao, Peng, Wei and Yao [7 ] established global-in-time uniform bounds and obtained the vanishing viscosity limit with convergence rates. The damping term plays a stabilizing role in their analysis and allows global-in-time estimates. The present paper is concerned with a different setting: we consider the compressible Navier–Stokes– Stokes–Poisson system without damping term in a general three-dimensional bounded domain, equipped with the Navier–slip boundary condition for the velocity and the homogeneous Neumann boundary condition for the electrostatic potential. Therefore, the problem is local in time, but it retains the essential difficulties caused by the curved boundary, the compressibility and the nonlocal Poisson coupling.
The goal of this paper is to establish uniform conormal regularity estimates for the Navier–Stokes–Poisson system, independently of the viscosity, and then to justify the vanishing viscosity limit to the corresponding Euler–Poisson system. Compared with the compressible Navier–Stokes theory [34 , 33 ] , the new difficulty is the electric force ρ ∇ ϕ \rho\nabla\phi ρ ∇ ϕ ∇ ϕ \nabla\phi ∇ ϕ L ∞ L^\infty L ∞ ϕ \phi ϕ
We now state the main results. The system and boundary conditions have been given above. Throughout this paper, for each 0 < ε ≤ 1 0 < \varepsilon\le1 0 < ε ≤ 1 ϕ 0 \phi_0 ϕ 0
Δ ϕ 0 = ρ 0 − 1 , ∂ n ϕ 0 = 0 on ∂ Ω . \Delta\phi_0=\rho_0-1,\qquad\partial_n\phi_0=0\quad\text{on }\partial\Omega. Δ ϕ 0 = ρ 0 − 1 , ∂ n ϕ 0 = 0 on ∂ Ω. We also assume the compatibility condition
∫ Ω ( ρ 0 − 1 ) d x = 0. \int_\Omega(\rho_0-1)\,dx=0. ∫ Ω ( ρ 0 − 1 ) d x = 0. We define the functional space adapted to the viscous Navier–Stokes–Poisson system. For 0 < ε ≤ 1 0 < \varepsilon\le1 0 < ε ≤ 1
∥ ( p , u , ∇ ϕ ) ( t ) ∥ X m ε 2 : = ∥ ( p , u , ∇ ϕ ) ( t ) ∥ H m 2 + ∥ ∇ u ( t ) ∥ H m − 1 2 + ∑ k = 0 m − 2 ∥ ∂ t k ∇ p ( t ) ∥ m − 1 − k 2 + ∥ Δ p ( t ) ∥ H 1 2 + ∥ ( ∇ u , ∇ ϕ ) ( t ) ∥ H 1 , ∞ 2 + ε ∥ ∇ ∂ t m − 1 p ( t ) ∥ L 2 2 + ε ∥ Δ p ( t ) ∥ H 2 2 . (6) \begin{aligned}
\|(p,u,\nabla\phi)(t)\|_{X_m^\varepsilon}^2 &:=
\|(p,u,\nabla\phi)(t)\|_{\mathcal{H}^m}^2+\|\nabla u(t)\|_{\mathcal{H}^{m-1}}^2\\
&\quad+\sum_{k=0}^{m-2}\|\partial_t^k\nabla p(t)\|_{m-1-k}^2+\|\Delta p(t)\|_{\mathcal{H}^1}^2\\
&\quad+\|(\nabla u,\nabla\phi)(t)\|_{\mathcal{H}^{1,\infty}}^2+\varepsilon\|\nabla\partial_t^{m-1}p(t)\|_{L^2}^2+\varepsilon\|\Delta p(t)\|_{\mathcal{H}^2}^2. \tag*{(6)}
\end{aligned} ∥ ( p , u , ∇ ϕ ) ( t ) ∥ X m ε 2 := ∥ ( p , u , ∇ ϕ ) ( t ) ∥ H m 2 + ∥∇ u ( t ) ∥ H m − 1 2 + k = 0 ∑ m − 2 ∥ ∂ t k ∇ p ( t ) ∥ m − 1 − k 2 + ∥Δ p ( t ) ∥ H 1 2 + ∥ ( ∇ u , ∇ ϕ ) ( t ) ∥ H 1 , ∞ 2 + ε ∥∇ ∂ t m − 1 p ( t ) ∥ L 2 2 + ε ∥Δ p ( t ) ∥ H 2 2 . (6) For the limiting Euler–Poisson system (4) , we use the notation
∥ ( p , u , ∇ ϕ ) ( t ) ∥ X m 0 2 : = ∥ ( p , u , ∇ ϕ ) ( t ) ∥ H m 2 + ∥ ∇ u ( t ) ∥ H m − 1 2 + ∑ k = 0 m − 2 ∥ ∂ t k ∇ p ( t ) ∥ m − 1 − k 2 + ∥ Δ p ( t ) ∥ H 1 2 + ∥ ( ∇ u , ∇ ϕ ) ( t ) ∥ H 1 , ∞ 2 . (7) \begin{aligned}
\|(p,u,\nabla\phi)(t)\|_{X_m^0}^2 &:=
\|(p,u,\nabla\phi)(t)\|_{\mathcal{H}^m}^2+\|\nabla u(t)\|_{\mathcal{H}^{m-1}}^2\\
&\quad+\sum_{k=0}^{m-2}\|\partial_t^k\nabla p(t)\|_{m-1-k}^2+\|\Delta p(t)\|_{\mathcal{H}^1}^2\\
&\quad+\|(\nabla u,\nabla\phi)(t)\|_{\mathcal{H}^{1,\infty}}^2. \tag*{(7)}
\end{aligned} ∥ ( p , u , ∇ ϕ ) ( t ) ∥ X m 0 2 := ∥ ( p , u , ∇ ϕ ) ( t ) ∥ H m 2 + ∥∇ u ( t ) ∥ H m − 1 2 + k = 0 ∑ m − 2 ∥ ∂ t k ∇ p ( t ) ∥ m − 1 − k 2 + ∥Δ p ( t ) ∥ H 1 2 + ∥ ( ∇ u , ∇ ϕ ) ( t ) ∥ H 1 , ∞ 2 . (7) Here ∥ ⋅ ∥ H m \|\cdot\|_{\mathcal{H}^m} ∥ ⋅ ∥ H m ∥ ⋅ ∥ k \|\cdot\|_k ∥ ⋅ ∥ k ∥ ⋅ ∥ H 1 , ∞ \|\cdot\|_{\mathcal{H}^{1,\infty}} ∥ ⋅ ∥ H 1 , ∞ L ∞ L^\infty L ∞ § 2 .
Compared with the space used for the compressible Navier–Stokes equations, the space in (1.6) contains the additional electric-field term ∥ ∇ ϕ ( t ) ∥ H m 2 \|\nabla\phi(t)\|_{\mathcal{H}^m}^2 ∥∇ ϕ ( t ) ∥ H m 2 ρ ∇ ϕ \rho\nabla\phi ρ ∇ ϕ
To complete the setup of our functional framework, we define the uniform solution space X m ε ( T ) X_m^\varepsilon(T) X m ε ( T )
X m ε ( T ) = { ( p , u , ∇ ϕ ) ∈ L ∞ ( [ 0 , T ] ; L 2 ( Ω ) ) | e s s s u p 0 ≤ t ≤ T ∥ ( p , u , ∇ ϕ ) ( t ) ∥ X m ε < + ∞ } . (8) X_m^\varepsilon(T)=\left\{(p,u,\nabla\phi)\in L^\infty([0,T];L^2(\Omega))\ \middle|\ \mathop{\mathrm{ess\,sup}}_{0\le t\le T}\|(p,u,\nabla\phi)(t)\|_{X_m^\varepsilon}<+\infty\right\}. \tag*{(8)} X m ε ( T ) = { ( p , u , ∇ ϕ ) ∈ L ∞ ([ 0 , T ] ; L 2 ( Ω )) ess sup 0 ≤ t ≤ T ∥ ( p , u , ∇ ϕ ) ( t ) ∥ X m ε < + ∞ } . (8) We assume that the initial data satisfy
0 < 1 C ^ 0 ≤ ρ 0 ( x ) ≤ C ^ 0 < ∞ , (9) 0 < \frac{1}{\widehat{C}_0} \leq\rho_0(x) \leq\widehat{C}_0 < \infty, \tag*{(9)} 0 < C 0 1 ≤ ρ 0 ( x ) ≤ C 0 < ∞ , (9) and
sup 0 ≤ ε ≤ 1 ∥ ( p 0 , u 0 , ∇ ϕ 0 ) ∥ X ε m ≤ C ~ 0 , (10) \sup_{0\leq\varepsilon\leq1}\left\|(p_0,u_0,\nabla\phi_0)\right\|_{X_\varepsilon^m}\leq\widetilde{C}_0, \tag*{(10)} 0 ≤ ε ≤ 1 sup ∥ ( p 0 , u 0 , ∇ ϕ 0 ) ∥ X ε m ≤ C 0 , (10) where
p 0 = ( ρ 0 ) γ . p_0=(\rho_0)^\gamma. p 0 = ( ρ 0 ) γ . Theorem 1.1 (Uniform regularity). Let m ≥ 6 m\geq6 m ≥ 6 Ω \Omega Ω C m + 2 C^{m+2} C m + 2 A A A
A ∈ C m + 1 ( ∂ Ω ) . A\in C^{m+1}(\partial\Omega). A ∈ C m + 1 ( ∂ Ω ) . Let
( p 0 , u 0 , ∇ ϕ 0 ) (p_0,u_0,\nabla\phi_0) ( p 0 , u 0 , ∇ ϕ 0 ) be initial data satisfying (1.9)–(1.10) and the compatibility conditions. Then there exist a time T 0 > 0 T_0>0 T 0 > 0 C ~ 1 > 0 \widetilde{C}_1>0 C 1 > 0 ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ]
( ρ ε , u ε , ∇ ϕ ε ) (\rho^\varepsilon,u^\varepsilon,\nabla\phi^\varepsilon) ( ρ ε , u ε , ∇ ϕ ε ) on [ 0 , T 0 ] [0,T_0] [ 0 , T 0 ]
1 2 C ^ 0 ≤ ρ ε ( t , x ) ≤ 2 C ^ 0 , ( t , x ) ∈ [ 0 , T 0 ] × Ω , (11) \frac{1}{2\widehat{C}_0}\leq\rho^\varepsilon(t,x)\leq2\widehat{C}_0,\qquad(t,x)\in[0,T_0]\times\Omega, \tag*{(11)} 2 C 0 1 ≤ ρ ε ( t , x ) ≤ 2 C 0 , ( t , x ) ∈ [ 0 , T 0 ] × Ω , (11) and the following uniform estimate:
sup 0 ≤ t ≤ T 0 { ∥ ( p ε , u ε , ∇ ϕ ε ) ( t ) ∥ H m 2 + ∥ ∇ u ε ( t ) ∥ H m − 1 2 + ∑ k = 0 m − 2 ∥ ∂ t k ∇ p ε ( t ) ∥ H m − 1 − k 2 + ∥ Δ p ε ( t ) ∥ H 1 2 + ∥ ( ∇ u ε , ∇ ϕ ε ) ( t ) ∥ H 1 , ∞ 2 + ε ∥ ∇ ∂ t m − 1 p ε ( t ) ∥ L 2 2 + ε ∥ Δ p ε ( t ) ∥ H 2 2 } + ∫ 0 T 0 ∥ ∇ ∂ t m − 1 p ε ( t ) ∥ L 2 2 d t + ∫ 0 T 0 ∥ Δ p ε ( t ) ∥ H 2 2 d t + ε ∫ 0 T 0 ∥ ∇ u ε ( t ) ∥ H m 2 d t + ε ∑ k = 0 m − 2 ∫ 0 T 0 ∥ ∇ 2 ∂ t k u ε ( t ) ∥ H m − 1 − k 2 d t + ε 2 ∫ 0 T 0 ∥ ∇ 2 ∂ t m − 1 u ε ( t ) ∥ L 2 2 d t ≤ C ~ 1 . (12) \begin{aligned}
\sup_{0\leq t\leq T_0}\Big\{&
\left\|(p^\varepsilon,u^\varepsilon,\nabla\phi^\varepsilon)(t)\right\|_{H^m}^2
+\left\|\nabla u^\varepsilon(t)\right\|_{H^{m-1}}^2
+\sum_{k=0}^{m-2}\left\|\partial_t^k\nabla p^\varepsilon(t)\right\|_{H^{m-1-k}}^2\\
&+\left\|\Delta p^\varepsilon(t)\right\|_{H^1}^2
+\left\|(\nabla u^\varepsilon,\nabla\phi^\varepsilon)(t)\right\|_{H^{1,\infty}}^2
+\varepsilon\left\|\nabla\partial_t^{m-1}p^\varepsilon(t)\right\|_{L^2}^2\\
&+\varepsilon\left\|\Delta p^\varepsilon(t)\right\|_{H^2}^2\Big\}
+\int_0^{T_0}\left\|\nabla\partial_t^{m-1}p^\varepsilon(t)\right\|_{L^2}^2\,dt\\
&+\int_0^{T_0}\left\|\Delta p^\varepsilon(t)\right\|_{H^2}^2\,dt
+\varepsilon\int_0^{T_0}\left\|\nabla u^\varepsilon(t)\right\|_{H^m}^2\,dt\\
&+\varepsilon\sum_{k=0}^{m-2}\int_0^{T_0}\left\|\nabla^2\partial_t^k u^\varepsilon(t)\right\|_{H^{m-1-k}}^2\,dt
+\varepsilon^2\int_0^{T_0}\left\|\nabla^2\partial_t^{m-1}u^\varepsilon(t)\right\|_{L^2}^2\,dt
\leq\widetilde{C}_1. \tag*{(12)}
\end{aligned} 0 ≤ t ≤ T 0 sup { ∥ ( p ε , u ε , ∇ ϕ ε ) ( t ) ∥ H m 2 + ∥ ∇ u ε ( t ) ∥ H m − 1 2 + k = 0 ∑ m − 2 ∂ t k ∇ p ε ( t ) H m − 1 − k 2 + ∥ Δ p ε ( t ) ∥ H 1 2 + ∥ ( ∇ u ε , ∇ ϕ ε ) ( t ) ∥ H 1 , ∞ 2 + ε ∇ ∂ t m − 1 p ε ( t ) L 2 2 + ε ∥ Δ p ε ( t ) ∥ H 2 2 } + ∫ 0 T 0 ∇ ∂ t m − 1 p ε ( t ) L 2 2 d t + ∫ 0 T 0 ∥ Δ p ε ( t ) ∥ H 2 2 d t + ε ∫ 0 T 0 ∥ ∇ u ε ( t ) ∥ H m 2 d t + ε k = 0 ∑ m − 2 ∫ 0 T 0 ∇ 2 ∂ t k u ε ( t ) H m − 1 − k 2 d t + ε 2 ∫ 0 T 0 ∇ 2 ∂ t m − 1 u ε ( t ) L 2 2 d t ≤ C 1 . (12) Here C ~ 1 \widetilde{C}_1 C 1 C ^ 0 \widehat{C}_0 C 0 C ~ 0 \widetilde{C}_0 C 0 m m m Ω \Omega Ω A A A
For 0 < ε ≤ 1 0<\varepsilon\leq1 0 < ε ≤ 1
( ρ ε , u ε , ∇ ϕ ε ) (\rho^\varepsilon,u^\varepsilon,\nabla\phi^\varepsilon) ( ρ ε , u ε , ∇ ϕ ε ) the solution on [ 0 , T 0 ] [0,T_0] [ 0 , T 0 ] ε → 0 \varepsilon\to0 ε → 0
Theorem 1.2. Let m ≥ 6 m\geq6 m ≥ 6 Ω \Omega Ω C m + 2 C^{m+2} C m + 2 A ∈ C m + 1 ( ∂ Ω ) A\in C^{m+1}(\partial\Omega) A ∈ C m + 1 ( ∂ Ω ) ( p 0 , u 0 , ∇ ϕ 0 ) (p_0,u_0,\nabla\phi_0) ( p 0 , u 0 , ∇ ϕ 0 ) 0 < ε ≤ 1 0<\varepsilon\leq1 0 < ε ≤ 1 ( ρ ε , u ε , ∇ ϕ ε ) (\rho^\varepsilon,u^\varepsilon,\nabla\phi^\varepsilon) ( ρ ε , u ε , ∇ ϕ ε ) [ 0 , T 0 ] [0,T_0] [ 0 , T 0 ]
with initial data ( p 0 , u 0 , ∇ ϕ 0 ) (p_0,u_0,\nabla\phi_0) ( p 0 , u 0 , ∇ ϕ 0 ) Theorem 1.1 . Then the Euler–Poisson system (1.4) with the boundary conditions
u ⋅ n = 0 , ∂ n ϕ = 0 on ∂ Ω u\cdot n=0,\qquad\partial_n\phi=0\quad\text{on }\partial\Omega u ⋅ n = 0 , ∂ n ϕ = 0 on ∂ Ω and the initial condition
( ρ , u , ∇ ϕ ) ( x , 0 ) = ( ρ 0 , u 0 , ∇ ϕ 0 ) ( x ) , x ∈ Ω , (\rho,u,\nabla\phi)(x,0)=(\rho_0,u_0,\nabla\phi_0)(x),\qquad x\in\Omega, ( ρ , u , ∇ ϕ ) ( x , 0 ) = ( ρ 0 , u 0 , ∇ ϕ 0 ) ( x ) , x ∈ Ω , admits a unique solution ( ρ , u , ∇ ϕ ) (\rho,u,\nabla\phi) ( ρ , u , ∇ ϕ ) [ 0 , T 0 ] [0,T_0] [ 0 , T 0 ]
sup 0 ≤ t ≤ T 0 ∥ ( p , u , ∇ ϕ ) ( t ) ∥ X m m < ∞ , \sup_{0\leq t\leq T_0}\|(p,u,\nabla\phi)(t)\|_{X_m^m}<\infty, 0 ≤ t ≤ T 0 sup ∥ ( p , u , ∇ ϕ ) ( t ) ∥ X m m < ∞ , where p = ρ γ p=\rho^\gamma p = ρ γ
lim ε → 0 sup 0 ≤ t ≤ T 0 ( ∥ ( p ε , u ε , ∇ ϕ ε ) − ( p , u , ∇ ϕ ) ∥ L 2 ( Ω ) + ∥ ( p ε , u ε , ∇ ϕ ε ) − ( p , u , ∇ ϕ ) ∥ L ∞ ( Ω ) ) = 0. \lim_{\varepsilon\to0}\sup_{0\leq t\leq T_0}\left(\|(p^\varepsilon,u^\varepsilon,\nabla\phi^\varepsilon)-(p,u,\nabla\phi)\|_{L^2(\Omega)}
+\|(p^\varepsilon,u^\varepsilon,\nabla\phi^\varepsilon)-(p,u,\nabla\phi)\|_{L^\infty(\Omega)}\right)=0. ε → 0 lim 0 ≤ t ≤ T 0 sup ( ∥ ( p ε , u ε , ∇ ϕ ε ) − ( p , u , ∇ ϕ ) ∥ L 2 ( Ω ) + ∥ ( p ε , u ε , ∇ ϕ ε ) − ( p , u , ∇ ϕ ) ∥ L ∞ ( Ω ) ) = 0. Remark 1.3. The term
∥ ∇ ϕ ε ( t ) ∥ H m 2 \|\nabla\phi^\varepsilon(t)\|_{H_m^2} ∥∇ ϕ ε ( t ) ∥ H m 2 is the new component in the conormal energy space compared with the compressible Navier–Stokes case. It is generated by the electric force ρ ε ∇ ϕ ε \rho^\varepsilon\nabla\phi^\varepsilon ρ ε ∇ ϕ ε
Remark 1.4. While this paper employs the homogeneous Neumann condition (1.3), the conormal framework can also be adapted to the insulated boundary setting through the Dirichlet condition
ϕ ∣ ∂ Ω = 0. \phi|_{\partial\Omega}=0. ϕ ∣ ∂ Ω = 0. Under this alternative boundary condition, the normal derivative ∂ n ϕ \partial_n\phi ∂ n ϕ ∂ Ω \partial\Omega ∂ Ω L 2 L^2 L 2 div u \operatorname{div}u div u
∫ ∂ Ω ρ ∂ n ϕ div u d S , \int_{\partial\Omega}\rho\,\partial_n\phi\,\operatorname{div}u\,dS, ∫ ∂ Ω ρ ∂ n ϕ div u d S , which cannot be controlled directly by the low-order energy.
To overcome this difficulty, one may temporarily postpone Lemma 3.2 and first proceed with the high-order conormal estimates, Lemma 3.3, together with the normal derivative estimates. These estimates do not introduce additional uncontrolled boundary terms, since the conormal derivatives Z α Z^\alpha Z α ε \varepsilon ε div u \operatorname{div}u div u ∂ Ω \partial\Omega ∂ Ω
We now outline the proof of Theorem 1.1 . The proof is based on a sequence of uniform a priori estimates in conormal Sobolev spaces. The basic and high-order conormal energy estimates are first obtained for ( p ε , u ε , ∇ ϕ ε ) (p^\varepsilon,u^\varepsilon,\nabla\phi^\varepsilon) ( p ε , u ε , ∇ ϕ ε ) div u ε \operatorname{div}u^\varepsilon div u ε div u ε \operatorname{div}u^\varepsilon div u ε ∇ p ε \nabla p^\varepsilon ∇ p ε L ∞ L^\infty L ∞ Δ p ε \Delta p^\varepsilon Δ p ε Theorem 1.1 by a standard approximation and continuity argument.
The paper is organized as follows. In § 2 , we introduce the conormal Sobolev spaces, the local coordinate system near the boundary, and several basic inequalities which will be used throughout the paper. § 3 is devoted to the uniform a priori estimates. We first derive the basic energy estimates and the high-order conormal estimates, then establish the normal derivative estimates, the L ∞ L^\infty L ∞ Δ p ε \Delta p^\varepsilon Δ p ε § 4 , we complete the proofs of Theorem 1.1 and Theorem 1.2 by a standard approximation, compactness and continuity argument.
Preliminaries
In this section, we collect some notation and inequalities which will be used throughout the paper. Since boundary layers may appear in the presence of physical boundaries, it is in general impossible to obtain uniform estimates in standard Sobolev spaces. Following the conormal framework developed by Wang, Xin and Yong [34 ] for the compressible Navier–Stokes equations with general Navier–slip boundary conditions, we derive uniform estimates for the Navier–Stokes–Poisson system. Compared with [34 ] , the main new difficulty comes from the Poisson coupling and the electric force ρ ∇ ϕ \rho\nabla\phi ρ ∇ ϕ
Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3 Ω \Omega Ω
Ω ⊂ Ω 0 ∪ ⋃ k = 1 n Ω k , (13) \Omega\subset\Omega_{0} \cup\bigcup_{k=1}^{n}\Omega_{k}, \tag*{(13)} Ω ⊂ Ω 0 ∪ k = 1 ⋃ n Ω k , (13) where Ω ‾ 0 ⊂ Ω \overline{\Omega}_{0} \subset\Omega Ω 0 ⊂ Ω Ω k \Omega_{k} Ω k ψ k \psi_{k} ψ k
Ω ∩ Ω k = { x = ( x 1 , x 2 , x 3 ) ∣ x 3 > ψ k ( x 1 , x 2 ) } ∩ Ω k , \Omega\cap\Omega_{k}=\{x=(x_{1},x_{2},x_{3})\mid x_{3}>\psi_{k}(x_{1},x_{2})\}\cap\Omega_{k}, Ω ∩ Ω k = { x = ( x 1 , x 2 , x 3 ) ∣ x 3 > ψ k ( x 1 , x 2 )} ∩ Ω k , and
∂ Ω ∩ Ω k = { x = ( x 1 , x 2 , x 3 ) ∣ x 3 = ψ k ( x 1 , x 2 ) } ∩ Ω k . \partial\Omega\cap\Omega_{k}=\{x=(x_{1},x_{2},x_{3})\mid x_{3}=\psi_{k}(x_{1},x_{2})\}\cap\Omega_{k}. ∂ Ω ∩ Ω k = { x = ( x 1 , x 2 , x 3 ) ∣ x 3 = ψ k ( x 1 , x 2 )} ∩ Ω k . We say that Ω \Omega Ω C m C^{m} C m ψ k \psi_{k} ψ k C m C^{m} C m
To define the Sobolev conormal spaces, we consider ( Z k ) 1 ≤ k ≤ N (Z_{k})_{1\leq k\leq N} ( Z k ) 1 ≤ k ≤ N ∂ Ω \partial\Omega ∂ Ω
H c o m = { f ∈ L 2 ( Ω ) ∣ Z I f ∈ L 2 ( Ω ) , ∣ I ∣ ≤ m } , H^{m}_{co}=\{f\in L^{2}(\Omega)\mid Z^{I}f\in L^{2}(\Omega),\ |I|\leq m\}, H co m = { f ∈ L 2 ( Ω ) ∣ Z I f ∈ L 2 ( Ω ) , ∣ I ∣ ≤ m } , where I = ( k 1 , … , k l ) I=(k_{1},\ldots,k_{l}) I = ( k 1 , … , k l )
∥ u ∥ m 2 = ∥ u ∥ H c o m 2 = ∑ j = 1 3 ∑ ∣ I ∣ ≤ m ∥ Z I u j ∥ L 2 2 , ∥ u ∥ m , ∞ = ∑ j = 1 3 ∑ ∣ I ∣ ≤ m ∥ Z I u j ∥ L ∞ 2 . \|u\|_{m}^{2}=\|u\|_{H^{m}_{co}}^{2}=\sum_{j=1}^{3}\sum_{|I|\leq m}\|Z^{I}u_{j}\|_{L^{2}}^{2},\qquad\|u\|_{m,\infty}=\sum_{j=1}^{3}\sum_{|I|\leq m}\|Z^{I}u_{j}\|_{L^{\infty}}^{2}. ∥ u ∥ m 2 = ∥ u ∥ H co m 2 = j = 1 ∑ 3 ∣ I ∣ ≤ m ∑ ∥ Z I u j ∥ L 2 2 , ∥ u ∥ m , ∞ = j = 1 ∑ 3 ∣ I ∣ ≤ m ∑ ∥ Z I u j ∥ L ∞ 2 . and
∥ ∇ Z m u ∥ 2 = ∑ ∣ I ∣ = m ∥ ∇ Z I u ∥ L 2 2 . \lVert\nabla Z^m u \rVert^2 = \sum_{|I|=m} \lVert\nabla Z^I u \rVert_{L^2}^2. ∥ ∇ Z m u ∥ 2 = ∣ I ∣ = m ∑ ∥ ∇ Z I u ∥ L 2 2 . For scalar functions, the same notation is used without the summation over the components.
Noting that by using the covering of Ω \Omega Ω Ω i \Omega_i Ω i Ω 0 \Omega_0 Ω 0 ∥ ⋅ ∥ m \lVert\cdot\rVert_m ∥ ⋅ ∥ m H m H^m H m Ω i ∩ ∂ Ω ≠ ∅ \Omega_i \cap\partial\Omega\ne\emptyset Ω i ∩ ∂ Ω = ∅
Denote by C k C_k C k ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] C k C^k C k ψ j \psi_j ψ j j = 1 , … , n j=1,\ldots,n j = 1 , … , n ∂ Ω \partial\Omega ∂ Ω x 3 = ψ ( x 1 , x 2 ) x_3=\psi(x_1,x_2) x 3 = ψ ( x 1 , x 2 )
Ψ : ( y , z ) ↦ ( y , ψ ( y ) + z ) = x , y = ( y 1 , y 2 ) . \Psi:(y,z)\mapsto(y,\psi(y)+z)=x,\qquad y=(y_1,y_2). Ψ : ( y , z ) ↦ ( y , ψ ( y ) + z ) = x , y = ( y 1 , y 2 ) . A local basis is given by
e y 1 = ( 1 , 0 , ∂ 1 ψ ) t , e y 2 = ( 0 , 1 , ∂ 2 ψ ) t , e z = ( 0 , 0 , − 1 ) t . e_{y_1}=(1,0,\partial_1\psi)^t,\qquad e_{y_2}=(0,1,\partial_2\psi)^t,\qquad e_z=(0,0,-1)^t. e y 1 = ( 1 , 0 , ∂ 1 ψ ) t , e y 2 = ( 0 , 1 , ∂ 2 ψ ) t , e z = ( 0 , 0 , − 1 ) t . On the boundary, e y 1 e_{y_1} e y 1 e y 2 e_{y_2} e y 2 ∂ Ω \partial\Omega ∂ Ω
Z i = ∂ y i = ∂ i + ∂ i ψ ∂ z , i = 1 , 2 , Z 3 = φ ( z ) ∂ z , (14) Z_i=\partial_{y_i}=\partial_i+\partial_i\psi\,\partial_z,\quad i=1,2,\qquad Z_3=\varphi(z)\partial_z, \tag*{(14)} Z i = ∂ y i = ∂ i + ∂ i ψ ∂ z , i = 1 , 2 , Z 3 = φ ( z ) ∂ z , (14) where
φ ( z ) = z 1 + z \varphi(z)=\frac{z}{1+z} φ ( z ) = 1 + z z is smooth on R + \mathbb{R}_+ R + φ ( 0 ) = 0 \varphi(0)=0 φ ( 0 ) = 0 φ ′ ( 0 ) > 0 \varphi'(0)>0 φ ′ ( 0 ) > 0 φ ( z ) > 0 \varphi(z)>0 φ ( z ) > 0 z > 0 z>0 z > 0
Z 0 = ∂ t . Z_0=\partial_t. Z 0 = ∂ t . It is easy to check that
Z k Z j = Z j Z k , j , k = 1 , 2 , 3 , Z_kZ_j=Z_jZ_k,\qquad j,k=1,2,3, Z k Z j = Z j Z k , j , k = 1 , 2 , 3 , and
∂ z Z i = Z i ∂ z , i = 1 , 2 , ∂ z Z 3 ≠ Z 3 ∂ z . \partial_zZ_i=Z_i\partial_z,\qquad i=1,2,\qquad\partial_zZ_3\ne Z_3\partial_z. ∂ z Z i = Z i ∂ z , i = 1 , 2 , ∂ z Z 3 = Z 3 ∂ z . For a vector field u u u ( e y 1 , e y 2 , e z ) (e_{y_1},e_{y_2},e_z) ( e y 1 , e y 2 , e z ) u i u^i u i
u = u 1 e y 1 + u 2 e y 2 + u 3 e z . u=u^1e_{y_1}+u^2e_{y_2}+u^3e_z. u = u 1 e y 1 + u 2 e y 2 + u 3 e z . We use u i u_i u i u u u R 3 \mathbb{R}^3 R 3
n ( x ) = n ( Ψ ( y , z ) ) = 1 1 + ∣ ∇ ψ ( y ) ∣ 2 ( ∂ 1 ψ ( y ) ∂ 2 ψ ( y ) − 1 ) . n(x)=n(\Psi(y,z))=\frac{1}{\sqrt{1+\lvert\nabla\psi(y)\rvert^2}}
\begin{pmatrix}
\partial_1\psi(y)\\
\partial_2\psi(y)\\
-1
\end{pmatrix}. n ( x ) = n ( Ψ ( y , z )) = 1 + ∣ ∇ ψ ( y ) ∣ 2 1 ∂ 1 ψ ( y ) ∂ 2 ψ ( y ) − 1 . We also define the orthogonal projection onto the tangent plane by
Π u = u − ( u ⋅ n ) n . \Pi u=u-(u\cdot n)n. Π u = u − ( u ⋅ n ) n . In the local coordinate patch, both n n n Π \Pi Π z z z
For a space-time multi-index
α = ( α 0 , α 1 ) , α 1 = ( α 11 , α 12 , α 13 ) , \alpha=(\alpha_0,\alpha_1),\qquad\alpha_1=(\alpha_{11},\alpha_{12},\alpha_{13}), α = ( α 0 , α 1 ) , α 1 = ( α 11 , α 12 , α 13 ) , we write
Z α = ∂ t α 0 Z 1 α 11 Z 2 α 12 Z 3 α 13 , ∣ α ∣ = α 0 + α 11 + α 12 + α 13 . Z^\alpha=\partial_t^{\alpha_0}Z_1^{\alpha_{11}}Z_2^{\alpha_{12}}Z_3^{\alpha_{13}},\qquad\lvert\alpha\rvert=\alpha_0+\alpha_{11}+\alpha_{12}+\alpha_{13}. Z α = ∂ t α 0 Z 1 α 11 Z 2 α 12 Z 3 α 13 , ∣ α ∣ = α 0 + α 11 + α 12 + α 13 . After introducing Z 0 = ∂ t Z_0=\partial_t Z 0 = ∂ t ∥ Z k f ∥ L 2 2 \|Z^k f\|_{L^2}^2 ∥ Z k f ∥ L 2 2 ∥ ∇ Z k f ∥ L 2 2 \|\nabla Z^k f\|_{L^2}^2 ∥∇ Z k f ∥ L 2 2 ∑ ∣ α ∣ = k ∥ Z α f ∥ L 2 2 \sum_{|\alpha|=k}\|Z^\alpha f\|_{L^2}^2 ∑ ∣ α ∣ = k ∥ Z α f ∥ L 2 2 ∑ ∣ α ∣ = k ∥ ∇ Z α f ∥ L 2 2 \sum_{|\alpha|=k}\|\nabla Z^\alpha f\|_{L^2}^2 ∑ ∣ α ∣ = k ∥∇ Z α f ∥ L 2 2 ∥ ⋅ ∥ k \|\cdot\|_k ∥ ⋅ ∥ k ∂ t k p 0 \partial_t^k p_0 ∂ t k p 0 ( ∂ t k p ) ∣ t = 0 (\partial_t^k p)|_{t=0} ( ∂ t k p ) ∣ t = 0 u 0 u_0 u 0 ϕ 0 \phi_0 ϕ 0
For a smooth function f = f ( t , x ) f=f(t,x) f = f ( t , x )
∥ f ( t ) ∥ H m 2 : = ∑ ∣ α ∣ ≤ m ∥ Z α f ( t ) ∥ L 2 ( Ω ) 2 . (15) \|f(t)\|_{H^m}^2 := \sum_{|\alpha|\le m}\|Z^\alpha f(t)\|_{L^2(\Omega)}^2. \tag*{(15)} ∥ f ( t ) ∥ H m 2 := ∣ α ∣ ≤ m ∑ ∥ Z α f ( t ) ∥ L 2 ( Ω ) 2 . (15) and
∥ f ( t ) ∥ H k , ∞ 2 : = ∑ ∣ α ∣ ≤ k ∥ Z α f ( t ) ∥ L ∞ ( Ω ) 2 . (16) \|f(t)\|_{H^{k,\infty}}^2 := \sum_{|\alpha|\le k}\|Z^\alpha f(t)\|_{L^\infty(\Omega)}^2. \tag*{(16)} ∥ f ( t ) ∥ H k , ∞ 2 := ∣ α ∣ ≤ k ∑ ∥ Z α f ( t ) ∥ L ∞ ( Ω ) 2 . (16) Thus, ∥ ⋅ ∥ m \|\cdot\|_m ∥ ⋅ ∥ m Z 1 , Z 2 , Z 3 Z_1,Z_2,Z_3 Z 1 , Z 2 , Z 3 ∥ ⋅ ∥ H m \|\cdot\|_{H^m} ∥ ⋅ ∥ H m Z 0 = ∂ t Z_0=\partial_t Z 0 = ∂ t ∥ ⋅ ∥ \|\cdot\| ∥ ⋅ ∥ Ω \Omega Ω ∣ ⋅ ∣ H s ( ∂ Ω ) |\cdot|_{H^s(\partial\Omega)} ∣ ⋅ ∣ H s ( ∂ Ω ) ∂ Ω \partial\Omega ∂ Ω
Proposition 2.1. Let m ∈ N + m\in\mathbb{N}_+ m ∈ N + u ∈ H m ( Ω ) u\in H^m(\Omega) u ∈ H m ( Ω ) C > 0 C>0 C > 0 u u u
∥ u ∥ H m ≤ C ( ∥ ∇ × u ∥ H m − 1 + ∥ div u ∥ H m − 1 + ∥ u ∥ H m − 1 + ∣ u ⋅ n ∣ H m − 1 2 ( ∂ Ω ) ) , (17) \|u\|_{H^m} \le C\left(\|\nabla\times u\|_{H^{m-1}}+\|\operatorname{div}u\|_{H^{m-1}}+\|u\|_{H^{m-1}}+|u\cdot n|_{H^{m-\frac{1}{2}}(\partial\Omega)}\right), \tag*{(17)} ∥ u ∥ H m ≤ C ( ∥∇ × u ∥ H m − 1 + ∥ div u ∥ H m − 1 + ∥ u ∥ H m − 1 + ∣ u ⋅ n ∣ H m − 2 1 ( ∂ Ω ) ) , (17) and
∥ u ∥ H m ≤ C ( ∥ ∇ × u ∥ H m − 1 + ∥ div u ∥ H m − 1 + ∥ u ∥ H m − 1 + ∣ n × u ∣ H m − 1 2 ( ∂ Ω ) ) . (18) \|u\|_{H^m} \le C\left(\|\nabla\times u\|_{H^{m-1}}+\|\operatorname{div}u\|_{H^{m-1}}+\|u\|_{H^{m-1}}+|n\times u|_{H^{m-\frac{1}{2}}(\partial\Omega)}\right). \tag*{(18)} ∥ u ∥ H m ≤ C ( ∥∇ × u ∥ H m − 1 + ∥ div u ∥ H m − 1 + ∥ u ∥ H m − 1 + ∣ n × u ∣ H m − 2 1 ( ∂ Ω ) ) . (18) We shall also use the following conormal product estimate.
Proposition 2.2. For u , v ∈ L ∞ ( Ω ) ∩ H c o k ( Ω ) u,v\in L^\infty(\Omega)\cap H^k_{\mathrm{co}}(\Omega) u , v ∈ L ∞ ( Ω ) ∩ H co k ( Ω ) α 1 , α 2 \alpha_1,\alpha_2 α 1 , α 2
∥ Z α 1 u Z α 2 v ∥ L 2 ≤ C ( ∥ u ∥ L ∞ ∥ v ∥ H c o k + ∥ v ∥ L ∞ ∥ u ∥ H c o k ) , ∣ α 1 ∣ + ∣ α 2 ∣ = k . (19) \|Z^{\alpha_1}u\,Z^{\alpha_2}v\|_{L^2}\le C\left(\|u\|_{L^\infty}\|v\|_{H^k_{\mathrm{co}}}+\|v\|_{L^\infty}\|u\|_{H^k_{\mathrm{co}}}\right),\qquad|\alpha_1|+|\alpha_2|=k. \tag*{(19)} ∥ Z α 1 u Z α 2 v ∥ L 2 ≤ C ( ∥ u ∥ L ∞ ∥ v ∥ H co k + ∥ v ∥ L ∞ ∥ u ∥ H co k ) , ∣ α 1 ∣ + ∣ α 2 ∣ = k . (19) For space–time conormal derivatives, we use the time-integrated version of this estimate; see [34 Proposition 2.2] . For smooth f , g f,g f , g ∣ β ∣ + ∣ ν ∣ ≤ k |\beta|+|\nu|\le k ∣ β ∣ + ∣ ν ∣ ≤ k 0 < t ≤ T 0<t\le T 0 < t ≤ T
∫ 0 t ∥ Z β f Z ν g ∥ L 2 2 d τ ≤ C ( ∥ f ∥ L ∞ ( [ 0 , t ] × Ω ) 2 ∫ 0 t ∥ g ∥ H k 2 d τ + ∥ g ∥ L ∞ ( [ 0 , t ] × Ω ) 2 ∫ 0 t ∥ f ∥ H k 2 d τ ) . \int_0^t\|Z^\beta f\,Z^\nu g\|_{L^2}^2\,\mathrm{d}\tau
\le C\left(\|f\|_{L^\infty([0,t]\times\Omega)}^2\int_0^t\|g\|_{H^k}^2\,\mathrm{d}\tau
+\|g\|_{L^\infty([0,t]\times\Omega)}^2\int_0^t\|f\|_{H^k}^2\,\mathrm{d}\tau\right). ∫ 0 t ∥ Z β f Z ν g ∥ L 2 2 d τ ≤ C ( ∥ f ∥ L ∞ ([ 0 , t ] × Ω ) 2 ∫ 0 t ∥ g ∥ H k 2 d τ + ∥ g ∥ L ∞ ([ 0 , t ] × Ω ) 2 ∫ 0 t ∥ f ∥ H k 2 d τ ) . The following anisotropic Sobolev embedding and trace estimates will be used frequently.
Proposition 2.3. Let m 1 ≥ 0 m_1\ge0 m 1 ≥ 0 m 2 ≥ 0 m_2\ge0 m 2 ≥ 0 f ∈ H c o m 1 ( Ω ) ∩ H c o m 2 ( Ω ) f\in H_{\mathrm{co}}^{m_1}(\Omega)\cap H_{\mathrm{co}}^{m_2}(\Omega) f ∈ H co m 1 ( Ω ) ∩ H co m 2 ( Ω ) ∇ f ∈ H c o m 2 ( Ω ) \nabla f\in H_{\mathrm{co}}^{m_2}(\Omega) ∇ f ∈ H co m 2 ( Ω )
The following anisotropic Sobolev embedding holds:
∥ f ∥ L ∞ 2 ≤ C ( ∥ ∇ f ∥ H c o m 2 + ∥ f ∥ H c o m 2 ) ∥ f ∥ H c o m 1 , (20) \|f\|_{L^\infty}^2\le C\left(\|\nabla f\|_{H_{\mathrm{co}}^{m_2}}+\|f\|_{H_{\mathrm{co}}^{m_2}}\right)\|f\|_{H_{\mathrm{co}}^{m_1}}, \tag*{(20)} ∥ f ∥ L ∞ 2 ≤ C ( ∥∇ f ∥ H co m 2 + ∥ f ∥ H co m 2 ) ∥ f ∥ H co m 1 , (20) provided that m 1 + m 2 ≥ 3 m_1+m_2\ge3 m 1 + m 2 ≥ 3
The following trace estimate holds:
∣ f ∣ H s ( ∂ Ω ) 2 ≤ C ( ∥ ∇ f ∥ H c o m 2 + ∥ f ∥ H c o m 2 ) ∥ f ∥ H c o m 1 , (21) |f|_{H^s(\partial\Omega)}^2\le C\left(\|\nabla f\|_{H_{\mathrm{co}}^{m_2}}+\|f\|_{H_{\mathrm{co}}^{m_2}}\right)\|f\|_{H_{\mathrm{co}}^{m_1}}, \tag*{(21)} ∣ f ∣ H s ( ∂ Ω ) 2 ≤ C ( ∥∇ f ∥ H co m 2 + ∥ f ∥ H co m 2 ) ∥ f ∥ H co m 1 , (21) provided that m 1 + m 2 ≥ 2 s ≥ 0 m_1+m_2\ge2s\ge0 m 1 + m 2 ≥ 2 s ≥ 0
Proposition 2.4. Assume that Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3 ϕ \phi ϕ
Δ ϕ = q in Ω , ∇ ϕ ⋅ n = 0 on ∂ Ω . \Delta\phi=q \quad\text{in }\Omega,\qquad\nabla\phi\cdot n=0 \quad\text{on }\partial\Omega. Δ ϕ = q in Ω , ∇ ϕ ⋅ n = 0 on ∂ Ω. Then there exists a positive constant C C C Ω \Omega Ω
∥ ∇ 2 ϕ ∥ L 2 ≤ C ( ∥ q ∥ L 2 + ∥ ∇ ϕ ∥ L 2 ) . (22) \lVert\nabla^{2}\phi\rVert_{L^{2}}\leq C\left(\lVert q\rVert_{L^{2}}+\lVert\nabla\phi\rVert_{L^{2}}\right). \tag*{(22)} ∥ ∇ 2 ϕ ∥ L 2 ≤ C ( ∥ q ∥ L 2 + ∥ ∇ ϕ ∥ L 2 ) . (22) Moreover, for the conormal Sobolev space, one has
∥ ∇ 2 ϕ ∥ H s ≤ C s + 1 ( ∥ q ∥ H s + ∥ ∇ 2 ϕ ∥ H s − 1 + ∥ ∇ ϕ ∥ L 2 ) ≤ C s + 1 ( ∥ q ∥ H s + ∥ ∇ ϕ ∥ L 2 ) , (23) \lVert\nabla^{2}\phi\rVert_{H^{s}}\leq C_{s+1}\left(\lVert q\rVert_{H^{s}}+\lVert\nabla^{2}\phi\rVert_{H^{s-1}}+\lVert\nabla\phi\rVert_{L^{2}}\right)\leq C_{s+1}\left(\lVert q\rVert_{H^{s}}+\lVert\nabla\phi\rVert_{L^{2}}\right), \tag*{(23)} ∥ ∇ 2 ϕ ∥ H s ≤ C s + 1 ( ∥ q ∥ H s + ∥ ∇ 2 ϕ ∥ H s − 1 + ∥ ∇ ϕ ∥ L 2 ) ≤ C s + 1 ( ∥ q ∥ H s + ∥ ∇ ϕ ∥ L 2 ) , (23) for s = 1 , … , m s=1,\ldots,m s = 1 , … , m
A Priori Estimates
In this section, we derive a priori estimates for smooth solutions to (1.1) together with the boundary conditions (1.2) and (1.3). For notational convenience, we drop the superscript ε \varepsilon ε ε \varepsilon ε
By the continuity equation, for any 0 ≤ t ≤ T 0\leq t\leq T 0 ≤ t ≤ T
∣ ρ 0 ( x ) ∣ exp ( − ∫ 0 t ∥ div u ∥ L ∞ d τ ) ≤ ρ ( x , t ) ≤ ∣ ρ 0 ( x ) ∣ exp ( ∫ 0 t ∥ div u ∥ L ∞ d τ ) . \lvert\rho_{0}(x)\rvert\exp\left(-\int_{0}^{t}\lVert\operatorname{div}u\rVert_{L^{\infty}}\,\mathrm{d}\tau\right)\leq\rho(x,t)\leq\lvert\rho_{0}(x)\rvert\exp\left(\int_{0}^{t}\lVert\operatorname{div}u\rVert_{L^{\infty}}\,\mathrm{d}\tau\right). ∣ ρ 0 ( x )∣ exp ( − ∫ 0 t ∥ div u ∥ L ∞ d τ ) ≤ ρ ( x , t ) ≤ ∣ ρ 0 ( x )∣ exp ( ∫ 0 t ∥ div u ∥ L ∞ d τ ) . Throughout this section, we shall work on the interval of time [ 0 , T ] [0,T] [ 0 , T ]
0 < c 0 ≤ ρ ( t , x ) ≤ 1 c 0 < ∞ , ( t , x ) ∈ [ 0 , T ] × Ω . 0<c_{0}\leq\rho(t,x)\leq\frac{1}{c_{0}}<\infty,\qquad(t,x)\in[0,T]\times\Omega. 0 < c 0 ≤ ρ ( t , x ) ≤ c 0 1 < ∞ , ( t , x ) ∈ [ 0 , T ] × Ω. We point out that the generic constant C C C 1 / c 0 1/c_{0} 1/ c 0
Let
ω = ∇ × u . \omega=\nabla\times u. ω = ∇ × u . Using
Δ u = ∇ div u − ∇ × ω , \Delta u=\nabla\operatorname{div}u-\nabla\times\omega, Δ u = ∇ div u − ∇ × ω , the momentum equation can be rewritten as
ρ u t + ρ u ⋅ ∇ u + ∇ p = − μ ε ∇ × ω + ( 2 μ + λ ) ε ∇ div u + ρ ∇ ϕ . (24) \rho u_{t}+\rho u\cdot\nabla u+\nabla p=-\mu\varepsilon\nabla\times\omega+(2\mu+\lambda)\varepsilon\nabla\operatorname{div}u+\rho\nabla\phi. \tag*{(24)} ρ u t + ρ u ⋅ ∇ u + ∇ p = − μ ε ∇ × ω + ( 2 μ + λ ) ε ∇ div u + ρ ∇ ϕ . (24) Lemma 3.1 (Basic energy estimate). For a smooth solution of (1.1) subject to (1.2) and (1.3), it holds that, for 0 < ε ≤ 1 0<\varepsilon\leq1 0 < ε ≤ 1
sup 0 ≤ τ ≤ t { ∫ Ω ( 1 2 ρ ∣ u ∣ 2 + 1 γ − 1 ρ γ ) ( τ ) d x + 1 2 ∥ ∇ ϕ ( τ ) ∥ L 2 2 } + c ε ∫ 0 t ∥ ∇ u ( τ ) ∥ L 2 2 d τ ≤ ∫ Ω ( 1 2 ρ 0 ∣ u 0 ∣ 2 + 1 γ − 1 ρ 0 γ ) d x + 1 2 ∥ ∇ ϕ 0 ∥ L 2 2 + C ∫ 0 t ∥ u ( τ ) ∥ L 2 2 d τ , (25) \sup_{0\leq\tau\leq t}\left\{\int_{\Omega}\left(\frac{1}{2}\rho\lvert u\rvert^{2}+\frac{1}{\gamma-1}\rho^{\gamma}\right)(\tau)\,\mathrm{d}x+\frac{1}{2}\lVert\nabla\phi(\tau)\rVert_{L^{2}}^{2}\right\}+c\varepsilon\int_{0}^{t}\lVert\nabla u(\tau)\rVert_{L^{2}}^{2}\,\mathrm{d}\tau
\leq\int_{\Omega}\left(\frac{1}{2}\rho_{0}\lvert u_{0}\rvert^{2}+\frac{1}{\gamma-1}\rho_{0}^{\gamma}\right)\,\mathrm{d}x+\frac{1}{2}\lVert\nabla\phi_{0}\rVert_{L^{2}}^{2}+C\int_{0}^{t}\lVert u(\tau)\rVert_{L^{2}}^{2}\,\mathrm{d}\tau, \tag*{(25)} 0 ≤ τ ≤ t sup { ∫ Ω ( 2 1 ρ ∣ u ∣ 2 + γ − 1 1 ρ γ ) ( τ ) d x + 2 1 ∥ ∇ ϕ ( τ ) ∥ L 2 2 } + c ε ∫ 0 t ∥ ∇ u ( τ ) ∥ L 2 2 d τ ≤ ∫ Ω ( 2 1 ρ 0 ∣ u 0 ∣ 2 + γ − 1 1 ρ 0 γ ) d x + 2 1 ∥ ∇ ϕ 0 ∥ L 2 2 + C ∫ 0 t ∥ u ( τ ) ∥ L 2 2 d τ , (25) where c > 0 c>0 c > 0 μ \mu μ 2 μ + λ 2\mu+\lambda 2 μ + λ
Proof. Multiplying (3.1) by u u u Ω \Omega Ω
∫ Ω ρ u t ⋅ u d x + ∫ Ω ρ ( u ⋅ ∇ u ) ⋅ u d x + ∫ Ω ∇ p ⋅ u d x = − μ ε ∫ Ω ( ∇ × ω ) ⋅ u d x + ( 2 μ + λ ) ε ∫ Ω ∇ div u ⋅ u d x + ∫ Ω ρ ∇ ϕ ⋅ u d x . (26) \int_{\Omega}\rho u_{t}\cdot u\,\mathrm{d}x+\int_{\Omega}\rho(u\cdot\nabla u)\cdot u\,\mathrm{d}x+\int_{\Omega}\nabla p\cdot u\,\mathrm{d}x
=-\mu\varepsilon\int_{\Omega}(\nabla\times\omega)\cdot u\,\mathrm{d}x+(2\mu+\lambda)\varepsilon\int_{\Omega}\nabla\operatorname{div}u\cdot u\,\mathrm{d}x+\int_{\Omega}\rho\nabla\phi\cdot u\,\mathrm{d}x. \tag*{(26)} ∫ Ω ρ u t ⋅ u d x + ∫ Ω ρ ( u ⋅ ∇ u ) ⋅ u d x + ∫ Ω ∇ p ⋅ u d x = − μ ε ∫ Ω ( ∇ × ω ) ⋅ u d x + ( 2 μ + λ ) ε ∫ Ω ∇ div u ⋅ u d x + ∫ Ω ρ ∇ ϕ ⋅ u d x . (26) Using the continuity equation and (2) , we have
∫ Ω ρ u t ⋅ u d x + ∫ Ω ρ ( u ⋅ ∇ u ) ⋅ u d x = 1 2 d d t ∫ Ω ρ ∣ u ∣ 2 d x . (27) \int_{\Omega}\rho u_t\cdot u\,dx+\int_{\Omega}\rho(u\cdot\nabla u)\cdot u\,dx=\frac{1}{2}\frac{d}{dt}\int_{\Omega}\rho|u|^2\,dx. \tag*{(27)} ∫ Ω ρ u t ⋅ u d x + ∫ Ω ρ ( u ⋅ ∇ u ) ⋅ u d x = 2 1 d t d ∫ Ω ρ ∣ u ∣ 2 d x . (27) and, since p = ρ γ p=\rho^\gamma p = ρ γ
∫ Ω ∇ p ⋅ u d x = d d t ∫ Ω 1 γ − 1 ρ γ d x . (28) \int_{\Omega}\nabla p\cdot u\,dx=\frac{d}{dt}\int_{\Omega}\frac{1}{\gamma-1}\rho^\gamma\,dx. \tag*{(28)} ∫ Ω ∇ p ⋅ u d x = d t d ∫ Ω γ − 1 1 ρ γ d x . (28) By (1) , (2) , and (3) , we get
∫ Ω ρ ∇ ϕ ⋅ u d x = − ∫ Ω ϕ div ( ρ u ) d x = ∫ Ω ϕ Δ ϕ t d x = − 1 2 d d t ∥ ∇ ϕ ∥ L 2 2 . (29) \int_{\Omega}\rho\nabla\phi\cdot u\,dx=-\int_{\Omega}\phi\operatorname{div}(\rho u)\,dx=\int_{\Omega}\phi\Delta\phi_t\,dx=-\frac{1}{2}\frac{d}{dt}\|\nabla\phi\|_{L^2}^2. \tag*{(29)} ∫ Ω ρ ∇ ϕ ⋅ u d x = − ∫ Ω ϕ div ( ρ u ) d x = ∫ Ω ϕ Δ ϕ t d x = − 2 1 d t d ∥∇ ϕ ∥ L 2 2 . (29) We next estimate the viscous terms. Integrating by parts with the boundary conditions (2) yields
− ε ∫ Ω ( ∇ × ω ) ⋅ u d x = − ε ∥ ω ∥ L 2 2 − ε ∫ ∂ Ω ( n × ω ) ⋅ u d S ≤ − ε ∥ ω ∥ L 2 2 + C ε ∣ u ∣ L 2 ( ∂ Ω ) 2 , (30) \begin{aligned}
-\varepsilon\int_{\Omega}(\nabla\times\omega)\cdot u\,dx=-\varepsilon\|\omega\|_{L^2}^2-\varepsilon\int_{\partial\Omega}(n\times\omega)\cdot u\,dS \\
&\leq-\varepsilon\|\omega\|_{L^2}^2+C\varepsilon|u|_{L^2(\partial\Omega)}^2, \tag*{(30)}
\end{aligned} − ε ∫ Ω ( ∇ × ω ) ⋅ u d x = − ε ∥ ω ∥ L 2 2 − ε ∫ ∂ Ω ( n × ω ) ⋅ u d S ≤ − ε ∥ ω ∥ L 2 2 + C ε ∣ u ∣ L 2 ( ∂ Ω ) 2 , (30) and
ε ∫ Ω ∇ div u ⋅ u d x = − ε ∥ div u ∥ L 2 2 . (31) \varepsilon\int_{\Omega}\nabla\operatorname{div}u\cdot u\,dx=-\varepsilon\|\operatorname{div}u\|_{L^2}^2. \tag*{(31)} ε ∫ Ω ∇ div u ⋅ u d x = − ε ∥ div u ∥ L 2 2 . (31) Combining (26) –(31) , we obtain
d d t { ∫ Ω ( 1 2 ρ ∣ u ∣ 2 + 1 γ − 1 ρ γ ) d x + 1 2 ∥ ∇ ϕ ∥ L 2 2 } + μ ε ∥ ω ∥ L 2 2 + ( 2 μ + λ ) ε ∥ div u ∥ L 2 2 ≤ C ε ∣ u ∣ L 2 ( ∂ Ω ) 2 . (32) \frac{d}{dt}\left\{\int_{\Omega}\left(\frac{1}{2}\rho|u|^2+\frac{1}{\gamma-1}\rho^\gamma\right)\,dx+\frac{1}{2}\|\nabla\phi\|_{L^2}^2\right\}
+\mu\varepsilon\|\omega\|_{L^2}^2+(2\mu+\lambda)\varepsilon\|\operatorname{div}u\|_{L^2}^2\leq C\varepsilon|u|_{L^2(\partial\Omega)}^2. \tag*{(32)} d t d { ∫ Ω ( 2 1 ρ ∣ u ∣ 2 + γ − 1 1 ρ γ ) d x + 2 1 ∥∇ ϕ ∥ L 2 2 } + μ ε ∥ ω ∥ L 2 2 + ( 2 μ + λ ) ε ∥ div u ∥ L 2 2 ≤ C ε ∣ u ∣ L 2 ( ∂ Ω ) 2 . (32) By Proposition 2.1 and (2) , we have
μ ∥ ω ∥ L 2 2 + ( 2 μ + λ ) ∥ div u ∥ L 2 2 ≥ 2 c ∥ ∇ u ∥ L 2 2 − C ∥ u ∥ L 2 2 . (33) \mu\|\omega\|_{L^2}^2+(2\mu+\lambda)\|\operatorname{div}u\|_{L^2}^2\geq2c\|\nabla u\|_{L^2}^2-C\|u\|_{L^2}^2. \tag*{(33)} μ ∥ ω ∥ L 2 2 + ( 2 μ + λ ) ∥ div u ∥ L 2 2 ≥ 2 c ∥∇ u ∥ L 2 2 − C ∥ u ∥ L 2 2 . (33) Moreover, Proposition 2.3 gives
ε ∣ u ∣ L 2 ( ∂ Ω ) 2 ≤ C ε ( ∥ ∇ u ∥ L 2 + ∥ u ∥ L 2 ) ∥ u ∥ L 2 ≤ ε 2 ∥ ∇ u ∥ L 2 2 + C ε ∥ u ∥ L 2 2 . (34) \varepsilon|u|_{L^2(\partial\Omega)}^2\leq C\varepsilon\left(\|\nabla u\|_{L^2}+\|u\|_{L^2}\right)\|u\|_{L^2}\leq\frac{\varepsilon}{2}\|\nabla u\|_{L^2}^2+C\varepsilon\|u\|_{L^2}^2. \tag*{(34)} ε ∣ u ∣ L 2 ( ∂ Ω ) 2 ≤ C ε ( ∥∇ u ∥ L 2 + ∥ u ∥ L 2 ) ∥ u ∥ L 2 ≤ 2 ε ∥∇ u ∥ L 2 2 + C ε ∥ u ∥ L 2 2 . (34) Since 0 < ε ≤ 1 0<\varepsilon\leq1 0 < ε ≤ 1 (32) –(34) that
d d t { ∫ Ω ( 1 2 ρ ∣ u ∣ 2 + 1 γ − 1 ρ γ ) d x + 1 2 ∥ ∇ ϕ ∥ L 2 2 } + c ε ∥ ∇ u ∥ L 2 2 ≤ C ∥ u ∥ L 2 2 . (35) \frac{d}{dt}\left\{\int_{\Omega}\left(\frac{1}{2}\rho|u|^2+\frac{1}{\gamma-1}\rho^\gamma\right)\,dx+\frac{1}{2}\|\nabla\phi\|_{L^2}^2\right\}+c\varepsilon\|\nabla u\|_{L^2}^2\leq C\|u\|_{L^2}^2. \tag*{(35)} d t d { ∫ Ω ( 2 1 ρ ∣ u ∣ 2 + γ − 1 1 ρ γ ) d x + 2 1 ∥∇ ϕ ∥ L 2 2 } + c ε ∥∇ u ∥ L 2 2 ≤ C ∥ u ∥ L 2 2 . (35) Integrating (35) over [ 0 , t ] [0,t] [ 0 , t ] (25) . This completes the proof.
We shall derive higher-order conormal estimates for ( p , u ) (p,u) ( p , u )
E m ( t ) : = ∥ ( p , u ) ( t ) ∥ H m 2 + ∥ ∇ u ( t ) ∥ H m − 1 2 + ∑ k = 0 m − 2 ∥ ∇ ∂ t k p ( t ) ∥ H m − 1 − k 2 + ε ∥ ∇ ∂ t m − 1 p ( t ) ∥ L 2 2 + ∥ ∇ ϕ ( t ) ∥ H m 2 , (36) \mathcal{E}_m(t):=\|(p,u)(t)\|_{\mathcal{H}^m}^2+\|\nabla u(t)\|_{\mathcal{H}^{m-1}}^2+\sum_{k=0}^{m-2}\|\nabla\partial_t^k p(t)\|_{\mathcal{H}^{m-1-k}}^2+\varepsilon\|\nabla\partial_t^{m-1}p(t)\|_{L^2}^2+\|\nabla\phi(t)\|_{\mathcal{H}^m}^2, \tag*{(36)} E m ( t ) := ∥ ( p , u ) ( t ) ∥ H m 2 + ∥∇ u ( t ) ∥ H m − 1 2 + k = 0 ∑ m − 2 ∥∇ ∂ t k p ( t ) ∥ H m − 1 − k 2 + ε ∥∇ ∂ t m − 1 p ( t ) ∥ L 2 2 + ∥∇ ϕ ( t ) ∥ H m 2 , (36) and
Q ( t ) : = sup 0 ≤ τ ≤ t { ∥ ( ∇ p , ∇ u ) ( τ ) ∥ H 1 , ∞ 2 + ∥ ( p , u , p t , u t ) ( τ ) ∥ L ∞ 2 + ∥ ∇ ϕ ( τ ) ∥ H 1 , ∞ 2 } . (37) Q(t):=\sup_{0\leq\tau\leq t}\left\{\|(\nabla p,\nabla u)(\tau)\|_{\mathcal{H}^{1,\infty}}^2+\|(p,u,p_t,u_t)(\tau)\|_{L^\infty}^2+\|\nabla\phi(\tau)\|_{\mathcal{H}^{1,\infty}}^2\right\}. \tag*{(37)} Q ( t ) := 0 ≤ τ ≤ t sup { ∥ ( ∇ p , ∇ u ) ( τ ) ∥ H 1 , ∞ 2 + ∥ ( p , u , p t , u t ) ( τ ) ∥ L ∞ 2 + ∥∇ ϕ ( τ ) ∥ H 1 , ∞ 2 } . (37) Here the norm ∥ ⋅ ∥ m − 1 − k \|\cdot\|_{m-1-k} ∥ ⋅ ∥ m − 1 − k E m ( t ) \mathcal{E}_m(t) E m ( t ) ∇ ∂ t k p \nabla\partial_t^k p ∇ ∂ t k p
Lemma 3.2 (Estimate for div u \operatorname{div}u div u ∇ p \nabla p ∇ p m ≥ 0 m \ge0 m ≥ 0
sup 0 ≤ τ ≤ t ( ∫ Ω ρ ∣ div u ∣ 2 ( τ ) d x + ∫ Ω 1 γ p ∣ ∇ p ∣ 2 ( τ ) d x ) + c ε ∫ 0 t ∥ ∇ div u ( τ ) ∥ L 2 2 d τ ≤ C ( ∫ Ω ρ 0 ∣ div u 0 ∣ 2 d x + ∫ Ω 1 γ p 0 ∣ ∇ p 0 ∣ 2 d x ) + C 3 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (38) \sup_{0\le\tau\le t}\left(\int_{\Omega}\rho|\operatorname{div}u|^2(\tau)\,dx+\int_{\Omega}\frac{1}{\gamma p}|\nabla p|^2(\tau)\,dx\right)+c\varepsilon\int_0^t\|\nabla\operatorname{div}u(\tau)\|_{L^2}^2\,d\tau
\le C\left(\int_{\Omega}\rho_0|\operatorname{div}u_0|^2\,dx+\int_{\Omega}\frac{1}{\gamma p_0}|\nabla p_0|^2\,dx\right)+C_3[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(38)} 0 ≤ τ ≤ t sup ( ∫ Ω ρ ∣ div u ∣ 2 ( τ ) d x + ∫ Ω γ p 1 ∣∇ p ∣ 2 ( τ ) d x ) + c ε ∫ 0 t ∥∇ div u ( τ ) ∥ L 2 2 d τ ≤ C ( ∫ Ω ρ 0 ∣ div u 0 ∣ 2 d x + ∫ Ω γ p 0 1 ∣∇ p 0 ∣ 2 d x ) + C 3 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (38) Here P ( ⋅ ) P(\cdot) P ( ⋅ ) C 3 C_3 C 3 ε \varepsilon ε t t t
Proof. Multiplying (24) by ∇ div u \nabla\operatorname{div}u ∇ div u Ω × ( 0 , t ) \Omega\times(0,t) Ω × ( 0 , t )
∫ 0 t ∫ Ω ( ρ u t + ρ u ⋅ ∇ u ) ⋅ ∇ div u d x d τ + ∫ 0 t ∫ Ω ∇ p ⋅ ∇ div u d x d τ = − μ ε ∫ 0 t ∫ Ω ( ∇ × ω ) ⋅ ∇ div u d x d τ + ( 2 μ + λ ) ε ∫ 0 t ∥ ∇ div u ∥ L 2 2 d τ + ∫ 0 t ∫ Ω ρ ∇ ϕ ⋅ ∇ div u d x d τ . (39) \int_0^t\int_{\Omega}(\rho u_t+\rho u\cdot\nabla u)\cdot\nabla\operatorname{div}u\,dx\,d\tau+\int_0^t\int_{\Omega}\nabla p\cdot\nabla\operatorname{div}u\,dx\,d\tau
=-\mu\varepsilon\int_0^t\int_{\Omega}(\nabla\times\omega)\cdot\nabla\operatorname{div}u\,dx\,d\tau+(2\mu+\lambda)\varepsilon\int_0^t\|\nabla\operatorname{div}u\|_{L^2}^2\,d\tau
+\int_0^t\int_{\Omega}\rho\nabla\phi\cdot\nabla\operatorname{div}u\,dx\,d\tau. \tag*{(39)} ∫ 0 t ∫ Ω ( ρ u t + ρ u ⋅ ∇ u ) ⋅ ∇ div u d x d τ + ∫ 0 t ∫ Ω ∇ p ⋅ ∇ div u d x d τ = − μ ε ∫ 0 t ∫ Ω ( ∇ × ω ) ⋅ ∇ div u d x d τ + ( 2 μ + λ ) ε ∫ 0 t ∥∇ div u ∥ L 2 2 d τ + ∫ 0 t ∫ Ω ρ ∇ ϕ ⋅ ∇ div u d x d τ . (39) We first estimate the electric force term. Using ∂ n ϕ = 0 \partial_n\phi=0 ∂ n ϕ = 0 Δ ϕ = ρ − 1 \Delta\phi=\rho-1 Δ ϕ = ρ − 1
∣ I E ∣ : = ∣ ∫ 0 t ∫ Ω ρ ∇ ϕ ⋅ ∇ div u d x d τ ∣ = ∣ − ∫ 0 t ∫ Ω div ( ρ ∇ ϕ ) div u d x d τ ∣ |I_E|:=\left|\int_0^t\int_{\Omega}\rho\nabla\phi\cdot\nabla\operatorname{div}u\,dx\,d\tau\right|=\left|-\int_0^t\int_{\Omega}\operatorname{div}(\rho\nabla\phi)\operatorname{div}u\,dx\,d\tau\right| ∣ I E ∣ := ∫ 0 t ∫ Ω ρ ∇ ϕ ⋅ ∇ div u d x d τ = − ∫ 0 t ∫ Ω div ( ρ ∇ ϕ ) div u d x d τ = ∣ − ∫ 0 t ∫ Ω ( ∇ ρ ⋅ ∇ ϕ + ρ 2 − ρ ) div u d x d τ ∣ ≤ C [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (40) =\left|-\int_0^t\int_{\Omega}(\nabla\rho\cdot\nabla\phi+\rho^2-\rho)\operatorname{div}u\,dx\,d\tau\right|\le C[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(40)} = − ∫ 0 t ∫ Ω ( ∇ ρ ⋅ ∇ ϕ + ρ 2 − ρ ) div u d x d τ ≤ C [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (40) Using integration by parts, the identity ( u ⋅ ∇ ) u ⋅ n = − u ⋅ ( ( u ⋅ ∇ ) n ) (u\cdot\nabla)u\cdot n=-u\cdot((u\cdot\nabla)n) ( u ⋅ ∇ ) u ⋅ n = − u ⋅ (( u ⋅ ∇ ) n ) ∂ Ω \partial\Omega ∂ Ω Proposition 2.3 , we obtain
∫ 0 t ∫ Ω ( ρ u t + ρ u ⋅ ∇ u ) ⋅ ∇ div u d x d τ = − 1 2 ∫ Ω ρ ∣ div u ∣ 2 ( t ) d x + 1 2 ∫ Ω ρ 0 ∣ div u 0 ∣ 2 d x − ∫ 0 t ∫ Ω ( ∇ ρ ⋅ u t + ∇ ( ρ u ) T : ∇ u ) div u d x d τ + ∫ 0 t ∫ ∂ Ω ρ ( u ⋅ ∇ ) u ⋅ n div u d S d τ \int_0^t\int_{\Omega}(\rho u_t+\rho u\cdot\nabla u)\cdot\nabla\operatorname{div}u\,dx\,d\tau
=-\frac{1}{2}\int_{\Omega}\rho|\operatorname{div}u|^2(t)\,dx+\frac{1}{2}\int_{\Omega}\rho_0|\operatorname{div}u_0|^2\,dx
-\int_0^t\int_{\Omega}(\nabla\rho\cdot u_t+\nabla(\rho u)^{T}:\nabla u)\operatorname{div}u\,dx\,d\tau
+\int_0^t\int_{\partial\Omega}\rho(u\cdot\nabla)u\cdot n\operatorname{div}u\,dS\,d\tau ∫ 0 t ∫ Ω ( ρ u t + ρ u ⋅ ∇ u ) ⋅ ∇ div u d x d τ = − 2 1 ∫ Ω ρ ∣ div u ∣ 2 ( t ) d x + 2 1 ∫ Ω ρ 0 ∣ div u 0 ∣ 2 d x − ∫ 0 t ∫ Ω ( ∇ ρ ⋅ u t + ∇ ( ρ u ) T : ∇ u ) div u d x d τ + ∫ 0 t ∫ ∂ Ω ρ ( u ⋅ ∇ ) u ⋅ n div u d S d τ ≤ − 1 2 ∫ Ω ρ ∣ div u ∣ 2 ( t ) d x + 1 2 ∫ Ω ρ 0 ∣ div u 0 ∣ 2 d x + C 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (41) \le-\frac{1}{2}\int_{\Omega}\rho|\operatorname{div}u|^2(t)\,dx+\frac{1}{2}\int_{\Omega}\rho_0|\operatorname{div}u_0|^2\,dx+C_2[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(41)} ≤ − 2 1 ∫ Ω ρ ∣ div u ∣ 2 ( t ) d x + 2 1 ∫ Ω ρ 0 ∣ div u 0 ∣ 2 d x + C 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (41) Since p = ρ γ p=\rho^\gamma p = ρ γ
p t + u ⋅ ∇ p + γ p div u = 0 , div u = − p t + u ⋅ ∇ p γ p . (42) p_t+u\cdot\nabla p+\gamma p\operatorname{div}u=0,\qquad\operatorname{div}u=-\frac{p_t+u\cdot\nabla p}{\gamma p}. \tag*{(42)} p t + u ⋅ ∇ p + γ p div u = 0 , div u = − γ p p t + u ⋅ ∇ p . (42) Using (42) and integrating by parts, we obtain
∫ 0 t ∫ Ω ∇ p ⋅ ∇ div u d x d τ = − ∫ 0 t ∫ Ω ∇ p ⋅ ∇ ( p t γ p ) d x d τ − ∫ 0 t ∫ Ω ∇ p ⋅ ∇ ( u ⋅ ∇ p γ p ) d x d τ \int_0^t\int_{\Omega}\nabla p\cdot\nabla\operatorname{div}u\,dx\,d\tau
=-\int_0^t\int_{\Omega}\nabla p\cdot\nabla\left(\frac{p_t}{\gamma p}\right)\,dx\,d\tau-\int_0^t\int_{\Omega}\nabla p\cdot\nabla\left(\frac{u\cdot\nabla p}{\gamma p}\right)\,dx\,d\tau ∫ 0 t ∫ Ω ∇ p ⋅ ∇ div u d x d τ = − ∫ 0 t ∫ Ω ∇ p ⋅ ∇ ( γ p p t ) d x d τ − ∫ 0 t ∫ Ω ∇ p ⋅ ∇ ( γ p u ⋅ ∇ p ) d x d τ ≤ − ∫ Ω 1 2 γ p ∣ ∇ p ∣ 2 ( t ) d x + ∫ Ω 1 2 γ p 0 ∣ ∇ p 0 ∣ 2 d x + C [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (43) \le-\int_{\Omega}\frac{1}{2\gamma p}|\nabla p|^2(t)\,dx+\int_{\Omega}\frac{1}{2\gamma p_0}|\nabla p_0|^2\,dx+C[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(43)} ≤ − ∫ Ω 2 γ p 1 ∣∇ p ∣ 2 ( t ) d x + ∫ Ω 2 γ p 0 1 ∣∇ p 0 ∣ 2 d x + C [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (43) For the vorticity term, using ∇ × ∇ div u = 0 \nabla\times\nabla\operatorname{div} u = 0 ∇ × ∇ div u = 0 (2) , integration by parts along the boundary, Proposition 2.3 , and Young’s inequality, we obtain
ε ∣ ∫ 0 t ∫ Ω ( ∇ × ω ) ⋅ ∇ div u d x d τ ∣ = ε ∣ ∫ 0 t ∫ ∂ Ω ( B u ) ⋅ Π ( ∇ div u ) d S d τ ∣ = ε ∣ ∫ 0 t ∫ ∂ Ω ( B u ) ⋅ Z y div u d S d τ ∣ ≤ C 2 ε ∫ 0 t ∣ u ∣ H 1 2 ( ∂ Ω ) ∣ div u ∣ H 1 2 ( ∂ Ω ) d τ ≤ 2 μ + λ 4 ε ∫ 0 t ∥ ∇ div u ( τ ) ∥ L 2 2 d τ + C 3 ε ∫ 0 t ( ∥ ∇ u ( τ ) ∥ L 2 2 + ∥ u ( τ ) ∥ L 2 2 ) d τ . (44) \begin{aligned}
\varepsilon\left|\int_0^t \int_{\Omega}(\nabla\times\omega)\cdot\nabla\operatorname{div}u\,dx\,d\tau\right|
&=\varepsilon\left|\int_0^t \int_{\partial\Omega}(Bu)\cdot\Pi(\nabla\operatorname{div}u)\,dS\,d\tau\right|\\
&=\varepsilon\left|\int_0^t \int_{\partial\Omega}(Bu)\cdot Z_y\operatorname{div}u\,dS\,d\tau\right|\\
&\leq C_2\varepsilon\int_0^t \lvert u\rvert_{H^{\frac12}(\partial\Omega)}\lvert\operatorname{div}u\rvert_{H^{\frac12}(\partial\Omega)}\,d\tau\\
&\leq\frac{2\mu+\lambda}{4}\varepsilon\int_0^t\lVert\nabla\operatorname{div}u(\tau)\rVert_{L^2}^2\,d\tau+C_3\varepsilon\int_0^t\left(\lVert\nabla u(\tau)\rVert_{L^2}^2+\lVert u(\tau)\rVert_{L^2}^2\right)\,d\tau. \tag*{(44)}
\end{aligned} ε ∫ 0 t ∫ Ω ( ∇ × ω ) ⋅ ∇ div u d x d τ = ε ∫ 0 t ∫ ∂ Ω ( B u ) ⋅ Π ( ∇ div u ) d S d τ = ε ∫ 0 t ∫ ∂ Ω ( B u ) ⋅ Z y div u d S d τ ≤ C 2 ε ∫ 0 t ∣ u ∣ H 2 1 ( ∂ Ω ) ∣ div u ∣ H 2 1 ( ∂ Ω ) d τ ≤ 4 2 μ + λ ε ∫ 0 t ∥ ∇ div u ( τ ) ∥ L 2 2 d τ + C 3 ε ∫ 0 t ( ∥ ∇ u ( τ ) ∥ L 2 2 + ∥ u ( τ ) ∥ L 2 2 ) d τ . (44) Substituting (40) , (41) , (43) , and (44) into (39) , we obtain (38) . This completes the proof.
Lemma 3.3 (Higher-order conormal estimate). Let m ≥ 1 m\geq1 m ≥ 1 For any sufficiently small δ > 0 \delta>0 δ > 0 it holds that
sup 0 ≤ τ ≤ t ( ∥ ( p , u ) ( τ ) ∥ H m 2 + ∥ ∇ φ ( τ ) ∥ H m 2 ) + c ε ∫ 0 t ∥ ∇ u ( τ ) ∥ H m 2 d τ ≤ C m + 2 { E m ( 0 ) + δ ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + δ ∫ 0 t ∥ ∇ ∂ t m − 1 p ( τ ) ∥ L 2 2 d τ + C δ [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ } . (45) \sup_{0\leq\tau\leq t}\left(\lVert(p,u)(\tau)\rVert_{\mathcal{H}^m}^2+\lVert\nabla\varphi(\tau)\rVert_{\mathcal{H}^m}^2\right)+c\varepsilon\int_0^t\lVert\nabla u(\tau)\rVert_{\mathcal{H}^m}^2\,d\tau
\leq C_{m+2}\left\{\mathcal{E}_m(0)+\delta\varepsilon^2\int_0^t\lVert\nabla^2u(\tau)\rVert_{\mathcal{H}^{m-1}}^2\,d\tau+\delta\int_0^t\lVert\nabla\partial_t^{m-1}p(\tau)\rVert_{L^2}^2\,d\tau+C_\delta[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau\right\}. \tag*{(45)} 0 ≤ τ ≤ t sup ( ∥( p , u ) ( τ ) ∥ H m 2 + ∥ ∇ φ ( τ ) ∥ H m 2 ) + c ε ∫ 0 t ∥ ∇ u ( τ ) ∥ H m 2 d τ ≤ C m + 2 { E m ( 0 ) + δ ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + δ ∫ 0 t ∥ ∇ ∂ t m − 1 p ( τ ) ∥ L 2 2 d τ + C δ [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ } . (45) Here C δ C_\delta C δ δ \delta δ ε \varepsilon ε
Proof. The case m = 0 m=0 m = 0 m − 1 m-1 m − 1 ∣ α ∣ = m . A p p l y i n g |\alpha|=m. Applying ∣ α ∣ = m . A ppl y in g ∣ α ∣ = m |\alpha|=m ∣ α ∣ = m (24) , multiplying the resulting equation by Z α u Z^\alpha u Z α u Ω \Omega Ω
1 2 d d t ∫ Ω ρ ∣ Z α u ∣ 2 d x + ∫ Ω Z α ∇ p ⋅ Z α u d x = − μ ε ∫ Ω Z α ∇ × ω ⋅ Z α u d x + ( 2 μ + λ ) ε ∫ Ω Z α ∇ div u ⋅ Z α u d x + ∫ Ω ρ ∇ Z α ϕ ⋅ Z α u d x + ∫ Ω ( C 1 α + C 2 α + C 3 α ) ⋅ Z α u d x . (46) \frac{1}{2}\frac{d}{dt}\int_{\Omega}\rho\lvert Z^\alpha u\rvert^2\,dx+\int_{\Omega}Z^\alpha\nabla p\cdot Z^\alpha u\,dx
=-\mu\varepsilon\int_{\Omega}Z^\alpha\nabla\times\omega\cdot Z^\alpha u\,dx+(2\mu+\lambda)\varepsilon\int_{\Omega}Z^\alpha\nabla\operatorname{div}u\cdot Z^\alpha u\,dx+\int_{\Omega}\rho\nabla Z^\alpha\phi\cdot Z^\alpha u\,dx+\int_{\Omega}(C_1^\alpha+C_2^\alpha+C_3^\alpha)\cdot Z^\alpha u\,dx. \tag*{(46)} 2 1 d t d ∫ Ω ρ ∣ Z α u ∣ 2 d x + ∫ Ω Z α ∇ p ⋅ Z α u d x = − μ ε ∫ Ω Z α ∇ × ω ⋅ Z α u d x + ( 2 μ + λ ) ε ∫ Ω Z α ∇ div u ⋅ Z α u d x + ∫ Ω ρ ∇ Z α ϕ ⋅ Z α u d x + ∫ Ω ( C 1 α + C 2 α + C 3 α ) ⋅ Z α u d x . (46) where
C 1 α : = − [ Z α , ρ ] u t = − ∑ ∣ β ∣ ≥ 1 β + ν = α C α β Z β ρ Z ν u t , C_1^\alpha:=-[Z^\alpha,\rho]u_t=-\sum_{\substack{|\beta|\geq1\\\beta+\nu=\alpha}}C_{\alpha\beta}Z^\beta\rho\,Z^\nu u_t, C 1 α := − [ Z α , ρ ] u t = − ∣ β ∣ ≥ 1 β + ν = α ∑ C α β Z β ρ Z ν u t , C 2 α : = − [ Z α , ρ u ⋅ ∇ ] u = − ∑ ∣ β ∣ ≥ 1 β + ν = α C α β Z β ( ρ u ) Z ν ∇ u − ρ u ⋅ [ Z α , ∇ ] u , C_2^\alpha:=-[Z^\alpha,\rho u\cdot\nabla]u=-\sum_{\substack{|\beta|\geq1\\\beta+\nu=\alpha}}C_{\alpha\beta}Z^\beta(\rho u)Z^\nu\nabla u-\rho u\cdot[Z^\alpha,\nabla]u, C 2 α := − [ Z α , ρ u ⋅ ∇ ] u = − ∣ β ∣ ≥ 1 β + ν = α ∑ C α β Z β ( ρ u ) Z ν ∇ u − ρ u ⋅ [ Z α , ∇ ] u , C 3 α : = [ Z α , ρ ] ∇ ϕ + ρ [ Z α , ∇ ] ϕ . (47) C_3^\alpha:=[Z^\alpha,\rho]\nabla\phi+\rho[Z^\alpha,\nabla]\phi. \tag*{(47)} C 3 α := [ Z α , ρ ] ∇ ϕ + ρ [ Z α , ∇ ] ϕ . (47) Here C 1 α C_1^\alpha C 1 α C 2 α C_2^\alpha C 2 α C 3 α C_3^\alpha C 3 α Z α Z^\alpha Z α C α β C_{\alpha\beta} C α β
J α : = ∫ Ω ρ ∇ Z α ϕ ⋅ Z α u d x . \mathcal{J}^\alpha:=\int_{\Omega}\rho\nabla Z^\alpha\phi\cdot Z^\alpha u\,dx. J α := ∫ Ω ρ ∇ Z α ϕ ⋅ Z α u d x . By integration by parts, the continuity equation, and the Poisson equation, we obtain
J α = − ∫ Ω Z α ϕ div ( ρ Z α u ) d x + ∫ ∂ Ω ρ Z α ϕ Z α u ⋅ n d S = − 1 2 d d t ∥ ∇ Z α ϕ ∥ L 2 2 + ∫ ∂ Ω ρ Z α ϕ Z α u ⋅ n d S + ∫ ∂ Ω Z α ϕ ∂ n Z α ϕ t d S + ∫ Ω Z α ϕ [ Z α , div ] ( ρ u ) d x − ∫ Ω ∇ Z α ϕ ⋅ [ Z α , ρ ] u d x + ∫ ∂ Ω Z α ϕ [ Z α , ρ ] u ⋅ n d S + ∫ Ω Z α ϕ [ Z α , Δ ] ϕ t d x ≜ − 1 2 d d t ∥ ∇ Z α ϕ ∥ L 2 2 + B 1 α + B 2 α + R div α + R ρ α + R Δ α . (48) \begin{aligned}
\mathcal{J}^{\alpha}
&= -\int_{\Omega} Z^{\alpha}\phi\operatorname{div}(\rho Z^{\alpha}u)\,dx+\int_{\partial\Omega}\rho Z^{\alpha}\phi Z^{\alpha}u\cdot n\,dS\\
&= -\frac{1}{2}\frac{d}{dt}\lVert\nabla Z^{\alpha}\phi\rVert_{L^2}^2+\int_{\partial\Omega}\rho Z^{\alpha}\phi Z^{\alpha}u\cdot n\,dS\\
&\quad+\int_{\partial\Omega}Z^{\alpha}\phi\,\partial_n Z^{\alpha}\phi_t\,dS+\int_{\Omega}Z^{\alpha}\phi[Z^{\alpha},\operatorname{div}](\rho u)\,dx\\
&\quad-\int_{\Omega}\nabla Z^{\alpha}\phi\cdot[Z^{\alpha},\rho]u\,dx+\int_{\partial\Omega}Z^{\alpha}\phi[Z^{\alpha},\rho]u\cdot n\,dS\\
&\quad+\int_{\Omega}Z^{\alpha}\phi[Z^{\alpha},\Delta]\phi_t\,dx\\
&\triangleq-\frac{1}{2}\frac{d}{dt}\lVert\nabla Z^{\alpha}\phi\rVert_{L^2}^2+B_1^{\alpha}+B_2^{\alpha}+R_{\operatorname{div}}^{\alpha}+R_{\rho}^{\alpha}+R_{\Delta}^{\alpha}. \tag*{(48)}
\end{aligned} J α = − ∫ Ω Z α ϕ div ( ρ Z α u ) d x + ∫ ∂ Ω ρ Z α ϕ Z α u ⋅ n d S = − 2 1 d t d ∥ ∇ Z α ϕ ∥ L 2 2 + ∫ ∂ Ω ρ Z α ϕ Z α u ⋅ n d S + ∫ ∂ Ω Z α ϕ ∂ n Z α ϕ t d S + ∫ Ω Z α ϕ [ Z α , div ] ( ρ u ) d x − ∫ Ω ∇ Z α ϕ ⋅ [ Z α , ρ ] u d x + ∫ ∂ Ω Z α ϕ [ Z α , ρ ] u ⋅ n d S + ∫ Ω Z α ϕ [ Z α , Δ ] ϕ t d x ≜ − 2 1 d t d ∥ ∇ Z α ϕ ∥ L 2 2 + B 1 α + B 2 α + R div α + R ρ α + R Δ α . (48) Here
div ( ρ Z α u ) = Z α div ( ρ u ) − [ Z α , div ] ( ρ u ) − div ( [ Z α , ρ ] u ) . \operatorname{div}(\rho Z^{\alpha}u)=Z^{\alpha}\operatorname{div}(\rho u)-[Z^{\alpha},\operatorname{div}](\rho u)-\operatorname{div}([Z^{\alpha},\rho]u). div ( ρ Z α u ) = Z α div ( ρ u ) − [ Z α , div ] ( ρ u ) − div ([ Z α , ρ ] u ) . We estimate B 1 α B_1^{\alpha} B 1 α Z α Z^{\alpha} Z α Z α u ⋅ n = 0 Z^{\alpha}u\cdot n=0 Z α u ⋅ n = 0 ∂ Ω \partial\Omega ∂ Ω u ⋅ n = 0 u\cdot n=0 u ⋅ n = 0 ∂ Ω \partial\Omega ∂ Ω
∣ Z α u ⋅ n ∣ L 2 ( ∂ Ω ) ≤ C m + 1 ∣ u ∣ H m − 1 ( ∂ Ω ) . \lvert Z^{\alpha}u\cdot n\rvert_{L^2(\partial\Omega)}\leq C_{m+1}\lvert u\rvert_{H^{m-1}(\partial\Omega)}. ∣ Z α u ⋅ n ∣ L 2 ( ∂ Ω ) ≤ C m + 1 ∣ u ∣ H m − 1 ( ∂ Ω ) . Moreover,
∣ Z α ϕ ∣ L 2 ( ∂ Ω ) ≤ C ∣ ∇ ϕ ∣ H m − 1 ( ∂ Ω ) . \lvert Z^{\alpha}\phi\rvert_{L^2(\partial\Omega)}\leq C\lvert\nabla\phi\rvert_{H^{m-1}(\partial\Omega)}. ∣ Z α ϕ ∣ L 2 ( ∂ Ω ) ≤ C ∣ ∇ ϕ ∣ H m − 1 ( ∂ Ω ) . Thus, by Proposition 2.3,
∣ B 1 α ∣ ≤ C m + 1 ∣ ∇ ϕ ∣ H m − 1 ( ∂ Ω ) ∣ u ∣ H m − 1 ( ∂ Ω ) ≤ C m + 2 ( ∥ ∇ 2 ϕ ∥ H m − 1 + ∥ ∇ ϕ ∥ H m − 1 ) 1 / 2 ∥ ∇ ϕ ∥ H m − 1 1 / 2 ( ∥ ∇ u ∥ H m − 1 + ∥ u ∥ H m ) . \begin{aligned}
\lvert B_1^{\alpha}\rvert\leq C_{m+1}\lvert\nabla\phi\rvert_{H^{m-1}(\partial\Omega)}\lvert u\rvert_{H^{m-1}(\partial\Omega)} \\
&\leq C_{m+2}\left(\lVert\nabla^2\phi\rVert_{H^{m-1}}+\lVert\nabla\phi\rVert_{H^{m-1}}\right)^{1/2}\lVert\nabla\phi\rVert_{H^{m-1}}^{1/2}\left(\lVert\nabla u\rVert_{H^{m-1}}+\lVert u\rVert_{H^m}\right).
\end{aligned} ∣ B 1 α ∣ ≤ C m + 1 ∣ ∇ ϕ ∣ H m − 1 ( ∂ Ω ) ∣ u ∣ H m − 1 ( ∂ Ω ) ≤ C m + 2 ( ∥ ∇ 2 ϕ ∥ H m − 1 + ∥ ∇ ϕ ∥ H m − 1 ) 1/2 ∥ ∇ ϕ ∥ H m − 1 1/2 ( ∥ ∇ u ∥ H m − 1 + ∥ u ∥ H m ) . By Proposition 2.4,
∥ ∇ 2 ϕ ∥ H m − 1 ≤ C m + 2 ( ∥ ρ − 1 ∥ H m − 1 + ∥ ∇ ϕ ∥ L 2 ) . (49) \lVert\nabla^2\phi\rVert_{H^{m-1}}\leq C_{m+2}\left(\lVert\rho-1\rVert_{H^{m-1}}+\lVert\nabla\phi\rVert_{L^2}\right). \tag*{(49)} ∥ ∇ 2 ϕ ∥ H m − 1 ≤ C m + 2 ( ∥ ρ − 1 ∥ H m − 1 + ∥ ∇ ϕ ∥ L 2 ) . (49) Therefore,
∑ ∣ α ∣ = m ∫ 0 t ∣ B 1 α ( τ ) ∣ d τ ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (50) \sum_{\lvert\alpha\rvert=m}\int_0^t\lvert B_1^{\alpha}(\tau)\rvert\,d\tau\leq C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(50)} ∣ α ∣ = m ∑ ∫ 0 t ∣ B 1 α ( τ )∣ d τ ≤ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (50) The term B 2 α B_2^{\alpha} B 2 α Z α Z^{\alpha} Z α ∂ n Z α ϕ t = 0 \partial_n Z^{\alpha}\phi_t=0 ∂ n Z α ϕ t = 0 ∂ Ω \partial\Omega ∂ Ω
∂ n Z α ϕ t = [ ∂ n , Z α ] ϕ t on ∂ Ω . \partial_n Z^{\alpha}\phi_t=[\partial_n,Z^{\alpha}]\phi_t\quad\text{on }\partial\Omega. ∂ n Z α ϕ t = [ ∂ n , Z α ] ϕ t on ∂ Ω. Applying Proposition 2.3 as above, and using Proposition 2.4 for the time-differentiated Poisson equation, we get
∥ ∇ 2 ϕ t ∥ H m − 1 ≤ C m + 2 ( ∥ ρ t ∥ H m − 1 + ∥ ∇ ϕ t ∥ L 2 ) . (51) \lVert\nabla^2\phi_t\rVert_{H^{m-1}}\leq C_{m+2}\left(\lVert\rho_t\rVert_{H^{m-1}}+\lVert\nabla\phi_t\rVert_{L^2}\right). \tag*{(51)} ∥ ∇ 2 ϕ t ∥ H m − 1 ≤ C m + 2 ( ∥ ρ t ∥ H m − 1 + ∥ ∇ ϕ t ∥ L 2 ) . (51) Consequently, as in (50) ,
∑ ∣ α ∣ = m ∫ 0 t ∣ B 2 α ( τ ) ∣ d τ ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (52) \sum_{\lvert\alpha\rvert=m}\int_0^t\lvert B_2^{\alpha}(\tau)\rvert\,d\tau\leq C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(52)} ∣ α ∣ = m ∑ ∫ 0 t ∣ B 2 α ( τ )∣ d τ ≤ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (52) For R div α R_{\operatorname{div}}^{\alpha} R div α [ Z α , div ] [Z^{\alpha},\operatorname{div}] [ Z α , div ] ρ u \rho u ρ u Z α Z^{\alpha} Z α Proposition 2.2 ,
∑ ∣ α ∣ = m ∫ 0 t ∣ R div α ( τ ) ∣ d τ ≤ ∑ ∣ α ∣ = m ∫ 0 t ∥ Z α ϕ ∥ L 2 ∥ [ Z α , div ] ( ρ u ) ∥ L 2 d τ ≤ C m + 1 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (53) \begin{aligned}
\sum_{|\alpha|=m}\int_0^t \left|R^\alpha_{\operatorname{div}}(\tau)\right|\,d\tau\le\sum_{|\alpha|=m}\int_0^t \left\|Z^\alpha\phi\right\|_{L^2}\left\|[Z^\alpha,\operatorname{div}](\rho u)\right\|_{L^2}\,d\tau\\
&\le C_{m+1}[1+P(Q(t))]\int_0^t \mathcal{E}_m(\tau)\,d\tau. \tag*{(53)}
\end{aligned} ∣ α ∣ = m ∑ ∫ 0 t ∣ R div α ( τ ) ∣ d τ ≤ ∣ α ∣ = m ∑ ∫ 0 t ∥ Z α ϕ ∥ L 2 ∥ [ Z α , div ] ( ρ u ) ∥ L 2 d τ ≤ C m + 1 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (53) For R ρ α R^\alpha_\rho R ρ α (48) gives
R ρ α = − ∫ Ω ∇ Z α ϕ ⋅ [ Z α , ρ ] u d x + ∫ ∂ Ω Z α ϕ [ Z α , ρ ] u ⋅ n d S . R^\alpha_\rho=-\int_\Omega\nabla Z^\alpha\phi\cdot[Z^\alpha,\rho]u\,dx+\int_{\partial\Omega}Z^\alpha\phi[Z^\alpha,\rho]u\cdot n\,dS. R ρ α = − ∫ Ω ∇ Z α ϕ ⋅ [ Z α , ρ ] u d x + ∫ ∂ Ω Z α ϕ [ Z α , ρ ] u ⋅ n d S . Proposition 2.2 and Proposition 2.3 imply
∑ ∣ α ∣ = m ∫ 0 t ∣ R ρ α ( τ ) ∣ d τ ≤ ∑ ∣ α ∣ = m ∫ 0 t ∥ ∇ Z α ϕ ∥ L 2 ∥ [ Z α , ρ ] u ∥ L 2 d τ + ∑ ∣ α ∣ = m ∫ 0 t ∣ Z α ϕ ∣ L 2 ( ∂ Ω ) ∣ [ Z α , ρ ] u ⋅ n ∣ L 2 ( ∂ Ω ) d τ ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (54) \begin{aligned}
\sum_{|\alpha|=m}\int_0^t\left|R^\alpha_\rho(\tau)\right|\,d\tau\le\sum_{|\alpha|=m}\int_0^t\left\|\nabla Z^\alpha\phi\right\|_{L^2}\left\|[Z^\alpha,\rho]u\right\|_{L^2}\,d\tau\\
&+\sum_{|\alpha|=m}\int_0^t\left|Z^\alpha\phi\right|_{L^2(\partial\Omega)}\left|[Z^\alpha,\rho]u\cdot n\right|_{L^2(\partial\Omega)}\,d\tau\\
&\le C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(54)}
\end{aligned} ∣ α ∣ = m ∑ ∫ 0 t R ρ α ( τ ) d τ ≤ ∣ α ∣ = m ∑ ∫ 0 t ∥ ∇ Z α ϕ ∥ L 2 ∥ [ Z α , ρ ] u ∥ L 2 d τ + ∣ α ∣ = m ∑ ∫ 0 t ∣ Z α ϕ ∣ L 2 ( ∂ Ω ) ∣ [ Z α , ρ ] u ⋅ n ∣ L 2 ( ∂ Ω ) d τ ≤ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (54) Finally, for R Δ α R^\alpha_\Delta R Δ α [ Z α , Δ ] ϕ t [Z^\alpha,\Delta]\phi_t [ Z α , Δ ] ϕ t
∥ [ Z α , Δ ] ϕ t ∥ L 2 ≤ C m + 2 ( ∥ ∇ 2 ϕ t ∥ H m − 1 + ∥ ∇ ϕ t ∥ H m − 1 ) . \left\|[Z^\alpha,\Delta]\phi_t\right\|_{L^2}\le C_{m+2}\left(\left\|\nabla^2\phi_t\right\|_{H^{m-1}}+\left\|\nabla\phi_t\right\|_{H^{m-1}}\right). ∥ [ Z α , Δ ] ϕ t ∥ L 2 ≤ C m + 2 ( ∇ 2 ϕ t H m − 1 + ∥ ∇ ϕ t ∥ H m − 1 ) . Thus, by Proposition 2.2 , (51) ,
∑ ∣ α ∣ = m ∫ 0 t ∣ R Δ α ( τ ) ∣ d τ ≤ ∑ ∣ α ∣ = m ∫ 0 t ∥ Z α ϕ ∥ L 2 ∥ [ Z α , Δ ] ϕ t ∥ L 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (55) \begin{aligned}
\sum_{|\alpha|=m}\int_0^t\left|R^\alpha_\Delta(\tau)\right|\,d\tau\le\sum_{|\alpha|=m}\int_0^t\left\|Z^\alpha\phi\right\|_{L^2}\left\|[Z^\alpha,\Delta]\phi_t\right\|_{L^2}\,d\tau\\
&\le C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(55)}
\end{aligned} ∣ α ∣ = m ∑ ∫ 0 t ∣ R Δ α ( τ ) ∣ d τ ≤ ∣ α ∣ = m ∑ ∫ 0 t ∥ Z α ϕ ∥ L 2 ∥ [ Z α , Δ ] ϕ t ∥ L 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (55) Combining (48) , (50) , (52) , (53) , (54) , and (55) , we obtain
∑ ∣ α ∣ = m ∫ 0 t J α ( τ ) d τ ≤ − 1 2 ∥ ∇ Z m ϕ ( t ) ∥ L 2 2 + 1 2 ∥ ∇ Z m ϕ 0 ∥ L 2 2 + C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (56) \begin{aligned}
\sum_{|\alpha|=m}\int_0^t\mathcal{J}^\alpha(\tau)\,d\tau\le-\frac{1}{2}\left\|\nabla Z^m\phi(t)\right\|_{L^2}^2+\frac{1}{2}\left\|\nabla Z^m\phi_0\right\|_{L^2}^2 \\
&+C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(56)}
\end{aligned} ∣ α ∣ = m ∑ ∫ 0 t J α ( τ ) d τ ≤ − 2 1 ∥ ∇ Z m ϕ ( t ) ∥ L 2 2 + 2 1 ∥ ∇ Z m ϕ 0 ∥ L 2 2 + C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (56) The viscous terms are considered as in Wang, Xin and Yong [34 ] , Lemma 3.4. More precisely, after integration by parts, using (2) , Proposition 2.1 , Proposition 2.3 , and Young’s inequality, one obtains
− μ ε ∫ Ω Z α ∇ × ω ⋅ Z α u d x + ( 2 μ + λ ) ε ∫ Ω Z α ∇ div u ⋅ Z α u d x ≤ − c ε ∥ ∇ Z α u ∥ L 2 2 + δ ε ∥ ∇ u ∥ H m 2 + δ ε 2 ∥ ∇ 2 u ∥ H m − 1 2 + C δ C m + 2 ( ∥ ∇ u ∥ H m − 1 2 + ∥ u ∥ H m 2 ) . (57) \begin{aligned}
-\mu\varepsilon\int_\Omega Z^\alpha\nabla\times\omega\cdot Z^\alpha u\,dx+(2\mu+\lambda)\varepsilon\int_\Omega Z^\alpha\nabla\operatorname{div}u\cdot Z^\alpha u\,dx \\
&\le-c\varepsilon\left\|\nabla Z^\alpha u\right\|_{L^2}^2+\delta\varepsilon\left\|\nabla u\right\|_{H^m}^2+\delta\varepsilon^2\left\|\nabla^2u\right\|_{H^{m-1}}^2 \\
&+C_\delta C_{m+2}\left(\left\|\nabla u\right\|_{H^{m-1}}^2+\left\|u\right\|_{H^m}^2\right). \tag*{(57)}
\end{aligned} − μ ε ∫ Ω Z α ∇ × ω ⋅ Z α u d x + ( 2 μ + λ ) ε ∫ Ω Z α ∇ div u ⋅ Z α u d x ≤ − c ε ∥ ∇ Z α u ∥ L 2 2 + δ ε ∥ ∇ u ∥ H m 2 + δ ε 2 ∇ 2 u H m − 1 2 + C δ C m + 2 ( ∥ ∇ u ∥ H m − 1 2 + ∥ u ∥ H m 2 ) . (57) We next estimate the pressure term. Integration by parts gives
∫ Ω Z α ∇ p ⋅ Z α u d x = − ∫ Ω Z α p Z α div u d x + ∫ ∂ Ω Z α p Z α u ⋅ n d S + ∫ Ω [ Z α , ∇ ] p ⋅ Z α u d x . (58) \begin{aligned}
\int_\Omega Z^\alpha\nabla p\cdot Z^\alpha u\,dx=-\int_\Omega Z^\alpha p Z^\alpha\operatorname{div}u\,dx+\int_{\partial\Omega}Z^\alpha p Z^\alpha u\cdot n\,dS \\
&+\int_\Omega[Z^\alpha,\nabla]p\cdot Z^\alpha u\,dx. \tag*{(58)}
\end{aligned} ∫ Ω Z α ∇ p ⋅ Z α u d x = − ∫ Ω Z α p Z α div u d x + ∫ ∂ Ω Z α p Z α u ⋅ n d S + ∫ Ω [ Z α , ∇ ] p ⋅ Z α u d x . (58) For the last two terms, if Z α Z^\alpha Z α Z α u ⋅ n = 0 Z^\alpha u\cdot n=0 Z α u ⋅ n = 0 ∂ Ω \partial\Omega ∂ Ω Z y α 1 ∂ t α 0 Z_y^{\alpha_1}\partial_t^{\alpha_0} Z y α 1 ∂ t α 0 ∂ Ω \partial\Omega ∂ Ω
∣ ∫ ∂ Ω Z α p Z α u ⋅ n d S ∣ ≤ C m + 2 ( ∥ ∇ Z y α 1 − 1 ∂ t α 0 p ∥ L 2 + ∥ Z y α 1 − 1 ∂ t α 0 p ∥ H 1 ) 1 / 2 × ∥ Z y α 1 − 1 ∂ t α 0 p ∥ H 1 1 / 2 ( ∥ ∇ u ∥ H m − 1 + ∥ u ∥ H m ) . \left|\int_{\partial\Omega} Z^\alpha p\,Z^\alpha u\cdot n\,dS\right|
\le C_{m+2}\left(\|\nabla Z_y^{\alpha_1-1}\partial_t^{\alpha_0}p\|_{L^2}+\|Z_y^{\alpha_1-1}\partial_t^{\alpha_0}p\|_{H^1}\right)^{1/2}
\times\|Z_y^{\alpha_1-1}\partial_t^{\alpha_0}p\|_{H^1}^{1/2}\left(\|\nabla u\|_{H^{m-1}}+\|u\|_{H^m}\right). ∫ ∂ Ω Z α p Z α u ⋅ n d S ≤ C m + 2 ( ∥∇ Z y α 1 − 1 ∂ t α 0 p ∥ L 2 + ∥ Z y α 1 − 1 ∂ t α 0 p ∥ H 1 ) 1/2 × ∥ Z y α 1 − 1 ∂ t α 0 p ∥ H 1 1/2 ( ∥∇ u ∥ H m − 1 + ∥ u ∥ H m ) . Hence, together with Proposition 2.3,
∣ ∫ ∂ Ω Z α p Z α u ⋅ n d S + ∫ Ω [ Z α , ∇ ] p ⋅ Z α u d x ∣ ≤ δ ∥ ∇ p ∥ H m − 1 2 + C δ C m + 2 ( ∥ ∇ u ∥ H m − 1 2 + ∥ ( p , u ) ∥ H m 2 ) . (59) \left|\int_{\partial\Omega} Z^\alpha p\,Z^\alpha u\cdot n\,dS+\int_\Omega[Z^\alpha,\nabla]p\cdot Z^\alpha u\,dx\right|
\le\delta\|\nabla p\|_{H^{m-1}}^2+C_\delta C_{m+2}\left(\|\nabla u\|_{H^{m-1}}^2+\|(p,u)\|_{H^m}^2\right). \tag*{(59)} ∫ ∂ Ω Z α p Z α u ⋅ n d S + ∫ Ω [ Z α , ∇ ] p ⋅ Z α u d x ≤ δ ∥∇ p ∥ H m − 1 2 + C δ C m + 2 ( ∥∇ u ∥ H m − 1 2 + ∥ ( p , u ) ∥ H m 2 ) . (59) It remains to estimate the first term on the right-hand side of (3.35). Applying Z α Z^\alpha Z α
Z α div u = − 1 γ p Z α p t − u γ p ⋅ Z α ∇ p − ∑ ∣ β ∣ ≥ 1 β + ν = α C α β Z β ( 1 γ p ) Z ν p t − ∑ ∣ β ∣ ≥ 1 β + ν = α C α β Z β ( u γ p ) ⋅ Z ν ∇ p . (60) Z^\alpha\operatorname{div}u=-\frac{1}{\gamma p}Z^\alpha p_t-\frac{u}{\gamma p}\cdot Z^\alpha\nabla p-\sum_{\substack{|\beta|\ge1\\ \beta+\nu=\alpha}}C_{\alpha\beta}Z^\beta\left(\frac{1}{\gamma p}\right)Z^\nu p_t-\sum_{\substack{|\beta|\ge1\\ \beta+\nu=\alpha}}C_{\alpha\beta}Z^\beta\left(\frac{u}{\gamma p}\right)\cdot Z^\nu\nabla p. \tag*{(60)} Z α div u = − γ p 1 Z α p t − γ p u ⋅ Z α ∇ p − ∣ β ∣ ≥ 1 β + ν = α ∑ C α β Z β ( γ p 1 ) Z ν p t − ∣ β ∣ ≥ 1 β + ν = α ∑ C α β Z β ( γ p u ) ⋅ Z ν ∇ p . (60) Substituting (3.37) gives
− ∫ 0 t ∫ Ω Z α p Z α div u d x d τ = ∫ 0 t ∫ Ω 1 γ p Z α p Z α p t d x d τ + ∫ 0 t ∫ Ω Z α p u γ p ⋅ Z α ∇ p d x d τ + R α p . (61) -\int_0^t\int_\Omega Z^\alpha p\,Z^\alpha\operatorname{div}u\,dx\,d\tau
=\int_0^t\int_\Omega\frac{1}{\gamma p}Z^\alpha p\,Z^\alpha p_t\,dx\,d\tau
+\int_0^t\int_\Omega Z^\alpha p\,\frac{u}{\gamma p}\cdot Z^\alpha\nabla p\,dx\,d\tau+R_\alpha^p. \tag*{(61)} − ∫ 0 t ∫ Ω Z α p Z α div u d x d τ = ∫ 0 t ∫ Ω γ p 1 Z α p Z α p t d x d τ + ∫ 0 t ∫ Ω Z α p γ p u ⋅ Z α ∇ p d x d τ + R α p . (61) For the first two terms,
∫ 0 t ∫ Ω 1 γ p Z α p Z α p t d x d τ + ∫ 0 t ∫ Ω Z α p u γ p ⋅ Z α ∇ p d x d τ ≥ ∫ Ω 1 2 γ p ∣ Z α p ∣ 2 ( t ) d x − ∫ Ω 1 2 γ p 0 ∣ Z α p 0 ∣ 2 d x − δ ∫ 0 t ∥ ∇ p ( τ ) ∥ H m − 1 2 d τ − C δ [ 1 + P ( Q ( t ) ) ] ∫ 0 t ∥ p ( τ ) ∥ H m 2 d τ . (62) \int_0^t\int_\Omega\frac{1}{\gamma p}Z^\alpha p\,Z^\alpha p_t\,dx\,d\tau+\int_0^t\int_\Omega Z^\alpha p\,\frac{u}{\gamma p}\cdot Z^\alpha\nabla p\,dx\,d\tau
\ge\int_\Omega\frac{1}{2\gamma p}|Z^\alpha p|^2(t)\,dx-\int_\Omega\frac{1}{2\gamma p_0}|Z^\alpha p_0|^2\,dx-\delta\int_0^t\|\nabla p(\tau)\|_{H^{m-1}}^2\,d\tau
-C_\delta[1+P(Q(t))]\int_0^t\|p(\tau)\|_{H^m}^2\,d\tau. \tag*{(62)} ∫ 0 t ∫ Ω γ p 1 Z α p Z α p t d x d τ + ∫ 0 t ∫ Ω Z α p γ p u ⋅ Z α ∇ p d x d τ ≥ ∫ Ω 2 γ p 1 ∣ Z α p ∣ 2 ( t ) d x − ∫ Ω 2 γ p 0 1 ∣ Z α p 0 ∣ 2 d x − δ ∫ 0 t ∥∇ p ( τ ) ∥ H m − 1 2 d τ − C δ [ 1 + P ( Q ( t ))] ∫ 0 t ∥ p ( τ ) ∥ H m 2 d τ . (62) The remaining terms in R α p R_\alpha^p R α p
∣ R α p ∣ ≤ δ ∫ 0 t ∥ ∇ p ( τ ) ∥ H m − 1 2 d τ + C δ [ 1 + P ( Q ( t ) ) ] ∫ 0 t ∥ ( p , u ) ( τ ) ∥ H m 2 d τ . (63) |R_\alpha^p|\le\delta\int_0^t\|\nabla p(\tau)\|_{H^{m-1}}^2\,d\tau+C_\delta[1+P(Q(t))]\int_0^t\|(p,u)(\tau)\|_{H^m}^2\,d\tau. \tag*{(63)} ∣ R α p ∣ ≤ δ ∫ 0 t ∥∇ p ( τ ) ∥ H m − 1 2 d τ + C δ [ 1 + P ( Q ( t ))] ∫ 0 t ∥ ( p , u ) ( τ ) ∥ H m 2 d τ . (63) Combining (3.35)–(3.40), we arrive at
∫ 0 t ∫ Ω Z α ∇ p ⋅ Z α u d x d τ ≥ ∫ Ω 1 2 γ p ∣ Z α p ∣ 2 ( t ) d x − ∫ Ω 1 2 γ p 0 ∣ Z α p 0 ∣ 2 d x − δ ∫ 0 t ∥ ∇ p ( τ ) ∥ H m − 1 2 d τ − C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (64) \int_0^t\int_\Omega Z^\alpha\nabla p\cdot Z^\alpha u\,dx\,d\tau
\ge\int_\Omega\frac{1}{2\gamma p}|Z^\alpha p|^2(t)\,dx-\int_\Omega\frac{1}{2\gamma p_0}|Z^\alpha p_0|^2\,dx-\delta\int_0^t\|\nabla p(\tau)\|_{H^{m-1}}^2\,d\tau
-C_\delta C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(64)} ∫ 0 t ∫ Ω Z α ∇ p ⋅ Z α u d x d τ ≥ ∫ Ω 2 γ p 1 ∣ Z α p ∣ 2 ( t ) d x − ∫ Ω 2 γ p 0 1 ∣ Z α p 0 ∣ 2 d x − δ ∫ 0 t ∥∇ p ( τ ) ∥ H m − 1 2 d τ − C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (64) The commutators C 1 α C_1^\alpha C 1 α C 2 α C_2^\alpha C 2 α C 3 α C_3^\alpha C 3 α
∑ ∣ α ∣ = m ∫ 0 t ∣ ∫ Ω ( C 1 α + C 2 α + C 3 α ) ⋅ Z α u d x ∣ d τ ≤ C [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (65) \sum_{|\alpha|=m}\int_0^t\left|\int_\Omega(C_1^\alpha+C_2^\alpha+C_3^\alpha)\cdot Z^\alpha u\,dx\right|\,d\tau
\le C[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(65)} ∣ α ∣ = m ∑ ∫ 0 t ∫ Ω ( C 1 α + C 2 α + C 3 α ) ⋅ Z α u d x d τ ≤ C [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (65) Substituting (56) , (57) , (64) , and (65) into (46) , summing over ∣ α ∣ = m |\alpha|=m ∣ α ∣ = m [ 0 , t ] [0,t] [ 0 , t ]
sup 0 ≤ τ ≤ t ( ∥ Z m ( p , u ) ( τ ) ∥ L 2 2 + ∥ ∇ Z m ϕ ( τ ) ∥ L 2 2 ) + ε ∫ 0 t ∥ ∇ Z m u ( τ ) ∥ L 2 2 d τ ≤ C m + 2 { E m ( 0 ) + δ ε ∫ 0 t ∥ ∇ u ( τ ) ∥ H m 2 d τ + δ ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + δ ∫ 0 t ∥ ∇ p ( τ ) ∥ H m − 1 2 d τ + C δ [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ } . \begin{aligned}
\sup_{0\leq\tau\leq t}(\|Z^m(p,u)(\tau)\|_{L^2}^2+\|\nabla Z^m\phi(\tau)\|_{L^2}^2)+\varepsilon\int_0^t\|\nabla Z^m u(\tau)\|_{L^2}^2\,d\tau
\leq C_{m+2}\{\mathcal{E}_m(0)+\delta\varepsilon\int_0^t\|\nabla u(\tau)\|_{H^m}^2\,d\tau+\delta\varepsilon^2\int_0^t\|\nabla^2u(\tau)\|_{H^{m-1}}^2\,d\tau\\
&+\delta\int_0^t\|\nabla p(\tau)\|_{H^{m-1}}^2\,d\tau+C_\delta[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau\}.
\end{aligned} 0 ≤ τ ≤ t sup ( ∥ Z m ( p , u ) ( τ ) ∥ L 2 2 + ∥∇ Z m ϕ ( τ ) ∥ L 2 2 ) + ε ∫ 0 t ∥∇ Z m u ( τ ) ∥ L 2 2 d τ ≤ C m + 2 { E m ( 0 ) + δ ε ∫ 0 t ∥∇ u ( τ ) ∥ H m 2 d τ + δ ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + δ ∫ 0 t ∥∇ p ( τ ) ∥ H m − 1 2 d τ + C δ [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ } . Taking δ > 0 \delta>0 δ > 0 (45) . This completes the proof.
Lemma 3.4 (High-order estimate for div u \operatorname{div}u div u ∇ p ) . L e t \nabla p). Let ∇ p ) . L e t Z α Z^\alpha Z α ∣ α ∣ = m − 1 |\alpha|=m-1 ∣ α ∣ = m − 1 α 0 ≤ m − 2 \alpha_0\leq m-2 α 0 ≤ m − 2 δ > 0 \delta>0 δ > 0
sup 0 ≤ τ ≤ t ∫ Ω ( ρ ∣ Z α div u ∣ 2 + 1 γ p ∣ Z α ∇ p ∣ 2 ) ( τ ) d x + c ε ∫ 0 t ∥ ∇ Z α div u ( τ ) ∥ L 2 2 d τ ≤ C ∫ Ω ( ρ 0 ∣ Z α div u 0 ∣ 2 + 1 γ p 0 ∣ Z α ∇ p 0 ∣ 2 ) d x \sup_{0\leq\tau\leq t}\int_\Omega\left(\rho|Z^\alpha\operatorname{div}u|^2+\frac{1}{\gamma p}|Z^\alpha\nabla p|^2\right)(\tau)\,dx+c\varepsilon\int_0^t\left\|\nabla Z^\alpha\operatorname{div}u(\tau)\right\|_{L^2}^2\,d\tau
\leq C\int_\Omega\left(\rho_0|Z^\alpha\operatorname{div}u_0|^2+\frac{1}{\gamma p_0}|Z^\alpha\nabla p_0|^2\right)\,dx 0 ≤ τ ≤ t sup ∫ Ω ( ρ ∣ Z α div u ∣ 2 + γ p 1 ∣ Z α ∇ p ∣ 2 ) ( τ ) d x + c ε ∫ 0 t ∥ ∇ Z α div u ( τ ) ∥ L 2 2 d τ ≤ C ∫ Ω ( ρ 0 ∣ Z α div u 0 ∣ 2 + γ p 0 1 ∣ Z α ∇ p 0 ∣ 2 ) d x + C m + 2 { δ ∫ 0 t ∥ ∇ Z t m − 1 p ( τ ) ∥ L 2 2 d τ + ( δ + ε ) ∫ 0 t ∥ ∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ + ε ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 2 2 d τ + C δ [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ } . (66) +C_{m+2}\left\{\delta\int_0^t\left\|\nabla Z^{m-1}_t p(\tau)\right\|_{L^2}^2\,d\tau+(\delta+\varepsilon)\int_0^t\left\|\nabla Z^{m-2}\operatorname{div}u(\tau)\right\|_{L^2}^2\,d\tau
+\varepsilon\int_0^t\left\|\nabla^2u(\tau)\right\|_{H^{m-2}}^2\,d\tau+C_\delta[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau\right\}. \tag*{(66)} + C m + 2 { δ ∫ 0 t ∇ Z t m − 1 p ( τ ) L 2 2 d τ + ( δ + ε ) ∫ 0 t ∇ Z m − 2 div u ( τ ) L 2 2 d τ + ε ∫ 0 t ∇ 2 u ( τ ) H m − 2 2 d τ + C δ [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ } . (66) Proof. We argue by induction on the order. The case ∣ α ∣ = 0 |\alpha|=0 ∣ α ∣ = 0 m − 2 m-2 m − 2 ∣ α ∣ = m − 1 |\alpha|=m-1 ∣ α ∣ = m − 1 α 0 ≤ m − 2 \alpha_0\leq m-2 α 0 ≤ m − 2
Applying Z α Z^\alpha Z α (24) , multiplying the resulting equation by ∇ Z α div u \nabla Z^\alpha\operatorname{div}u ∇ Z α div u Ω × ( 0 , t ) \Omega\times(0,t) Ω × ( 0 , t )
∫ 0 t ∫ Ω ( ρ Z α u t + ρ u ⋅ ∇ Z α u + Z α ∇ p ) ⋅ ∇ Z α div u d x d τ = − μ ε ∫ 0 t ∫ Ω Z α ∇ × ω ⋅ ∇ Z α div u d x d τ + ( 2 μ + λ ) ε ∫ 0 t ∥ ∇ Z α div u ∥ L 2 2 d τ + ∫ 0 t ∫ Ω Z α ( ρ ∇ ϕ ) ⋅ ∇ Z α div u d x d τ + ∫ 0 t ∫ Ω ( D 1 α + D 2 α ) ⋅ ∇ Z α div u d x d τ . (67) \begin{aligned}
\int_0^t\int_\Omega\left(\rho Z^\alpha u_t+\rho u\cdot\nabla Z^\alpha u+Z^\alpha\nabla p\right)\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau
=-\mu\varepsilon\int_0^t\int_\Omega Z^\alpha\nabla\times\omega\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau\\
&+(2\mu+\lambda)\varepsilon\int_0^t\left\|\nabla Z^\alpha\operatorname{div}u\right\|_{L^2}^2\,d\tau
+\int_0^t\int_\Omega Z^\alpha(\rho\nabla\phi)\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau
+\int_0^t\int_\Omega(D_1^\alpha+D_2^\alpha)\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau. \tag*{(67)}
\end{aligned} ∫ 0 t ∫ Ω ( ρ Z α u t + ρ u ⋅ ∇ Z α u + Z α ∇ p ) ⋅ ∇ Z α div u d x d τ = − μ ε ∫ 0 t ∫ Ω Z α ∇ × ω ⋅ ∇ Z α div u d x d τ + ( 2 μ + λ ) ε ∫ 0 t ∥ ∇ Z α div u ∥ L 2 2 d τ + ∫ 0 t ∫ Ω Z α ( ρ ∇ ϕ ) ⋅ ∇ Z α div u d x d τ + ∫ 0 t ∫ Ω ( D 1 α + D 2 α ) ⋅ ∇ Z α div u d x d τ . (67) where
D 1 α : = − [ Z α , ρ ] u t , D 2 α : = − [ Z α , ρ u ⋅ ∇ ] u . D_1^\alpha:=-[Z^\alpha,\rho]u_t,\qquad D_2^\alpha:=-[Z^\alpha,\rho u\cdot\nabla]u. D 1 α := − [ Z α , ρ ] u t , D 2 α := − [ Z α , ρ u ⋅ ∇ ] u . We first estimate the electric force term. We write
I E α : = ∫ 0 t ∫ Ω Z α ( ρ ∇ ϕ ) ⋅ ∇ Z α div u d x d τ = ∫ 0 t ∫ Ω ρ ∇ Z α ϕ ⋅ ∇ Z α div u d x d τ + ∫ 0 t ∫ Ω [ Z α , ρ ] ∇ ϕ ⋅ ∇ Z α div u d x d τ + ∫ 0 t ∫ Ω ρ [ Z α , ∇ ] ϕ ⋅ ∇ Z α div u d x d τ ≡ I 1 α + I 2 α + I 3 α . (68) \begin{aligned}
I_E^\alpha:=\int_0^t\int_\Omega Z^\alpha(\rho\nabla\phi)\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau\\
&=\int_0^t\int_\Omega\rho\nabla Z^\alpha\phi\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau
+\int_0^t\int_\Omega[Z^\alpha,\rho]\nabla\phi\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau\\
&+\int_0^t\int_\Omega\rho[Z^\alpha,\nabla]\phi\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau
\equiv I_1^\alpha+I_2^\alpha+I_3^\alpha. \tag*{(68)}
\end{aligned} I E α := ∫ 0 t ∫ Ω Z α ( ρ ∇ ϕ ) ⋅ ∇ Z α div u d x d τ = ∫ 0 t ∫ Ω ρ ∇ Z α ϕ ⋅ ∇ Z α div u d x d τ + ∫ 0 t ∫ Ω [ Z α , ρ ] ∇ ϕ ⋅ ∇ Z α div u d x d τ + ∫ 0 t ∫ Ω ρ [ Z α , ∇ ] ϕ ⋅ ∇ Z α div u d x d τ ≡ I 1 α + I 2 α + I 3 α . (68) For I 1 α I_1^\alpha I 1 α
I 1 α = − ∫ 0 t ∫ Ω div ( ρ ∇ Z α ϕ ) Z α div u d x d τ + ∫ 0 t ∫ ∂ Ω ρ ∂ n Z α ϕ Z α div u d S d τ = ≔ I 1 , i n t α + I 1 , b d α . (69) \begin{aligned}
I_1^\alpha&= -\int_0^t\int_\Omega\operatorname{div}(\rho\nabla Z^\alpha\phi)Z^\alpha\operatorname{div}u\,dx\,d\tau+\int_0^t\int_{\partial\Omega}\rho\partial_n Z^\alpha\phi\,Z^\alpha\operatorname{div}u\,dS\,d\tau\\
&\mathrel{\phantom{=}}\coloneqq I_{1,\mathrm{int}}^\alpha+I_{1,\mathrm{bd}}^\alpha. \tag*{(69)}
\end{aligned} I 1 α = − ∫ 0 t ∫ Ω div ( ρ ∇ Z α ϕ ) Z α div u d x d τ + ∫ 0 t ∫ ∂ Ω ρ ∂ n Z α ϕ Z α div u d S d τ = : = I 1 , int α + I 1 , bd α . (69) Since
div ( ρ ∇ Z α ϕ ) = ∇ ρ ⋅ ∇ Z α ϕ + ρ Z α ( ρ − 1 ) + ρ [ Δ , Z α ] ϕ , \operatorname{div}(\rho\nabla Z^\alpha\phi)=\nabla\rho\cdot\nabla Z^\alpha\phi+\rho Z^\alpha(\rho-1)+\rho[\Delta,Z^\alpha]\phi, div ( ρ ∇ Z α ϕ ) = ∇ ρ ⋅ ∇ Z α ϕ + ρ Z α ( ρ − 1 ) + ρ [ Δ , Z α ] ϕ , the first two terms are estimated by Proposition 2.2 and Proposition 2.4. The last one is treated in the same way as in (55) . Hence
∣ I 1 , i n t α ∣ ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (70) |I_{1,\mathrm{int}}^\alpha|\le C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(70)} ∣ I 1 , int α ∣ ≤ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (70) For the boundary part, using (3) , integration by parts along ∂ Ω \partial\Omega ∂ Ω
∣ I 1 , b d α ∣ ≤ δ ∫ 0 t ∥ ∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (71) |I_{1,\mathrm{bd}}^\alpha|\le\delta\int_0^t\|\nabla Z^{m-2}\operatorname{div}u(\tau)\|_{L^2}^2\,d\tau+C_\delta C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(71)} ∣ I 1 , bd α ∣ ≤ δ ∫ 0 t ∥∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (71) For I 2 α I_2^\alpha I 2 α
I 2 α = − ∫ 0 t ∫ Ω div ( [ Z α , ρ ] ∇ ϕ ) Z α div u d x d τ + ∫ 0 t ∫ ∂ Ω h α Z α div u d S d τ , h α ≔ n ⋅ [ Z α , ρ ] ∇ ϕ . I_2^\alpha=-\int_0^t\int_\Omega\operatorname{div}([Z^\alpha,\rho]\nabla\phi)Z^\alpha\operatorname{div}u\,dx\,d\tau+\int_0^t\int_{\partial\Omega}h_\alpha Z^\alpha\operatorname{div}u\,dS\,d\tau,\qquad h_\alpha\coloneqq n\cdot[Z^\alpha,\rho]\nabla\phi. I 2 α = − ∫ 0 t ∫ Ω div ([ Z α , ρ ] ∇ ϕ ) Z α div u d x d τ + ∫ 0 t ∫ ∂ Ω h α Z α div u d S d τ , h α : = n ⋅ [ Z α , ρ ] ∇ ϕ . The boundary term is not identically zero on a curved boundary. Indeed, for a tangential vector field Z i Z_i Z i n ⋅ Z i ∇ ϕ = − ( Z i n ) ⋅ ∇ ϕ n\cdot Z_i\nabla\phi=-(Z_i n)\cdot\nabla\phi n ⋅ Z i ∇ ϕ = − ( Z i n ) ⋅ ∇ ϕ Z α Z^\alpha Z α Z 3 Z_3 Z 3
h α = − ∑ 0 < β ≤ α ( α β ) ( Z β ρ ) ∑ 0 < ν ≤ α − β ( α − β ν ) ( Z ν n ) ⋅ Z α − β − ν ∇ ϕ . h_\alpha=-\sum_{0<\beta\le\alpha}\binom{\alpha}{\beta}(Z^\beta\rho)\sum_{0<\nu\le\alpha-\beta}\binom{\alpha-\beta}{\nu}(Z^\nu n)\cdot Z^{\alpha-\beta-\nu}\nabla\phi. h α = − 0 < β ≤ α ∑ ( β α ) ( Z β ρ ) 0 < ν ≤ α − β ∑ ( ν α − β ) ( Z ν n ) ⋅ Z α − β − ν ∇ ϕ . In particular, each nonzero term contains a derivative of the normal and at most m − 2 m-2 m − 2 α 0 ≤ m − 2 \alpha_0\le m-2 α 0 ≤ m − 2 T T T Z α Z^\alpha Z α Z α div u = T g Z^\alpha\operatorname{div}u=Tg Z α div u = T g g g g m − 2 m-2 m − 2 div u \operatorname{div}u div u
∫ ∂ Ω h α T g d S = − ∫ ∂ Ω ( T h α + ( div ∂ Ω T ) h α ) g d S . \int_{\partial\Omega}h_\alpha Tg\,dS=-\int_{\partial\Omega}\left(Th_\alpha+(\operatorname{div}_{\partial\Omega}T)h_\alpha\right)g\,dS. ∫ ∂ Ω h α T g d S = − ∫ ∂ Ω ( T h α + ( div ∂ Ω T ) h α ) g d S . The local cut-off terms have the same lower order. More explicitly, the product, trace, and Poisson estimates give
∫ 0 t ∣ T h α + ( div ∂ Ω T ) h α ∣ L 2 ( ∂ Ω ) 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m d τ , \int_0^t\left|Th_\alpha+(\operatorname{div}_{\partial\Omega}T)h_\alpha\right|_{L^2(\partial\Omega)}^2\,d\tau\le C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m\,d\tau, ∫ 0 t ∣ T h α + ( div ∂ Ω T ) h α ∣ L 2 ( ∂ Ω ) 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m d τ , ∫ 0 t ∣ g ∣ L 2 ( ∂ Ω ) 2 d τ ≤ C ∫ 0 t E m d τ + C ( ∫ 0 t E m d τ ) 1 / 2 ( ∫ 0 t ∥ ∇ Z m − 2 div u ∥ L 2 2 d τ ) 1 / 2 . \int_0^t|g|_{L^2(\partial\Omega)}^2\,d\tau\le C\int_0^t\mathcal{E}_m\,d\tau+C\left(\int_0^t\mathcal{E}_m\,d\tau\right)^{1/2}\left(\int_0^t\|\nabla Z^{m-2}\operatorname{div}u\|_{L^2}^2\,d\tau\right)^{1/2}. ∫ 0 t ∣ g ∣ L 2 ( ∂ Ω ) 2 d τ ≤ C ∫ 0 t E m d τ + C ( ∫ 0 t E m d τ ) 1/2 ( ∫ 0 t ∥∇ Z m − 2 div u ∥ L 2 2 d τ ) 1/2 . Here the first estimate uses only mixed pressure gradients with at most m − 2 m-2 m − 2 E m \mathcal{E}_m E m ∣ g ∣ L 2 ( ∂ Ω ) 2 ≤ C ∥ g ∥ L 2 ( ∥ g ∥ L 2 + ∥ ∇ g ∥ L 2 ) |g|_{L^2(\partial\Omega)}^2\le C\|g\|_{L^2}(\|g\|_{L^2}+\|\nabla g\|_{L^2}) ∣ g ∣ L 2 ( ∂ Ω ) 2 ≤ C ∥ g ∥ L 2 ( ∥ g ∥ L 2 + ∥∇ g ∥ L 2 ) Z α Z^\alpha Z α Z 3 Z_3 Z 3
∣ I 2 α ∣ ≤ δ ∫ 0 t ∥ ∇ Z m − 2 div u ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (72) \begin{aligned}
|I_2^\alpha|&\le\delta\int_0^t\|\nabla Z^{m-2}\operatorname{div}u\|_{L^2}^2\,d\tau\\
&\quad+C_\delta C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(72)}
\end{aligned} ∣ I 2 α ∣ ≤ δ ∫ 0 t ∥∇ Z m − 2 div u ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (72) For I 3 α I_3^\alpha I 3 α
I 3 α = − ∫ 0 t ∫ Ω div ( ρ [ Z α , ∇ ] φ ) Z α div u d x d τ + ∫ 0 t ∫ ∂ Ω ρ [ Z α , ∇ ] φ ⋅ n Z α div u d S d τ . (73) \begin{aligned}
I_3^\alpha={}&-\int_0^t\int_\Omega\operatorname{div}\left(\rho[Z^\alpha,\nabla]\varphi\right)Z^\alpha\operatorname{div}u\,dx\,d\tau\\
&+\int_0^t\int_{\partial\Omega}\rho[Z^\alpha,\nabla]\varphi\cdot n\,Z^\alpha\operatorname{div}u\,dS\,d\tau. \tag*{(73)}
\end{aligned} I 3 α = − ∫ 0 t ∫ Ω div ( ρ [ Z α , ∇ ] φ ) Z α div u d x d τ + ∫ 0 t ∫ ∂ Ω ρ [ Z α , ∇ ] φ ⋅ n Z α div u d S d τ . (73) Using Proposition 2.2 for the interior part and Proposition 2.3 for the boundary part, together with Proposition 2.4 , we obtain
∣ I 3 α ∣ ≤ δ ∫ 0 t ∥ ∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (74) |I_3^\alpha|\leq\delta\int_0^t\|\nabla Z^{m-2}\operatorname{div}u(\tau)\|_{L^2}^2\,d\tau+C_\delta C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(74)} ∣ I 3 α ∣ ≤ δ ∫ 0 t ∥∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (74) Combining (70) –(3.51)}, we obtain
∣ I E α ∣ ≤ δ ∫ 0 t ∥ ∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (75) |I_E^\alpha|\leq\delta\int_0^t\|\nabla Z^{m-2}\operatorname{div}u(\tau)\|_{L^2}^2\,d\tau+C_\delta C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(75)} ∣ I E α ∣ ≤ δ ∫ 0 t ∥∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (75) We next consider the inertial part. Integrating by parts, using the continuity equation and (1.2), and estimating the boundary contribution by Proposition 2.3 , we get
∫ 0 t ∫ Ω ( ρ Z α u t + ρ u ⋅ ∇ Z α u ) ⋅ ∇ Z α div u d x d τ ≤ − 1 2 ∫ Ω ρ ∣ Z α div u ∣ 2 ( t ) d x + 1 2 ∫ Ω ρ 0 ∣ Z α div u 0 ∣ 2 d x + δ ∫ 0 t ∥ ∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (76) \begin{aligned}
\int_0^t\int_\Omega(\rho Z^\alpha u_t+\rho u\cdot\nabla Z^\alpha u)\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau
\leq{}&-\frac{1}{2}\int_\Omega\rho|Z^\alpha\operatorname{div}u|^2(t)\,dx+\frac{1}{2}\int_\Omega\rho_0|Z^\alpha\operatorname{div}u_0|^2\,dx\\
&+\delta\int_0^t\|\nabla Z^{m-2}\operatorname{div}u(\tau)\|_{L^2}^2\,d\tau+C_\delta C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(76)}
\end{aligned} ∫ 0 t ∫ Ω ( ρ Z α u t + ρ u ⋅ ∇ Z α u ) ⋅ ∇ Z α div u d x d τ ≤ − 2 1 ∫ Ω ρ ∣ Z α div u ∣ 2 ( t ) d x + 2 1 ∫ Ω ρ 0 ∣ Z α div u 0 ∣ 2 d x + δ ∫ 0 t ∥∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (76) Next, we estimate the pressure term. Applying ∇ \nabla ∇ (61) –(64) , we obtain, after integration by parts in time and space,
∫ 0 t ∫ Ω Z α ∇ p ⋅ ∇ Z α div u d x d τ ≤ − ∫ Ω 1 2 γ p ∣ Z α ∇ p ∣ 2 ( t ) d x + ∫ Ω 1 2 γ p 0 ∣ Z α ∇ p 0 ∣ 2 d x + δ ∫ 0 t ∥ ∇ ∂ t m − 1 p ( τ ) ∥ L 2 2 d τ + C δ [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (77) \begin{aligned}
\int_0^t\int_\Omega Z^\alpha\nabla p\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau
\leq{}&-\int_\Omega\frac{1}{2\gamma p}|Z^\alpha\nabla p|^2(t)\,dx+\int_\Omega\frac{1}{2\gamma p_0}|Z^\alpha\nabla p_0|^2\,dx\\
&+\delta\int_0^t\|\nabla\partial_t^{m-1}p(\tau)\|_{L^2}^2\,d\tau+C_\delta[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(77)}
\end{aligned} ∫ 0 t ∫ Ω Z α ∇ p ⋅ ∇ Z α div u d x d τ ≤ − ∫ Ω 2 γ p 1 ∣ Z α ∇ p ∣ 2 ( t ) d x + ∫ Ω 2 γ p 0 1 ∣ Z α ∇ p 0 ∣ 2 d x + δ ∫ 0 t ∥∇ ∂ t m − 1 p ( τ ) ∥ L 2 2 d τ + C δ [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (77) The commutator terms generated by differentiating the coefficients ( γ p ) − 1 (\gamma p)^{-1} ( γ p ) − 1 u ( γ p ) − 1 u(\gamma p)^{-1} u ( γ p ) − 1 Proposition 2.2 , as in the pressure commutator estimate of Lemma 3.3.
For the viscous terms, we use ∇ × ∇ Z α div u = 0 \nabla\times\nabla Z^\alpha\operatorname{div}u=0 ∇ × ∇ Z α div u = 0 ∂ Ω \partial\Omega ∂ Ω Proposition 2.3 , and Young’s inequality. This yields
− μ ε ∫ 0 t ∫ Ω Z α ∇ × ω ⋅ ∇ Z α div u d x d τ + ( 2 μ + λ ) ε ∫ 0 t ∥ ∇ Z α div u ∥ L 2 2 d τ ≥ c ε ∫ 0 t ∥ ∇ Z α div u ( τ ) ∥ L 2 2 d τ − C ε ∫ 0 t ∥ ∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ − C ε ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 2 2 d τ − C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (78) \begin{aligned}
-\mu\varepsilon\int_0^t\int_\Omega Z^\alpha\nabla\times\omega\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau
+(2\mu+\lambda)\varepsilon\int_0^t\|\nabla Z^\alpha\operatorname{div}u\|_{L^2}^2\,d\tau
\geq{}&c\varepsilon\int_0^t\|\nabla Z^\alpha\operatorname{div}u(\tau)\|_{L^2}^2\,d\tau\\
&-C\varepsilon\int_0^t\|\nabla Z^{m-2}\operatorname{div}u(\tau)\|_{L^2}^2\,d\tau\\
&-C\varepsilon\int_0^t\|\nabla^2u(\tau)\|_{H^{m-2}}^2\,d\tau-C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(78)}
\end{aligned} − μ ε ∫ 0 t ∫ Ω Z α ∇ × ω ⋅ ∇ Z α div u d x d τ + ( 2 μ + λ ) ε ∫ 0 t ∥∇ Z α div u ∥ L 2 2 d τ ≥ c ε ∫ 0 t ∥∇ Z α div u ( τ ) ∥ L 2 2 d τ − C ε ∫ 0 t ∥∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ − C ε ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 2 2 d τ − C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (78) The commutators D 1 α D_1^\alpha D 1 α D 2 α D_2^\alpha D 2 α Proposition 2.2 , integrating by parts,
∣ ∫ 0 t ∫ Ω ( D 1 α + D 2 α ) ⋅ ∇ Z α div u d x d τ ∣ ≤ δ ∫ 0 t ∥ ∇ Z α div u ( τ ) ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (79) \left|\int_0^t\int_\Omega(D_1^\alpha+D_2^\alpha)\cdot\nabla Z^\alpha\operatorname{div}u\,dx\,d\tau\right|
\leq\delta\int_0^t\|\nabla Z^\alpha\operatorname{div}u(\tau)\|_{L^2}^2\,d\tau+C_\delta C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(79)} ∫ 0 t ∫ Ω ( D 1 α + D 2 α ) ⋅ ∇ Z α div u d x d τ ≤ δ ∫ 0 t ∥∇ Z α div u ( τ ) ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (79) Substituting (75) , (76) , (77) , (78) , and (79) into (67) , we obtain (66) . This completes the proof.
It remains to handle the pure time derivative case, which is not covered by Lemma 3.4. In this case no spatial conormal derivative is available for tangential integration by parts, and the corresponding estimate is therefore taken with an additional factor ε \varepsilon ε
Lemma 3.5 (Pure time derivative estimate). Let m ≥ 1 m \ge1 m ≥ 1
sup 0 ≤ τ ≤ t ε ∫ Ω ( ρ ∣ ∂ t m − 1 div u ∣ 2 + 1 γ p ∣ ∂ t m − 1 ∇ p ∣ 2 ) ( τ ) d x + ε 2 ∫ 0 t ∥ ∇ ∂ t m − 1 div u ( τ ) ∥ L 2 2 d τ ≤ C ε ∫ Ω ( ρ 0 ∣ ∂ t m − 1 div u 0 ∣ 2 + 1 γ p 0 ∣ ∂ t m − 1 ∇ p 0 ∣ 2 ) d x + C m + 1 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (80) \begin{aligned}
\sup_{0 \le\tau\le t} \varepsilon\int_{\Omega} \left( \rho\left|\partial_t^{m-1}\operatorname{div} u\right|^2 + \frac{1}{\gamma p}\left|\partial_t^{m-1}\nabla p\right|^2 \right)(\tau)\,dx
&+ \varepsilon^2 \int_0^t \left\|\nabla\partial_t^{m-1}\operatorname{div}u(\tau)\right\|_{L^2}^2\,d\tau\\
&\le C\varepsilon\int_{\Omega} \left( \rho_0 \left|\partial_t^{m-1}\operatorname{div} u_0\right|^2 + \frac{1}{\gamma p_0}\left|\partial_t^{m-1}\nabla p_0\right|^2 \right)\,dx \\
&\quad+ C_{m+1}[1+P(Q(t))]\int_0^t \mathcal{E}_m(\tau)\,d\tau. \tag*{(80)}
\end{aligned} 0 ≤ τ ≤ t sup ε ∫ Ω ( ρ ∂ t m − 1 div u 2 + γ p 1 ∂ t m − 1 ∇ p 2 ) ( τ ) d x + ε 2 ∫ 0 t ∇ ∂ t m − 1 div u ( τ ) L 2 2 d τ ≤ C ε ∫ Ω ( ρ 0 ∂ t m − 1 div u 0 2 + γ p 0 1 ∂ t m − 1 ∇ p 0 2 ) d x + C m + 1 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (80) Proof. Applying ∂ t m − 1 \partial_t^{m-1} ∂ t m − 1 ε ∇ ∂ t m − 1 div u \varepsilon\nabla\partial_t^{m-1}\operatorname{div}u ε ∇ ∂ t m − 1 div u Ω × ( 0 , t ) \Omega\times(0,t) Ω × ( 0 , t )
ε ∫ 0 t ∫ Ω ( ρ ∂ t m u + ρ u ⋅ ∇ ∂ t m − 1 u + ∂ t m − 1 ∇ p ) ⋅ ∇ ∂ t m − 1 div u d x d τ = ε ∫ 0 t ∫ Ω ∂ t m − 1 ( ρ ∇ ϕ ) ⋅ ∇ ∂ t m − 1 div u d x d τ − μ ε 2 ∫ 0 t ∫ Ω ∇ × ∂ t m − 1 ω ⋅ ∇ ∂ t m − 1 div u d x d τ + ( 2 μ + λ ) ε 2 ∫ 0 t ∥ ∇ ∂ t m − 1 div u ∥ L 2 2 d τ + ε ∫ 0 t ∫ Ω C t m − 1 ⋅ ∇ ∂ t m − 1 div u d x d τ . (81) \begin{aligned}
\varepsilon\int_0^t\int_{\Omega} \left(\rho\partial_t^m u+\rho u\cdot\nabla\partial_t^{m-1}u+\partial_t^{m-1}\nabla p\right)\cdot\nabla\partial_t^{m-1}\operatorname{div}u\,dx\,d\tau
&=\varepsilon\int_0^t\int_{\Omega}\partial_t^{m-1}(\rho\nabla\phi)\cdot\nabla\partial_t^{m-1}\operatorname{div}u\,dx\,d\tau\\
&\quad-\mu\varepsilon^2\int_0^t\int_{\Omega}\nabla\times\partial_t^{m-1}\omega\cdot\nabla\partial_t^{m-1}\operatorname{div}u\,dx\,d\tau\\
&\quad+(2\mu+\lambda)\varepsilon^2\int_0^t\left\|\nabla\partial_t^{m-1}\operatorname{div}u\right\|_{L^2}^2\,d\tau\\
&\quad+\varepsilon\int_0^t\int_{\Omega}\mathcal{C}_t^{m-1}\cdot\nabla\partial_t^{m-1}\operatorname{div}u\,dx\,d\tau. \tag*{(81)}
\end{aligned} ε ∫ 0 t ∫ Ω ( ρ ∂ t m u + ρ u ⋅ ∇ ∂ t m − 1 u + ∂ t m − 1 ∇ p ) ⋅ ∇ ∂ t m − 1 div u d x d τ = ε ∫ 0 t ∫ Ω ∂ t m − 1 ( ρ ∇ ϕ ) ⋅ ∇ ∂ t m − 1 div u d x d τ − μ ε 2 ∫ 0 t ∫ Ω ∇ × ∂ t m − 1 ω ⋅ ∇ ∂ t m − 1 div u d x d τ + ( 2 μ + λ ) ε 2 ∫ 0 t ∇ ∂ t m − 1 div u L 2 2 d τ + ε ∫ 0 t ∫ Ω C t m − 1 ⋅ ∇ ∂ t m − 1 div u d x d τ . (81) where
C t m − 1 = − [ ∂ t m − 1 , ρ ] u t − [ ∂ t m − 1 , ρ u ⋅ ∇ ] u . \mathcal{C}_t^{m-1}=-[\partial_t^{m-1},\rho]u_t-[\partial_t^{m-1},\rho u\cdot\nabla]u. C t m − 1 = − [ ∂ t m − 1 , ρ ] u t − [ ∂ t m − 1 , ρ u ⋅ ∇ ] u . We first estimate the electric force term. Since time derivatives commute with spatial derivatives,
∂ t m − 1 ( ρ ∇ ϕ ) = ρ ∇ ∂ t m − 1 ϕ + [ ∂ t m − 1 , ρ ] ∇ ϕ . \partial_t^{m-1}(\rho\nabla\phi)=\rho\nabla\partial_t^{m-1}\phi+[\partial_t^{m-1},\rho]\nabla\phi. ∂ t m − 1 ( ρ ∇ ϕ ) = ρ ∇ ∂ t m − 1 ϕ + [ ∂ t m − 1 , ρ ] ∇ ϕ . Hence, by Young’s inequality and Proposition 2.2,
∣ ε ∫ 0 t ∫ Ω ∂ t m − 1 ( ρ ∇ ϕ ) ⋅ ∇ ∂ t m − 1 div u d x d τ ∣ ≤ ε ∫ 0 t ( ∥ ρ ∇ ∂ t m − 1 ϕ ∥ L 2 + ∥ [ ∂ t m − 1 , ρ ] ∇ ϕ ∥ L 2 ) ∥ ∇ ∂ t m − 1 div u ∥ L 2 d τ ≤ 2 μ + λ 8 ε 2 ∫ 0 t ∥ ∇ ∂ t m − 1 div u ( τ ) ∥ L 2 2 d τ + C m + 1 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (82) \begin{aligned}
\left|\varepsilon\int_0^t\int_{\Omega}\partial_t^{m-1}(\rho\nabla\phi)\cdot\nabla\partial_t^{m-1}\operatorname{div}u\,dx\,d\tau\right|
&\le\varepsilon\int_0^t\left(\|\rho\nabla\partial_t^{m-1}\phi\|_{L^2}+\|[\partial_t^{m-1},\rho]\nabla\phi\|_{L^2}\right)\left\|\nabla\partial_t^{m-1}\operatorname{div}u\right\|_{L^2}\,d\tau\\
&\le\frac{2\mu+\lambda}{8}\varepsilon^2\int_0^t\left\|\nabla\partial_t^{m-1}\operatorname{div}u(\tau)\right\|_{L^2}^2\,d\tau+C_{m+1}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(82)}
\end{aligned} ε ∫ 0 t ∫ Ω ∂ t m − 1 ( ρ ∇ ϕ ) ⋅ ∇ ∂ t m − 1 div u d x d τ ≤ ε ∫ 0 t ( ∥ ρ ∇ ∂ t m − 1 ϕ ∥ L 2 + ∥ [ ∂ t m − 1 , ρ ] ∇ ϕ ∥ L 2 ) ∇ ∂ t m − 1 div u L 2 d τ ≤ 8 2 μ + λ ε 2 ∫ 0 t ∇ ∂ t m − 1 div u ( τ ) L 2 2 d τ + C m + 1 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (82) By the same argument as in Lemmas 3.2 and 3.4, using (1.2), (3.19), Proposition 2.3, Proposition 2.2, and Young’s inequality, we estimate the standard terms as follows.
The inertial term can be estimated by
ε ∫ 0 t ∫ Ω ( ρ ∂ t m − 1 u t + ρ u ⋅ ∇ ∂ t m − 1 u ) ⋅ ∇ ∂ t m − 1 div u d x d τ \varepsilon\int_0^t \int_\Omega\left(\rho\partial_t^{m-1}u_t+\rho u\cdot\nabla\partial_t^{m-1}u\right)\cdot\nabla\partial_t^{m-1}\operatorname{div}u\,dx\,d\tau ε ∫ 0 t ∫ Ω ( ρ ∂ t m − 1 u t + ρ u ⋅ ∇ ∂ t m − 1 u ) ⋅ ∇ ∂ t m − 1 div u d x d τ = − ε ∫ 0 t ∫ Ω ρ ( ∂ t m − 1 div u t + u ⋅ ∇ ∂ t m − 1 div u ) ∂ t m − 1 div u d x d τ − ε ∫ 0 t ∫ Ω ( ∇ ρ ⋅ ∂ t m − 1 u t + ( ∇ ( ρ u ) ) T : ∇ ∂ t m − 1 u ) ∂ t m − 1 div u d x d τ − ε ∫ 0 t ∫ ∂ Ω ρ ( u ⋅ ∇ ) n ⋅ ∂ t m − 1 u ∂ t m − 1 div u d S d τ ≤ − ε 2 ∫ Ω ρ ∣ ∂ t m − 1 div u ∣ 2 ( t ) d x + ε 2 ∫ Ω ρ 0 ∣ ∂ t m − 1 div u 0 ∣ 2 d x + 2 μ + λ 16 ε 2 ∫ 0 t ∥ ∇ ∂ t m − 1 div u ∥ L 2 2 d τ + C m + 1 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (83) \begin{aligned}
={}&-\varepsilon\int_0^t\int_\Omega\rho\left(\partial_t^{m-1}\operatorname{div}u_t+u\cdot\nabla\partial_t^{m-1}\operatorname{div}u\right)\partial_t^{m-1}\operatorname{div}u\,dx\,d\tau\\
&-\varepsilon\int_0^t\int_\Omega\left(\nabla\rho\cdot\partial_t^{m-1}u_t+(\nabla(\rho u))^T:\nabla\partial_t^{m-1}u\right)\partial_t^{m-1}\operatorname{div}u\,dx\,d\tau\\
&-\varepsilon\int_0^t\int_{\partial\Omega}\rho(u\cdot\nabla)n\cdot\partial_t^{m-1}u\,\partial_t^{m-1}\operatorname{div}u\,dS\,d\tau\\
\le{}&-\frac{\varepsilon}{2}\int_\Omega\rho|\partial_t^{m-1}\operatorname{div}u|^2(t)\,dx+\frac{\varepsilon}{2}\int_\Omega\rho_0|\partial_t^{m-1}\operatorname{div}u_0|^2\,dx\\
&+\frac{2\mu+\lambda}{16}\varepsilon^2\int_0^t\|\nabla\partial_t^{m-1}\operatorname{div}u\|_{L^2}^2\,d\tau+C_{m+1}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(83)}
\end{aligned} = ≤ − ε ∫ 0 t ∫ Ω ρ ( ∂ t m − 1 div u t + u ⋅ ∇ ∂ t m − 1 div u ) ∂ t m − 1 div u d x d τ − ε ∫ 0 t ∫ Ω ( ∇ ρ ⋅ ∂ t m − 1 u t + ( ∇ ( ρ u ) ) T : ∇ ∂ t m − 1 u ) ∂ t m − 1 div u d x d τ − ε ∫ 0 t ∫ ∂ Ω ρ ( u ⋅ ∇ ) n ⋅ ∂ t m − 1 u ∂ t m − 1 div u d S d τ − 2 ε ∫ Ω ρ ∣ ∂ t m − 1 div u ∣ 2 ( t ) d x + 2 ε ∫ Ω ρ 0 ∣ ∂ t m − 1 div u 0 ∣ 2 d x + 16 2 μ + λ ε 2 ∫ 0 t ∥∇ ∂ t m − 1 div u ∥ L 2 2 d τ + C m + 1 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (83) The pressure term can be estimated by
ε ∫ 0 t ∫ Ω ∂ t m − 1 ∇ p ⋅ ∇ ∂ t m − 1 div u d x d τ \varepsilon\int_0^t\int_\Omega\partial_t^{m-1}\nabla p\cdot\nabla\partial_t^{m-1}\operatorname{div}u\,dx\,d\tau ε ∫ 0 t ∫ Ω ∂ t m − 1 ∇ p ⋅ ∇ ∂ t m − 1 div u d x d τ ≤ − ε 2 ∫ Ω 1 γ p ∣ ∂ t m − 1 ∇ p ∣ 2 ( t ) d x + ε 2 ∫ Ω 1 γ p 0 ∣ ∂ t m − 1 ∇ p 0 ∣ 2 d x + C m + 1 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (84) \begin{aligned}
\le{}&-\frac{\varepsilon}{2}\int_\Omega\frac{1}{\gamma p}|\partial_t^{m-1}\nabla p|^2(t)\,dx+\frac{\varepsilon}{2}\int_\Omega\frac{1}{\gamma p_0}|\partial_t^{m-1}\nabla p_0|^2\,dx\\
&+C_{m+1}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(84)}
\end{aligned} ≤ − 2 ε ∫ Ω γ p 1 ∣ ∂ t m − 1 ∇ p ∣ 2 ( t ) d x + 2 ε ∫ Ω γ p 0 1 ∣ ∂ t m − 1 ∇ p 0 ∣ 2 d x + C m + 1 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (84) For the viscous term, we have
− μ ε 2 ∫ 0 t ∫ Ω ∇ × ∂ t m − 1 ω ⋅ ∇ ∂ t m − 1 div u d x d τ ≤ 2 μ + λ 16 ε 2 ∫ 0 t ∥ ∇ ∂ t m − 1 div u ∥ L 2 2 d τ + C m + 1 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (85) \begin{aligned}
-\mu\varepsilon^2\int_0^t\int_\Omega\nabla\times\partial_t^{m-1}\omega\cdot\nabla\partial_t^{m-1}\operatorname{div}u\,dx\,d\tau\\
&\le\frac{2\mu+\lambda}{16}\varepsilon^2\int_0^t\|\nabla\partial_t^{m-1}\operatorname{div}u\|_{L^2}^2\,d\tau+C_{m+1}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(85)}
\end{aligned} − μ ε 2 ∫ 0 t ∫ Ω ∇ × ∂ t m − 1 ω ⋅ ∇ ∂ t m − 1 div u d x d τ ≤ 16 2 μ + λ ε 2 ∫ 0 t ∥∇ ∂ t m − 1 div u ∥ L 2 2 d τ + C m + 1 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (85) The commutator term is controlled by
∣ ε ∫ 0 t ∫ Ω C t m − 1 ⋅ ∇ ∂ t m − 1 div u d x d τ ∣ ≤ 2 μ + λ 16 ε 2 ∫ 0 t ∥ ∇ ∂ t m − 1 div u ∥ L 2 2 d τ + C m + 1 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (86) \begin{aligned}
\left|\varepsilon\int_0^t\int_\Omega C_t^{m-1}\cdot\nabla\partial_t^{m-1}\operatorname{div}u\,dx\,d\tau\right| \\
&\le\frac{2\mu+\lambda}{16}\varepsilon^2\int_0^t\|\nabla\partial_t^{m-1}\operatorname{div}u\|_{L^2}^2\,d\tau+C_{m+1}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(86)}
\end{aligned} ε ∫ 0 t ∫ Ω C t m − 1 ⋅ ∇ ∂ t m − 1 div u d x d τ ≤ 16 2 μ + λ ε 2 ∫ 0 t ∥∇ ∂ t m − 1 div u ∥ L 2 2 d τ + C m + 1 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (86) Substituting (3.59), (3.60), (3.61), (3.62), and (3.63) into (3.58), we obtain (3.57). This completes the proof.
The preceding estimates provide the conormal control of ( p , u ) (p,u) ( p , u ) ε \varepsilon ε div u \operatorname{div}u div u ∂ t m − 1 div u \partial_t^{m-1}\operatorname{div}u ∂ t m − 1 div u
Λ m 1 ( t ) : = ∥ ( p , u ) ( t ) ∥ H m 2 + ∑ ∣ β ∣ ≤ m − 2 ∥ Z β ∇ p ( t ) ∥ 1 2 + ∑ ∣ β ∣ ≤ m − 2 ∥ Z β ∇ u ( t ) ∥ 1 2 . (87) \Lambda_m^1(t):=\|(p,u)(t)\|_{H^m}^2+\sum_{|\beta|\le m-2}\|Z^\beta\nabla p(t)\|_1^2+\sum_{|\beta|\le m-2}\|Z^\beta\nabla u(t)\|_1^2. \tag*{(87)} Λ m 1 ( t ) := ∥ ( p , u ) ( t ) ∥ H m 2 + ∣ β ∣ ≤ m − 2 ∑ ∥ Z β ∇ p ( t ) ∥ 1 2 + ∣ β ∣ ≤ m − 2 ∑ ∥ Z β ∇ u ( t ) ∥ 1 2 . (87) This quantity does not contain the term ∥ ∂ t m − 1 div u ( t ) ∥ L 2 2 \|\partial_t^{m-1}\operatorname{div}u(t)\|_{L^2}^2 ∥ ∂ t m − 1 div u ( t ) ∥ L 2 2
Lemma 3.6. Let m ≥ 3 m\ge3 m ≥ 3
∥ ∂ t m − 1 div u ( t ) ∥ L 2 2 ≤ C 2 [ 1 + P ( Q ( t ) ) ] P ( Λ m 1 ( t ) ) . (88) \|\partial_t^{m-1}\operatorname{div}u(t)\|_{L^2}^2\le C_2[1+P(Q(t))]P\left(\Lambda_m^1(t)\right). \tag*{(88)} ∥ ∂ t m − 1 div u ( t ) ∥ L 2 2 ≤ C 2 [ 1 + P ( Q ( t ))] P ( Λ m 1 ( t ) ) . (88) Proof. By (42) ,
div u = − p t γ p − u ⋅ ∇ p γ p . \operatorname{div} u=-\frac{p_t}{\gamma p}-\frac{u\cdot\nabla p}{\gamma p}. div u = − γ p p t − γ p u ⋅ ∇ p . Applying ∂ t m − 1 \partial_t^{m-1} ∂ t m − 1
∂ t m − 1 div u = − 1 γ p ∂ t m − 1 p − u γ p ⋅ ∇ ∂ t m − 1 p − [ ∂ t m − 1 , 1 γ p ] p t − [ ∂ t m − 1 , u γ p ] ⋅ ∇ p . (89) \begin{aligned}
\partial_t^{m-1}\operatorname{div}u={}&-\frac{1}{\gamma p}\partial_t^{m-1}p-\frac{u}{\gamma p}\cdot\nabla\partial_t^{m-1}p\\
&-\left[\partial_t^{m-1},\frac{1}{\gamma p}\right]p_t-\left[\partial_t^{m-1},\frac{u}{\gamma p}\right]\cdot\nabla p. \tag*{(89)}
\end{aligned} ∂ t m − 1 div u = − γ p 1 ∂ t m − 1 p − γ p u ⋅ ∇ ∂ t m − 1 p − [ ∂ t m − 1 , γ p 1 ] p t − [ ∂ t m − 1 , γ p u ] ⋅ ∇ p . (89) The two leading terms are controlled by the definition of Λ m 1 ( t ) \Lambda_m^1(t) Λ m 1 ( t )
∥ 1 γ p ∂ t m − 1 p ∥ L 2 2 + ∥ u γ p ⋅ ∇ ∂ t m − 1 p ∥ L 2 2 ≤ C [ 1 + P ( Q ( t ) ) ] Λ m 1 ( t ) . (90) \left\|\frac{1}{\gamma p}\partial_t^{m-1}p\right\|_{L^2}^2+\left\|\frac{u}{\gamma p}\cdot\nabla\partial_t^{m-1}p\right\|_{L^2}^2\le C[1+P(Q(t))]\Lambda_m^1(t). \tag*{(90)} γ p 1 ∂ t m − 1 p L 2 2 + γ p u ⋅ ∇ ∂ t m − 1 p L 2 2 ≤ C [ 1 + P ( Q ( t ))] Λ m 1 ( t ) . (90) For the commutator terms, Proposition 2.2 yields
∥ [ ∂ t m − 1 , 1 γ p ] p t ∥ L 2 2 + ∥ [ ∂ t m − 1 , u γ p ] ⋅ ∇ p ∥ L 2 2 ≤ C 2 [ 1 + P ( Q ( t ) ) ] P ( Λ m 1 ( t ) ) . (91) \left\|\left[\partial_t^{m-1},\frac{1}{\gamma p}\right]p_t\right\|_{L^2}^2+\left\|\left[\partial_t^{m-1},\frac{u}{\gamma p}\right]\cdot\nabla p\right\|_{L^2}^2
\le C_2[1+P(Q(t))]P\left(\Lambda_m^1(t)\right). \tag*{(91)} [ ∂ t m − 1 , γ p 1 ] p t L 2 2 + [ ∂ t m − 1 , γ p u ] ⋅ ∇ p L 2 2 ≤ C 2 [ 1 + P ( Q ( t ))] P ( Λ m 1 ( t ) ) . (91) Combining (89) –(91) , we obtain (88) . This completes the proof.
Lemma 3.7. Let m ≥ 1 m\ge1 m ≥ 1
∫ 0 t ∥ ∇ ∂ t m − 1 p ( τ ) ∥ L 2 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ + C ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ . (92) \int_0^t\left\|\nabla\partial_t^{m-1}p(\tau)\right\|_{L^2}^2\,d\tau\le C_{m+2}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau+C\varepsilon^2\int_0^t\left\|\nabla^2u(\tau)\right\|_{H^{m-1}}^2\,d\tau. \tag*{(92)} ∫ 0 t ∇ ∂ t m − 1 p ( τ ) L 2 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ + C ε 2 ∫ 0 t ∇ 2 u ( τ ) H m − 1 2 d τ . (92) Proof. Applying ∂ t m − 1 \partial_t^{m-1} ∂ t m − 1 (24) and solving for ∇ ∂ t m − 1 p \nabla\partial_t^{m-1}p ∇ ∂ t m − 1 p
∇ ∂ t m − 1 p = − ∂ t m − 1 ( ρ u t ) − ∂ t m − 1 ( ρ u ⋅ ∇ u ) − μ ε ∂ t m − 1 ∇ × ω + ( 2 μ + λ ) ε ∂ t m − 1 ∇ div u + ∂ t m − 1 ( ρ ∇ ϕ ) . (93) \begin{aligned}
\nabla\partial_t^{m-1}p={}&-\partial_t^{m-1}(\rho u_t)-\partial_t^{m-1}(\rho u\cdot\nabla u)-\mu\varepsilon\partial_t^{m-1}\nabla\times\omega\\
&+(2\mu+\lambda)\varepsilon\partial_t^{m-1}\nabla\operatorname{div}u+\partial_t^{m-1}(\rho\nabla\phi). \tag*{(93)}
\end{aligned} ∇ ∂ t m − 1 p = − ∂ t m − 1 ( ρ u t ) − ∂ t m − 1 ( ρ u ⋅ ∇ u ) − μ ε ∂ t m − 1 ∇ × ω + ( 2 μ + λ ) ε ∂ t m − 1 ∇ div u + ∂ t m − 1 ( ρ ∇ ϕ ) . (93) We estimate each term on the right-hand side of (93) . First,
∂ t m − 1 ( ρ u t ) = ρ ∂ t m u + [ ∂ t m − 1 , ρ ] u t . \partial_t^{m-1}(\rho u_t)=\rho\partial_t^m u+[\partial_t^{m-1},\rho]u_t. ∂ t m − 1 ( ρ u t ) = ρ ∂ t m u + [ ∂ t m − 1 , ρ ] u t . Thus, by Proposition 2.2 ,
∫ 0 t ∥ ∂ t m − 1 ( ρ u t ) ( τ ) ∥ L 2 2 d τ ≤ C m + 1 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (94) \int_0^t\left\|\partial_t^{m-1}(\rho u_t)(\tau)\right\|_{L^2}^2\,d\tau\le C_{m+1}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(94)} ∫ 0 t ∂ t m − 1 ( ρ u t ) ( τ ) L 2 2 d τ ≤ C m + 1 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (94) Similarly,
∂ t m − 1 ( ρ u ⋅ ∇ u ) = ρ u ⋅ ∇ ∂ t m − 1 u + [ ∂ t m − 1 , ρ u ] ⋅ ∇ u , \partial_t^{m-1}(\rho u\cdot\nabla u)=\rho u\cdot\nabla\partial_t^{m-1}u+[\partial_t^{m-1},\rho u]\cdot\nabla u, ∂ t m − 1 ( ρ u ⋅ ∇ u ) = ρ u ⋅ ∇ ∂ t m − 1 u + [ ∂ t m − 1 , ρ u ] ⋅ ∇ u , and Proposition 2.2 gives
∫ 0 t ∥ ∂ t m − 1 ( ρ u ⋅ ∇ u ) ( τ ) ∥ L 2 2 d τ ≤ C m + 1 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (95) \int_0^t\left\|\partial_t^{m-1}(\rho u\cdot\nabla u)(\tau)\right\|_{L^2}^2\,d\tau\le C_{m+1}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(95)} ∫ 0 t ∂ t m − 1 ( ρ u ⋅ ∇ u ) ( τ ) L 2 2 d τ ≤ C m + 1 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (95) For the viscous terms, we directly have
ε 2 ∫ 0 t ∥ ∂ t m − 1 ( ∇ × ω , ∇ div u ) ( τ ) ∥ L 2 2 d τ ≤ C ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ . (96) \varepsilon^2\int_0^t\left\|\partial_t^{m-1}(\nabla\times\omega,\nabla\operatorname{div}u)(\tau)\right\|_{L^2}^2\,d\tau\le C\varepsilon^2\int_0^t\left\|\nabla^2u(\tau)\right\|_{H^{m-1}}^2\,d\tau. \tag*{(96)} ε 2 ∫ 0 t ∂ t m − 1 ( ∇ × ω , ∇ div u ) ( τ ) L 2 2 d τ ≤ C ε 2 ∫ 0 t ∇ 2 u ( τ ) H m − 1 2 d τ . (96) It remains to estimate the electric force term. Since
∂ t m − 1 ( ρ ∇ ϕ ) = ρ ∇ ∂ t m − 1 ϕ + [ ∂ t m − 1 , ρ ] ∇ ϕ , \partial_t^{m-1}(\rho\nabla\phi)=\rho\nabla\partial_t^{m-1}\phi+[\partial_t^{m-1},\rho]\nabla\phi, ∂ t m − 1 ( ρ ∇ ϕ ) = ρ ∇ ∂ t m − 1 ϕ + [ ∂ t m − 1 , ρ ] ∇ ϕ , [ ∂ t m − 1 , ρ ] ∇ ϕ = ∑ j = 1 m − 1 C m − 1 j ∂ t j ρ ∇ ∂ t m − 1 − j ϕ . [\partial_t^{m-1},\rho]\nabla\phi=\sum_{j=1}^{m-1}C_{m-1}^{j}\partial_t^{j}\rho\nabla\partial_t^{m-1-j}\phi. [ ∂ t m − 1 , ρ ] ∇ ϕ = j = 1 ∑ m − 1 C m − 1 j ∂ t j ρ ∇ ∂ t m − 1 − j ϕ . where C m − 1 j C_{m-1}^{j} C m − 1 j
∫ 0 t ∥ ∂ t m − 1 ( ρ ∇ ϕ ) ( τ ) ∥ L 2 2 d τ ≤ C m + 1 [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (97) \int_0^t\|\partial_t^{m-1}(\rho\nabla\phi)(\tau)\|_{L^2}^2\,\mathrm{d}\tau\leq C_{m+1}[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,\mathrm{d}\tau. \tag*{(97)} ∫ 0 t ∥ ∂ t m − 1 ( ρ ∇ ϕ ) ( τ ) ∥ L 2 2 d τ ≤ C m + 1 [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (97) Combining (93) –(97) , we obtain (92) . This completes the proof.
The estimates obtained above still contain the term ∥ ∇ Z m − 2 div u ∥ L 2 2 \|\nabla Z^{m-2}\operatorname{div}u\|_{L^2}^2 ∥∇ Z m − 2 div u ∥ L 2 2 div u \operatorname{div}u div u
Lemma 3.8. Let m ≥ 2 m\geq2 m ≥ 2
∫ 0 t ∥ ∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ ≤ C ∫ 0 t ∥ ∇ ∂ t m − 1 p ∥ L 2 2 d τ + C m [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (98) \int_0^t\|\nabla Z^{m-2}\operatorname{div}u(\tau)\|_{L^2}^2\,\mathrm{d}\tau\leq C\int_0^t\|\nabla\partial_t^{m-1}p\|_{L^2}^2\,\mathrm{d}\tau+C_m[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,\mathrm{d}\tau. \tag*{(98)} ∫ 0 t ∥∇ Z m − 2 div u ( τ ) ∥ L 2 2 d τ ≤ C ∫ 0 t ∥∇ ∂ t m − 1 p ∥ L 2 2 d τ + C m [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (98) Proof. By (42) , we may write
div u = − 1 γ ( ∂ t ln p + u ⋅ ∇ ln p ) . \operatorname{div}u=-\frac{1}{\gamma}(\partial_t\ln p+u\cdot\nabla\ln p). div u = − γ 1 ( ∂ t ln p + u ⋅ ∇ ln p ) . Let ∣ α ∣ ≤ m − 2 |\alpha|\leq m-2 ∣ α ∣ ≤ m − 2
u ⋅ ∇ ln p = u y ⋅ ∇ y ln p + ( u ⋅ N ) ∂ z ln p . u\cdot\nabla\ln p=u_y\cdot\nabla_y\ln p+(u\cdot N)\partial_z\ln p. u ⋅ ∇ ln p = u y ⋅ ∇ y ln p + ( u ⋅ N ) ∂ z ln p . Hence
∫ 0 t ∥ ∇ Z α div u ∥ L 2 2 d τ ≤ C ∫ 0 t ( ∥ ∇ Z α ∂ t ln p ∥ L 2 2 + ∥ ∇ Z α ( u y ⋅ ∇ y ln p ) ∥ L 2 2 + ∥ ∇ Z α ( ( u ⋅ N ) ∂ z ln p ) ∥ L 2 2 ) d τ . (99) \int_0^t\|\nabla Z^\alpha\operatorname{div}u\|_{L^2}^2\,\mathrm{d}\tau\leq C\int_0^t\left(\|\nabla Z^\alpha\partial_t\ln p\|_{L^2}^2+\|\nabla Z^\alpha(u_y\cdot\nabla_y\ln p)\|_{L^2}^2+\|\nabla Z^\alpha((u\cdot N)\partial_z\ln p)\|_{L^2}^2\right)\,\mathrm{d}\tau. \tag*{(99)} ∫ 0 t ∥∇ Z α div u ∥ L 2 2 d τ ≤ C ∫ 0 t ( ∥∇ Z α ∂ t ln p ∥ L 2 2 + ∥∇ Z α ( u y ⋅ ∇ y ln p ) ∥ L 2 2 + ∥∇ Z α (( u ⋅ N ) ∂ z ln p ) ∥ L 2 2 ) d τ . (99) The first two terms are estimated as in [34 Lemma 3.12, (3.98)–(3.100)] . The case Z α = ∂ t m − 2 Z^\alpha=\partial_t^{m-2} Z α = ∂ t m − 2 ∇ ∂ t m − 1 ln p \nabla\partial_t^{m-1}\ln p ∇ ∂ t m − 1 ln p ε \varepsilon ε E m \mathcal{E}_m E m
∫ 0 t ∥ ∇ Z α ∂ t ln p ∥ L 2 2 d τ + ∫ 0 t ∥ ∇ Z α ( u y ⋅ ∇ y ln p ) ∥ L 2 2 d τ ≤ C ∫ 0 t ∥ ∇ ∂ t m − 1 p ∥ L 2 2 d τ + C m [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (100) \int_0^t\|\nabla Z^\alpha\partial_t\ln p\|_{L^2}^2\,\mathrm{d}\tau+\int_0^t\|\nabla Z^\alpha(u_y\cdot\nabla_y\ln p)\|_{L^2}^2\,\mathrm{d}\tau
\leq C\int_0^t\|\nabla\partial_t^{m-1}p\|_{L^2}^2\,\mathrm{d}\tau+C_m[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,\mathrm{d}\tau. \tag*{(100)} ∫ 0 t ∥∇ Z α ∂ t ln p ∥ L 2 2 d τ + ∫ 0 t ∥∇ Z α ( u y ⋅ ∇ y ln p ) ∥ L 2 2 d τ ≤ C ∫ 0 t ∥∇ ∂ t m − 1 p ∥ L 2 2 d τ + C m [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (100) It remains to estimate the normal part. Since u ⋅ n = 0 u\cdot n=0 u ⋅ n = 0 ∂ Ω \partial\Omega ∂ Ω u ⋅ N u\cdot N u ⋅ N Z 3 = φ ( z ) ∂ z Z_3=\varphi(z)\partial_z Z 3 = φ ( z ) ∂ z φ ( z ) ∼ z \varphi(z)\sim z φ ( z ) ∼ z ∂ Ω \partial\Omega ∂ Ω
∫ 0 t ∥ ∇ Z α ( ( u ⋅ N ) ∂ z ln p ) ( τ ) ∥ L 2 2 d τ ≤ ∫ 0 t ∥ ∇ ( ( u ⋅ N ) ∂ z ln p ) ( τ ) ∥ H m − 2 2 d τ ≤ C ∫ 0 t ∥ ∇ ( u ⋅ N ) ∂ z ln p ∥ H m − 2 2 d τ + C ∫ 0 t ∥ u ⋅ N φ ( z ) Z 3 ∇ ln p ∥ H m − 2 2 d τ ≤ C m [ 1 + P ( Q ( t ) ) ] ∫ 0 t ( E m ( τ ) + ∑ ∣ β ∣ ≤ 1 ∥ Z β ( u ⋅ N φ ( z ) ) ( τ ) ∥ H m − 2 2 ) d τ ≤ C m [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (101) \begin{aligned}
\int_0^t \left\|\nabla Z^\alpha\bigl((u\cdot N)\partial_z\ln p\bigr)(\tau)\right\|_{L^2}^2\,d\tau
&\le\int_0^t \left\|\nabla\bigl((u\cdot N)\partial_z\ln p\bigr)(\tau)\right\|_{H^{m-2}}^2\,d\tau\\
&\le C\int_0^t \left\|\nabla(u\cdot N)\,\partial_z\ln p\right\|_{H^{m-2}}^2\,d\tau
+C\int_0^t \left\|\frac{u\cdot N}{\varphi(z)}Z_3\nabla\ln p\right\|_{H^{m-2}}^2\,d\tau\\
&\le C_m[1+P(Q(t))]\int_0^t\left(\mathcal{E}_m(\tau)+\sum_{|\beta|\le1}\left\|Z^\beta\left(\frac{u\cdot N}{\varphi(z)}\right)(\tau)\right\|_{H^{m-2}}^2\right)\,d\tau\\
&\le C_m[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(101)}
\end{aligned} ∫ 0 t ∇ Z α ( ( u ⋅ N ) ∂ z ln p ) ( τ ) L 2 2 d τ ≤ ∫ 0 t ∇ ( ( u ⋅ N ) ∂ z ln p ) ( τ ) H m − 2 2 d τ ≤ C ∫ 0 t ∥ ∇ ( u ⋅ N ) ∂ z ln p ∥ H m − 2 2 d τ + C ∫ 0 t φ ( z ) u ⋅ N Z 3 ∇ ln p H m − 2 2 d τ ≤ C m [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) + ∣ β ∣ ≤ 1 ∑ Z β ( φ ( z ) u ⋅ N ) ( τ ) H m − 2 2 d τ ≤ C m [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (101) Here N N N z z z φ ( z ) ∼ z \varphi(z)\sim z φ ( z ) ∼ z Z 3 = φ ( z ) ∂ z Z_3=\varphi(z)\partial_z Z 3 = φ ( z ) ∂ z
∑ ∣ β ∣ ≤ 1 ∥ Z β ( u ⋅ N φ ( z ) ) ∥ H m − 2 ≤ C m ( ∥ ∇ u ∥ H m − 1 + ∥ u ∥ H m − 1 ) , \sum_{|\beta|\le1}\left\|Z^\beta\left(\frac{u\cdot N}{\varphi(z)}\right)\right\|_{H^{m-2}}
\le C_m\left(\|\nabla u\|_{H^{m-1}}+\|u\|_{H^{m-1}}\right), ∣ β ∣ ≤ 1 ∑ Z β ( φ ( z ) u ⋅ N ) H m − 2 ≤ C m ( ∥∇ u ∥ H m − 1 + ∥ u ∥ H m − 1 ) , which follows from u ⋅ n = 0 u\cdot n=0 u ⋅ n = 0 ∂ Ω \partial\Omega ∂ Ω
Combining (99) –(101) , summing over ∣ α ∣ ≤ m − 2 |\alpha|\le m-2 ∣ α ∣ ≤ m − 2 (98) . This completes the proof.
Combining (98) with (92) , and using ε ∥ ∇ ∂ t m − 1 p ∥ L 2 2 ≤ E m \varepsilon\|\nabla\partial_t^{m-1}p\|_{L^2}^2\le\mathcal{E}_m ε ∥∇ ∂ t m − 1 p ∥ L 2 2 ≤ E m 0 < δ ≤ 1 0<\delta\le1 0 < δ ≤ 1
( δ + ε ) ∫ 0 t ∥ ∇ Z m − 2 div u ∥ L 2 2 d τ + δ ∫ 0 t ∥ ∇ ∂ t m − 1 p ∥ L 2 2 d τ ≤ C δ ε 2 ∫ 0 t ∥ ∇ 2 u ∥ H m − 1 2 d τ + C m [ 1 + P ( Q ( t ) ) ] ∫ 0 t E m ( τ ) d τ . (102) (\delta+\varepsilon)\int_0^t\|\nabla Z^{m-2}\operatorname{div}u\|_{L^2}^2\,d\tau+\delta\int_0^t\|\nabla\partial_t^{m-1}p\|_{L^2}^2\,d\tau
\le C\delta\varepsilon^2\int_0^t\|\nabla^2u\|_{H^{m-1}}^2\,d\tau+C_m[1+P(Q(t))]\int_0^t\mathcal{E}_m(\tau)\,d\tau. \tag*{(102)} ( δ + ε ) ∫ 0 t ∥∇ Z m − 2 div u ∥ L 2 2 d τ + δ ∫ 0 t ∥∇ ∂ t m − 1 p ∥ L 2 2 d τ ≤ C δ ε 2 ∫ 0 t ∥ ∇ 2 u ∥ H m − 1 2 d τ + C m [ 1 + P ( Q ( t ))] ∫ 0 t E m ( τ ) d τ . (102) Here C C C δ \delta δ ε \varepsilon ε
Normal derivative estimates
In order to estimate ∥ ∇ u ∥ H m − 1 \|\nabla u\|_{H^{m-1}} ∥∇ u ∥ H m − 1 χ \chi χ
∥ χ ∂ y i u ∥ H m − 1 ≤ C ∥ u ∥ H m , i = 1 , 2. \|\chi\partial_{y_i}u\|_{H^{m-1}}\le C\|u\|_{H^m},\qquad i=1,2. ∥ χ ∂ y i u ∥ H m − 1 ≤ C ∥ u ∥ H m , i = 1 , 2. Thus it suffices to estimate ∥ χ ∂ n u ∥ H m − 1 \|\chi\partial_nu\|_{H^{m-1}} ∥ χ ∂ n u ∥ H m − 1
In the local coordinates, one has
div u = ∂ n u ⋅ n + ( Π ∂ y 1 u ) 1 + ( Π ∂ y 2 u ) 2 , (103) \operatorname{div}u=\partial_nu\cdot n+(\Pi\partial_{y_1}u)_1+(\Pi\partial_{y_2}u)_2, \tag*{(103)} div u = ∂ n u ⋅ n + ( Π ∂ y 1 u ) 1 + ( Π ∂ y 2 u ) 2 , (103) and
∂ n u = ( ∂ n u ⋅ n ) n + Π ( ∂ n u ) . (104) \partial_nu=(\partial_nu\cdot n)n+\Pi(\partial_nu). \tag*{(104)} ∂ n u = ( ∂ n u ⋅ n ) n + Π ( ∂ n u ) . (104) It follows from (103) and (104) that
∥ χ ∂ n u ∥ H m − 1 ≤ ∥ χ ∂ n u ⋅ n ∥ H m − 1 + ∥ χ Π ( ∂ n u ) ∥ H m − 1 ≤ C m ( ∥ χ div u ∥ H m − 1 + ∥ χ Π ( ∂ n u ) ∥ H m − 1 + ∥ u ∥ H m ) . (105) \begin{aligned}
\|\chi\partial_nu\|_{H^{m-1}}
&\le\|\chi\partial_nu\cdot n\|_{H^{m-1}}+\|\chi\Pi(\partial_nu)\|_{H^{m-1}} \\
&\le C_m\left(\|\chi\operatorname{div}u\|_{H^{m-1}}+\|\chi\Pi(\partial_nu)\|_{H^{m-1}}+\|u\|_{H^m}\right). \tag*{(105)}
\end{aligned} ∥ χ ∂ n u ∥ H m − 1 ≤ ∥ χ ∂ n u ⋅ n ∥ H m − 1 + ∥ χ Π ( ∂ n u ) ∥ H m − 1 ≤ C m ( ∥ χ div u ∥ H m − 1 + ∥ χ Π ( ∂ n u ) ∥ H m − 1 + ∥ u ∥ H m ) . (105) The terms ∥ u ∥ H m \|u\|_{H^m} ∥ u ∥ H m ∥ χ div u ∥ H m − 1 \|\chi\operatorname{div}u\|_{H^{m-1}} ∥ χ div u ∥ H m − 1 ∥ χ Π ( ∂ n u ) ∥ H m − 1 \|\chi\Pi(\partial_nu)\|_{H^{m-1}} ∥ χ Π ( ∂ n u ) ∥ H m − 1
We extend the boundary matrix B B B B ( y , z ) = B ( y ) B(y,z)=B(y) B ( y , z ) = B ( y )
η : = χ ( ω × n + Π ( B u ) ) = χ ( Π ( ω × n ) + Π ( B u ) ) . (106) \eta:=\chi(\omega\times n+\Pi(Bu))=\chi\bigl(\Pi(\omega\times n)+\Pi(Bu)\bigr). \tag*{(106)} η := χ ( ω × n + Π ( B u )) = χ ( Π ( ω × n ) + Π ( B u ) ) . (106) By the Navier–slip boundary condition,
n × ω = [ B u ] τ on ∂ Ω , n\times\omega=[Bu]_\tau\quad\text{on }\partial\Omega, n × ω = [ B u ] τ on ∂ Ω , we have
η ∣ ∂ Ω = 0. (107) \eta|_{\partial\Omega}=0. \tag*{(107)} η ∣ ∂ Ω = 0. (107) Moreover, since
ω × n = ( ∇ u − ( ∇ u ) T ) n , \omega\times n=(\nabla u-(\nabla u)^{T})n, ω × n = ( ∇ u − ( ∇ u ) T ) n , we can rewrite η \eta η
η = χ [ Π ( ∂ n u ) − Π ( ∇ ( u ⋅ n ) ) + Π ( ( ∇ n ) T u ) + Π ( B u ) ] . (108) \eta=\chi\bigl[\Pi(\partial_nu)-\Pi(\nabla(u\cdot n))+\Pi((\nabla n)^{T}u)+\Pi(Bu)\bigr]. \tag*{(108)} η = χ [ Π ( ∂ n u ) − Π ( ∇ ( u ⋅ n )) + Π (( ∇ n ) T u ) + Π ( B u ) ] . (108) Therefore,
∥ χ Π ( ∂ n u ) ∥ H m − 1 ≤ C m + 1 ( ∥ η ∥ H m − 1 + ∥ u ∥ H m ) . (109) \|\chi\Pi(\partial_nu)\|_{H^{m-1}}\leq C_{m+1}\bigl(\|\eta\|_{H^{m-1}}+\|u\|_{H^m}\bigr). \tag*{(109)} ∥ χ Π ( ∂ n u ) ∥ H m − 1 ≤ C m + 1 ( ∥ η ∥ H m − 1 + ∥ u ∥ H m ) . (109) It remains to estimate η \eta η
Lemma 3.9. Let m ≥ 1 m\geq1 m ≥ 1
sup 0 ≤ τ ≤ t ∥ η ( τ ) ∥ H m − 1 2 + ε ∫ 0 t ∥ ∇ η ( τ ) ∥ H m − 1 2 d τ \sup_{0\leq\tau\leq t}\|\eta(\tau)\|_{H^{m-1}}^2+\varepsilon\int_0^t\|\nabla\eta(\tau)\|_{H^{m-1}}^2\,d\tau 0 ≤ τ ≤ t sup ∥ η ( τ ) ∥ H m − 1 2 + ε ∫ 0 t ∥∇ η ( τ ) ∥ H m − 1 2 d τ ≤ C m + 2 { ∥ η ( 0 ) ∥ H m − 1 2 + δ ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + C δ [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( E m ( τ ) ) d τ } . (110) \leq C_{m+2}\left\{\|\eta(0)\|_{H^{m-1}}^2+\delta\varepsilon^2\int_0^t\|\nabla^2u(\tau)\|_{H^{m-1}}^2\,d\tau+C_\delta[1+P(Q(t))]\int_0^tP(\mathcal E_m(\tau))\,d\tau\right\}. \tag*{(110)} ≤ C m + 2 { ∥ η ( 0 ) ∥ H m − 1 2 + δ ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + C δ [ 1 + P ( Q ( t ))] ∫ 0 t P ( E m ( τ )) d τ } . (110) Proof. Set
T f : = χ Π ( f × n ) , B f : = χ Π ( B f ) , L : = ρ ∂ t + ρ u ⋅ ∇ − μ ε Δ . Tf:=\chi\Pi(f\times n),\qquad\mathcal{B}f:=\chi\Pi(Bf),\qquad\mathcal{L}:=\rho\partial_t+\rho u\cdot\nabla-\mu\varepsilon\Delta. T f := χ Π ( f × n ) , B f := χ Π ( B f ) , L := ρ ∂ t + ρ u ⋅ ∇ − μ ε Δ. Then
η = T ω + B u . \eta=T\omega+\mathcal{B}u. η = T ω + B u . Taking curl to (24) , one obtains
ρ ω t + ρ u ⋅ ∇ ω − μ ε Δ ω = F ω 0 + ∇ ρ × ∇ ϕ , (111) \rho\omega_t+\rho u\cdot\nabla\omega-\mu\varepsilon\Delta\omega=F_\omega^0+\nabla\rho\times\nabla\phi, \tag*{(111)} ρ ω t + ρ u ⋅ ∇ ω − μ ε Δ ω = F ω 0 + ∇ ρ × ∇ ϕ , (111) where
F ω 0 : = − ∇ ρ × u t − ∇ ρ × ( u ⋅ ∇ u ) + ρ ( ω ⋅ ∇ ) u − ρ ( div u ) ω . F_\omega^0:=-\nabla\rho\times u_t-\nabla\rho\times(u\cdot\nabla u)+\rho(\omega\cdot\nabla)u-\rho(\operatorname{div}u)\omega. F ω 0 := − ∇ ρ × u t − ∇ ρ × ( u ⋅ ∇ u ) + ρ ( ω ⋅ ∇ ) u − ρ ( div u ) ω . Using (111) and (24) , we get
ρ η t + ρ u ⋅ ∇ η − μ ε Δ η = F η 0 + F η E , η ∣ ∂ Ω = 0 , (112) \rho\eta_t+\rho u\cdot\nabla\eta-\mu\varepsilon\Delta\eta=F_\eta^0+F_\eta^E,\qquad\eta|_{\partial\Omega}=0, \tag*{(112)} ρ η t + ρ u ⋅ ∇ η − μ ε Δ η = F η 0 + F η E , η ∣ ∂ Ω = 0 , (112) where
F η 0 : = T F ω 0 + B ( − ∇ p + ( μ + λ ) ε ∇ div u ) + R η , F_\eta^0:=TF_\omega^0+\mathcal{B}\bigl(-\nabla p+(\mu+\lambda)\varepsilon\nabla\operatorname{div}u\bigr)+R_\eta, F η 0 := T F ω 0 + B ( − ∇ p + ( μ + λ ) ε ∇ div u ) + R η , F η E : = T ( ∇ ρ × ∇ ϕ ) + B ( ρ ∇ ϕ ) . (113) F_\eta^E:=T(\nabla\rho\times\nabla\phi)+\mathcal{B}(\rho\nabla\phi). \tag*{(113)} F η E := T ( ∇ ρ × ∇ ϕ ) + B ( ρ ∇ ϕ ) . (113) and
R η : = [ L , T ] ω + [ L , B ] u . \mathcal{R}_{\eta}:=[\mathcal{L},\mathcal{T}]_{\omega}+[\mathcal{L},\mathcal{B}]_{u}. R η := [ L , T ] ω + [ L , B ] u . Here
[ L , T ] ω = L ( T ω ) − T ( L ω ) , [ L , B ] u = L ( B u ) − B ( L u ) . [\mathcal{L},\mathcal{T}]_{\omega}=\mathcal{L}(\mathcal{T}\omega)-\mathcal{T}(\mathcal{L}\omega),\qquad[\mathcal{L},\mathcal{B}]_{u}=\mathcal{L}(\mathcal{B}u)-\mathcal{B}(\mathcal{L}u). [ L , T ] ω = L ( T ω ) − T ( L ω ) , [ L , B ] u = L ( B u ) − B ( L u ) . Applying Z α Z^\alpha Z α ∣ α ∣ ≤ m − 1 |\alpha|\le m-1 ∣ α ∣ ≤ m − 1 (112) , we obtain
ρ ∂ t Z α η + ρ u ⋅ ∇ Z α η − μ ε Δ Z α η = Z α F η 0 + Z α F η E + C η α , (114) \rho\partial_t Z^\alpha\eta+\rho u\cdot\nabla Z^\alpha\eta-\mu\varepsilon\Delta Z^\alpha\eta=Z^\alpha F_\eta^0+Z^\alpha F_\eta^E+\mathcal{C}_\eta^\alpha, \tag*{(114)} ρ ∂ t Z α η + ρ u ⋅ ∇ Z α η − μ ε Δ Z α η = Z α F η 0 + Z α F η E + C η α , (114) where
C η α : = − [ Z α , ρ ] η t − [ Z α , ρ u ⋅ ∇ ] η + μ ε [ Z α , Δ ] η . (115) \mathcal{C}_\eta^\alpha:=-[Z^\alpha,\rho]\eta_t-[Z^\alpha,\rho u\cdot\nabla]\eta+\mu\varepsilon[Z^\alpha,\Delta]\eta. \tag*{(115)} C η α := − [ Z α , ρ ] η t − [ Z α , ρ u ⋅ ∇ ] η + μ ε [ Z α , Δ ] η . (115) Multiplying (114) by Z α η Z^\alpha\eta Z α η Ω \Omega Ω η ∣ ∂ Ω = 0 \eta|_{\partial\Omega}=0 η ∣ ∂ Ω = 0
1 2 d d t ∫ Ω ρ ∣ Z α η ∣ 2 d x + μ ε ∥ ∇ Z α η ∥ L 2 2 = ∫ Ω Z α F η 0 ⋅ Z α η d x + ∫ Ω Z α F η E ⋅ Z α η d x + ∫ Ω C η α ⋅ Z α η d x . (116) \frac{1}{2}\frac{d}{dt}\int_\Omega\rho|Z^\alpha\eta|^2\,dx+\mu\varepsilon\|\nabla Z^\alpha\eta\|_{L^2}^2
=
\int_\Omega Z^\alpha F_\eta^0\cdot Z^\alpha\eta\,dx+\int_\Omega Z^\alpha F_\eta^E\cdot Z^\alpha\eta\,dx+\int_\Omega\mathcal{C}_\eta^\alpha\cdot Z^\alpha\eta\,dx. \tag*{(116)} 2 1 d t d ∫ Ω ρ ∣ Z α η ∣ 2 d x + μ ε ∥∇ Z α η ∥ L 2 2 = ∫ Ω Z α F η 0 ⋅ Z α η d x + ∫ Ω Z α F η E ⋅ Z α η d x + ∫ Ω C η α ⋅ Z α η d x . (116) By Proposition 2.2, Proposition 2.3, and Young’s inequality, together with the estimates obtained above,
∑ ∣ α ∣ ≤ m − 1 ∫ 0 t ∣ ∫ Ω ( Z α F η 0 + C η α ) ⋅ Z α η d x ∣ d τ ≤ δ ε ∫ 0 t ∥ ∇ η ( τ ) ∥ H m − 1 2 d τ + δ ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( E m ( τ ) ) d τ . (117) \begin{aligned}
\sum_{|\alpha|\le m-1}\int_0^t\left|\int_\Omega\left(Z^\alpha F_\eta^0+\mathcal{C}_\eta^\alpha\right)\cdot Z^\alpha\eta\,dx\right|\,d\tau
&\le\delta\varepsilon\int_0^t\|\nabla\eta(\tau)\|_{\mathcal{H}^{m-1}}^2\,d\tau+\delta\varepsilon^2\int_0^t\|\nabla^2u(\tau)\|_{\mathcal{H}^{m-1}}^2\,d\tau\\
&\quad+C_\delta C_{m+2}[1+P(Q(t))]\int_0^t P(\mathcal{E}_m(\tau))\,d\tau. \tag*{(117)}
\end{aligned} ∣ α ∣ ≤ m − 1 ∑ ∫ 0 t ∫ Ω ( Z α F η 0 + C η α ) ⋅ Z α η d x d τ ≤ δ ε ∫ 0 t ∥∇ η ( τ ) ∥ H m − 1 2 d τ + δ ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t P ( E m ( τ )) d τ . (117) It remains to estimate the electric force term. The highest pure time derivative of ∇ p \nabla p ∇ p Lemma 3.8 . The time-integrated product estimate and the Poisson estimate give
∫ 0 t ∥ F η E ∥ H m − 1 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t ( ∥ ∇ ∂ t m − 1 p ∥ L 2 2 + P ( E m ( τ ) ) ) d τ . (118) \int_0^t\|F_\eta^E\|_{\mathcal{H}^{m-1}}^2\,d\tau\le C_{m+2}[1+P(Q(t))]\int_0^t\left(\|\nabla\partial_t^{m-1}p\|_{L^2}^2+P(\mathcal{E}_m(\tau))\right)\,d\tau. \tag*{(118)} ∫ 0 t ∥ F η E ∥ H m − 1 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t ( ∥∇ ∂ t m − 1 p ∥ L 2 2 + P ( E m ( τ )) ) d τ . (118) More precisely, the only derivatives not controlled by E m \mathcal{E}_m E m T ( ∇ ∂ t m − 1 ρ × ∇ ϕ ) \mathcal{T}(\nabla\partial_t^{m-1}\rho\times\nabla\phi) T ( ∇ ∂ t m − 1 ρ × ∇ ϕ ) p = ρ γ p=\rho^\gamma p = ρ γ ∇ ∂ t m − 1 p \nabla\partial_t^{m-1}p ∇ ∂ t m − 1 p ∥ η ∥ H m − 1 2 ≤ C E m \|\eta\|_{\mathcal{H}^{m-1}}^2\le C\mathcal{E}_m ∥ η ∥ H m − 1 2 ≤ C E m (92) , we obtain
∑ ∣ α ∣ ≤ m − 1 ∫ 0 t ∣ ∫ Ω Z α F η E ⋅ Z α η d x ∣ d τ ≤ δ ∫ 0 t ∥ ∇ ∂ t m − 1 p ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( E m ( τ ) ) d τ ≤ C δ ε 2 ∫ 0 t ∥ ∇ 2 u ∥ H m − 1 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( E m ( τ ) ) d τ . (119) \begin{aligned}
\sum_{|\alpha|\le m-1}\int_0^t\left|\int_\Omega Z^\alpha F_\eta^E\cdot Z^\alpha\eta\,dx\right|\,d\tau
&\le\delta\int_0^t\|\nabla\partial_t^{m-1}p\|_{L^2}^2\,d\tau+C_\delta C_{m+2}[1+P(Q(t))]\int_0^tP(\mathcal{E}_m(\tau))\,d\tau\\
&\le C\delta\varepsilon^2\int_0^t\|\nabla^2u\|_{\mathcal{H}^{m-1}}^2\,d\tau+C_\delta C_{m+2}[1+P(Q(t))]\int_0^tP(\mathcal{E}_m(\tau))\,d\tau. \tag*{(119)}
\end{aligned} ∣ α ∣ ≤ m − 1 ∑ ∫ 0 t ∫ Ω Z α F η E ⋅ Z α η d x d τ ≤ δ ∫ 0 t ∥∇ ∂ t m − 1 p ∥ L 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t P ( E m ( τ )) d τ ≤ C δ ε 2 ∫ 0 t ∥ ∇ 2 u ∥ H m − 1 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t P ( E m ( τ )) d τ . (119) Summing (116) over ∣ α ∣ ≤ m − 1 |\alpha|\le m-1 ∣ α ∣ ≤ m − 1 [ 0 , t ] [0,t] [ 0 , t ] (117) and (119) , we obtain
∥ η ( t ) ∥ H m − 1 2 + ε ∫ 0 t ∥ ∇ η ( τ ) ∥ H m − 1 2 d τ ≤ C m + 2 ( ∥ η ( 0 ) ∥ H m − 1 2 ) + δ ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( E m ( τ ) ) d τ + C ∫ 0 t ∥ η ( τ ) ∥ H m − 1 2 d τ . \begin{aligned}
\|\eta(t)\|_{\mathcal{H}^{m-1}}^2+\varepsilon\int_0^t\|\nabla\eta(\tau)\|_{\mathcal{H}^{m-1}}^2\,d\tau
&\le C_{m+2}\left(\|\eta(0)\|_{\mathcal{H}^{m-1}}^2\right)+\delta\varepsilon^2\int_0^t\|\nabla^2u(\tau)\|_{\mathcal{H}^{m-1}}^2\,d\tau\\
&\quad+C_\delta C_{m+2}[1+P(Q(t))]\int_0^tP(\mathcal{E}_m(\tau))\,d\tau+C\int_0^t\|\eta(\tau)\|_{\mathcal{H}^{m-1}}^2\,d\tau.
\end{aligned} ∥ η ( t ) ∥ H m − 1 2 + ε ∫ 0 t ∥∇ η ( τ ) ∥ H m − 1 2 d τ ≤ C m + 2 ( ∥ η ( 0 ) ∥ H m − 1 2 ) + δ ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t P ( E m ( τ )) d τ + C ∫ 0 t ∥ η ( τ ) ∥ H m − 1 2 d τ . Taking δ > 0 \delta>0 δ > 0 (110) . This completes the proof.
It follows from (103) –(105) that
∑ ∣ β ∣ ≤ m − 2 ∥ Z β ∇ u ∥ 1 2 ≤ C m + 1 ( ∥ u ∥ H m 2 + ∥ η ∥ H m − 1 2 + ∑ k = 0 m − 2 ∥ ∂ t k div u ∥ m − 1 − k 2 ) . (120) \sum_{|\beta|\le m-2}\left\|Z^\beta\nabla u\right\|_1^2 \le C_{m+1}\left(\|u\|_{\mathcal{H}^m}^2+\|\eta\|_{\mathcal{H}^{m-1}}^2+\sum_{k=0}^{m-2}\|\partial_t^k\operatorname{div}u\|_{m-1-k}^2\right). \tag*{(120)} ∣ β ∣ ≤ m − 2 ∑ Z β ∇ u 1 2 ≤ C m + 1 ( ∥ u ∥ H m 2 + ∥ η ∥ H m − 1 2 + k = 0 ∑ m − 2 ∥ ∂ t k div u ∥ m − 1 − k 2 ) . (120) Moreover,
∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ ≤ C m + 2 ∫ 0 t ( ∥ ∇ u ( τ ) ∥ H m 2 + ∥ ∇ η ( τ ) ∥ H m − 1 2 + ∥ ∇ div u ( τ ) ∥ H m − 1 2 + E m ( τ ) ) d τ . (121) \int_0^t\|\nabla^2u(\tau)\|_{\mathcal{H}^{m-1}}^2\,\mathrm{d}\tau\le C_{m+2}\int_0^t\left(\|\nabla u(\tau)\|_{\mathcal{H}^m}^2+\|\nabla\eta(\tau)\|_{\mathcal{H}^{m-1}}^2+\|\nabla\operatorname{div}u(\tau)\|_{\mathcal{H}^{m-1}}^2+\mathcal{E}_m(\tau)\right)\,\mathrm{d}\tau. \tag*{(121)} ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ ≤ C m + 2 ∫ 0 t ( ∥∇ u ( τ ) ∥ H m 2 + ∥∇ η ( τ ) ∥ H m − 1 2 + ∥∇ div u ( τ ) ∥ H m − 1 2 + E m ( τ ) ) d τ . (121) Similarly,
∑ k = 0 m − 2 ∫ 0 t ∥ ∇ 2 ∂ t k u ( τ ) ∥ m − 1 − k 2 d τ ≤ C m + 2 { ∫ 0 t ( ∥ ∇ u ( τ ) ∥ H m 2 + ∥ ∇ η ( τ ) ∥ H m − 1 2 ) d τ + ∑ k = 0 m − 2 ∫ 0 t ∥ ∂ t k ∇ div u ( τ ) ∥ m − 1 − k 2 d τ + ∫ 0 t E m ( τ ) d τ } . (122) \sum_{k=0}^{m-2}\int_0^t\|\nabla^2\partial_t^k u(\tau)\|_{m-1-k}^2\,\mathrm{d}\tau
\le C_{m+2}\left\{\int_0^t\left(\|\nabla u(\tau)\|_{\mathcal{H}^m}^2+\|\nabla\eta(\tau)\|_{\mathcal{H}^{m-1}}^2\right)\,\mathrm{d}\tau+\sum_{k=0}^{m-2}\int_0^t\|\partial_t^k\nabla\operatorname{div}u(\tau)\|_{m-1-k}^2\,\mathrm{d}\tau+\int_0^t\mathcal{E}_m(\tau)\,\mathrm{d}\tau\right\}. \tag*{(122)} k = 0 ∑ m − 2 ∫ 0 t ∥ ∇ 2 ∂ t k u ( τ ) ∥ m − 1 − k 2 d τ ≤ C m + 2 { ∫ 0 t ( ∥∇ u ( τ ) ∥ H m 2 + ∥∇ η ( τ ) ∥ H m − 1 2 ) d τ + k = 0 ∑ m − 2 ∫ 0 t ∥ ∂ t k ∇ div u ( τ ) ∥ m − 1 − k 2 d τ + ∫ 0 t E m ( τ ) d τ } . (122) In addition,
ε ∫ 0 t ∥ ∇ 2 Z m − 2 u ( τ ) ∥ L 2 2 d τ ≤ C m + 1 ε ∫ 0 t ∥ ∇ η ( τ ) ∥ H m − 2 2 d τ + C m + 1 ∫ 0 t E m ( τ ) d τ . (123) \varepsilon\int_0^t\|\nabla^2 Z^{m-2}u(\tau)\|_{L^2}^2\,\mathrm{d}\tau\le C_{m+1}\varepsilon\int_0^t\|\nabla\eta(\tau)\|_{\mathcal{H}^{m-2}}^2\,\mathrm{d}\tau+C_{m+1}\int_0^t\mathcal{E}_m(\tau)\,\mathrm{d}\tau. \tag*{(123)} ε ∫ 0 t ∥ ∇ 2 Z m − 2 u ( τ ) ∥ L 2 2 d τ ≤ C m + 1 ε ∫ 0 t ∥∇ η ( τ ) ∥ H m − 2 2 d τ + C m + 1 ∫ 0 t E m ( τ ) d τ . (123) Taking δ > 0 \delta>0 δ > 0 (45) , (66) , (80) , (88) , (92) , (102) , (110) , and (120) –(123) , we obtain
sup 0 ≤ τ ≤ t { E m ( τ ) + ∥ η ( τ ) ∥ H m − 1 2 + ε ∥ ( ∂ t m − 1 div u , ∂ t m − 1 ∇ p ) ( τ ) ∥ L 2 2 } + ε ∫ 0 t ∥ ∇ η ( τ ) ∥ H m − 1 2 d τ + ε ∫ 0 t ∥ ∇ u ( τ ) ∥ H m 2 d τ + ε ∑ k = 0 m − 2 ∫ 0 t ∥ ∂ t k ∇ div u ( τ ) ∥ m − 1 − k 2 d τ + ε 2 ∫ 0 t ∥ ∂ t m − 1 ∇ div u ( τ ) ∥ L 2 2 d τ + ε ∑ k = 0 m − 2 ∫ 0 t ∥ ∇ 2 ∂ t k u ( τ ) ∥ m − 1 − k 2 d τ + ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + ∫ 0 t ∥ ∇ ∂ t m − 1 p ( τ ) ∥ L 2 2 d τ ≤ C m + 2 { E m ( 0 ) + [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( E m ( τ ) ) d τ } . (124) \begin{aligned}
&\sup_{0\le\tau\le t}\left\{\mathcal{E}_m(\tau)+\|\eta(\tau)\|_{\mathcal{H}^{m-1}}^2+\varepsilon\left\|\left(\partial_t^{m-1}\operatorname{div}u,\partial_t^{m-1}\nabla p\right)(\tau)\right\|_{L^2}^2\right\}\\
&\quad+\varepsilon\int_0^t\|\nabla\eta(\tau)\|_{\mathcal{H}^{m-1}}^2\,\mathrm{d}\tau+\varepsilon\int_0^t\|\nabla u(\tau)\|_{\mathcal{H}^m}^2\,\mathrm{d}\tau\\
&\quad+\varepsilon\sum_{k=0}^{m-2}\int_0^t\|\partial_t^k\nabla\operatorname{div}u(\tau)\|_{m-1-k}^2\,\mathrm{d}\tau+\varepsilon^2\int_0^t\|\partial_t^{m-1}\nabla\operatorname{div}u(\tau)\|_{L^2}^2\,\mathrm{d}\tau\\
&\quad+\varepsilon\sum_{k=0}^{m-2}\int_0^t\|\nabla^2\partial_t^k u(\tau)\|_{m-1-k}^2\,\mathrm{d}\tau+\varepsilon^2\int_0^t\|\nabla^2u(\tau)\|_{\mathcal{H}^{m-1}}^2\,\mathrm{d}\tau\\
&\quad+\int_0^t\|\nabla\partial_t^{m-1}p(\tau)\|_{L^2}^2\,\mathrm{d}\tau\\
&\le C_{m+2}\left\{\mathcal{E}_m(0)+[1+P(Q(t))]\int_0^tP(\mathcal{E}_m(\tau))\,\mathrm{d}\tau\right\}. \tag*{(124)}
\end{aligned} 0 ≤ τ ≤ t sup { E m ( τ ) + ∥ η ( τ ) ∥ H m − 1 2 + ε ( ∂ t m − 1 div u , ∂ t m − 1 ∇ p ) ( τ ) L 2 2 } + ε ∫ 0 t ∥∇ η ( τ ) ∥ H m − 1 2 d τ + ε ∫ 0 t ∥∇ u ( τ ) ∥ H m 2 d τ + ε k = 0 ∑ m − 2 ∫ 0 t ∥ ∂ t k ∇ div u ( τ ) ∥ m − 1 − k 2 d τ + ε 2 ∫ 0 t ∥ ∂ t m − 1 ∇ div u ( τ ) ∥ L 2 2 d τ + ε k = 0 ∑ m − 2 ∫ 0 t ∥ ∇ 2 ∂ t k u ( τ ) ∥ m − 1 − k 2 d τ + ε 2 ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H m − 1 2 d τ + ∫ 0 t ∥∇ ∂ t m − 1 p ( τ ) ∥ L 2 2 d τ ≤ C m + 2 { E m ( 0 ) + [ 1 + P ( Q ( t ))] ∫ 0 t P ( E m ( τ )) d τ } . (124) L-infinity estimates
Lemma 3.10. Let m ≥ 6 m\ge6 m ≥ 6
∥ Z α ( ln ρ , p , u ) ( t ) ∥ L ∞ 2 ≤ C P ( Λ m 1 ( t ) ) , m ≥ 2 + ∣ α ∣ . (125) \|Z^\alpha(\ln\rho,p,u)(t)\|_{L^\infty}^2\le CP\left(\Lambda_m^1(t)\right),\qquad m\ge2+|\alpha|. \tag*{(125)} ∥ Z α ( ln ρ , p , u ) ( t ) ∥ L ∞ 2 ≤ C P ( Λ m 1 ( t ) ) , m ≥ 2 + ∣ α ∣. (125) Moreover,
∥ ∇ ( ln ρ , p ) ( t ) ∥ H 1 , ∞ 2 ≤ C 3 [ P ( Λ m 1 ( t ) ) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) ] , m ≥ 5. (126) \|\nabla(\ln\rho,p)(t)\|_{\mathcal{H}^{1,\infty}}^2\le C_3\left[P\left(\Lambda_m^1(t)\right)+P\left(\|\Delta p(t)\|_{\mathcal{H}^1}^2\right)\right],\qquad m\ge5. \tag*{(126)} ∥∇ ( ln ρ , p ) ( t ) ∥ H 1 , ∞ 2 ≤ C 3 [ P ( Λ m 1 ( t ) ) + P ( ∥Δ p ( t ) ∥ H 1 2 ) ] , m ≥ 5. (126) and
∥ ∇ ϕ ( t ) ∥ H 1 , ∞ 2 ≤ C 3 P ( E m ( t ) ) , m ≥ 5. (127) \lVert\nabla\phi(t) \rVert_{H^{1,\infty}}^{2} \le C_{3}P\left(\mathcal{E}_{m}(t)\right), \qquad m \ge5. \tag*{(127)} ∥ ∇ ϕ ( t ) ∥ H 1 , ∞ 2 ≤ C 3 P ( E m ( t ) ) , m ≥ 5. (127) Furthermore,
∥ div u ( t ) ∥ H 1 , ∞ 2 ≤ C 3 [ P ( Λ m 1 ( t ) ) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) ] , m ≥ 5 , (128) \lVert\operatorname{div} u(t) \rVert_{H^{1,\infty}}^{2} \le C_{3}\left[P\left(\Lambda_{m}^{1}(t)\right)+P\left(\lVert\Delta p(t) \rVert_{H^{1}}^{2}\right)\right], \qquad m \ge5, \tag*{(128)} ∥ div u ( t ) ∥ H 1 , ∞ 2 ≤ C 3 [ P ( Λ m 1 ( t ) ) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) ] , m ≥ 5 , (128) ∥ ∇ div u ( t ) ∥ L ∞ 2 ≤ C 3 P ( Q ( t ) ) , (129) \lVert\nabla\operatorname{div} u(t) \rVert_{L^{\infty}}^{2} \le C_{3}P(Q(t)), \tag*{(129)} ∥ ∇ div u ( t ) ∥ L ∞ 2 ≤ C 3 P ( Q ( t )) , (129) and, for any sufficiently small δ > 0 \delta> 0 δ > 0
∥ ∇ div u ( t ) ∥ H 1 , ∞ 2 ≤ C 4 [ 1 + P ( Q ( t ) ) ] [ C δ P ( Λ m 1 ( t ) ) + C δ P ( ∥ Δ p ( t ) ∥ H 2 2 ) + δ ∥ Δ p ( t ) ∥ H 2 2 ] , m ≥ 6. (130) \begin{aligned}
\lVert\nabla\operatorname{div} u(t) \rVert_{H^{1,\infty}}^{2}
&\le C_{4}\left[1+P(Q(t))\right]\left[C_{\delta}P\left(\Lambda_{m}^{1}(t)\right)+C_{\delta}P\left(\lVert\Delta p(t) \rVert_{H^{2}}^{2}\right)\right.\\
&\qquad\left.+\delta\lVert\Delta p(t) \rVert_{H^{2}}^{2}\right], \qquad m \ge6. \tag*{(130)}
\end{aligned} ∥ ∇ div u ( t ) ∥ H 1 , ∞ 2 ≤ C 4 [ 1 + P ( Q ( t )) ] [ C δ P ( Λ m 1 ( t ) ) + C δ P ( ∥ Δ p ( t ) ∥ H 2 2 ) + δ ∥ Δ p ( t ) ∥ H 2 2 ] , m ≥ 6. (130) Consequently,
Q ( t ) ≤ C 4 sup 0 ≤ τ ≤ t { P ( Λ m 1 ( τ ) ) + P ( E m ( τ ) ) + P ( ∥ Δ p ( τ ) ∥ H 1 2 ) + ∥ ∇ u ( τ ) ∥ H 1 , ∞ 2 + δ ∥ Δ p ( τ ) ∥ H 2 2 } . (131) Q(t) \le C_{4}\sup_{0\le\tau\le t}\left\{P\left(\Lambda_{m}^{1}(\tau)\right)+P\left(\mathcal{E}_{m}(\tau)\right)+P\left(\lVert\Delta p(\tau) \rVert_{H^{1}}^{2}\right)+\lVert\nabla u(\tau) \rVert_{H^{1,\infty}}^{2}+\delta\lVert\Delta p(\tau) \rVert_{H^{2}}^{2}\right\}. \tag*{(131)} Q ( t ) ≤ C 4 0 ≤ τ ≤ t sup { P ( Λ m 1 ( τ ) ) + P ( E m ( τ ) ) + P ( ∥ Δ p ( τ ) ∥ H 1 2 ) + ∥ ∇ u ( τ ) ∥ H 1 , ∞ 2 + δ ∥ Δ p ( τ ) ∥ H 2 2 } . (131) Proof. We first estimate the lower-order L ∞ L^{\infty} L ∞ m ≥ 2 + ∣ α ∣ m \ge2+|\alpha| m ≥ 2 + ∣ α ∣
∥ Z α ( ln ρ , p , u ) ∥ L ∞ 2 ≤ C ( ∥ ∇ Z α ( ln ρ , p , u ) ∥ H 1 + ∥ Z α ( ln ρ , p , u ) ∥ H 1 ) ∥ Z α ( ln ρ , p , u ) ∥ H 2 ≤ C P ( Λ m 1 ( t ) ) . (132) \lVert Z^{\alpha}(\ln\rho,p,u) \rVert_{L^{\infty}}^{2}
\le C\left(\lVert\nabla Z^{\alpha}(\ln\rho,p,u) \rVert_{H^{1}}+\lVert Z^{\alpha}(\ln\rho,p,u) \rVert_{H^{1}}\right)\lVert Z^{\alpha}(\ln\rho,p,u) \rVert_{H^{2}}
\le CP\left(\Lambda_{m}^{1}(t)\right). \tag*{(132)} ∥ Z α ( ln ρ , p , u ) ∥ L ∞ 2 ≤ C ( ∥ ∇ Z α ( ln ρ , p , u ) ∥ H 1 + ∥ Z α ( ln ρ , p , u ) ∥ H 1 ) ∥ Z α ( ln ρ , p , u ) ∥ H 2 ≤ C P ( Λ m 1 ( t ) ) . (132) where p = ρ γ p=\rho^{\gamma} p = ρ γ (125) .
Next, applying the same anisotropic Sobolev estimate to Z α ∇ p Z^{\alpha}\nabla p Z α ∇ p ∣ α ∣ ≤ 1 |\alpha|\le1 ∣ α ∣ ≤ 1
∥ Z α ∇ p ∥ L ∞ 2 ≤ C ( ∥ ∇ Z α ∇ p ∥ H 1 + ∥ Z α ∇ p ∥ H 1 ) ∥ Z α ∇ p ∥ H 2 ≤ C 3 [ P ( Λ m 1 ( t ) ) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) ] . (133) \lVert Z^{\alpha}\nabla p \rVert_{L^{\infty}}^{2}
\le C\left(\lVert\nabla Z^{\alpha}\nabla p \rVert_{H^{1}}+\lVert Z^{\alpha}\nabla p \rVert_{H^{1}}\right)\lVert Z^{\alpha}\nabla p \rVert_{H^{2}}
\le C_{3}\left[P\left(\Lambda_{m}^{1}(t)\right)+P\left(\lVert\Delta p(t) \rVert_{H^{1}}^{2}\right)\right]. \tag*{(133)} ∥ Z α ∇ p ∥ L ∞ 2 ≤ C ( ∥ ∇ Z α ∇ p ∥ H 1 + ∥ Z α ∇ p ∥ H 1 ) ∥ Z α ∇ p ∥ H 2 ≤ C 3 [ P ( Λ m 1 ( t ) ) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) ] . (133) Here the non-conormal second-order derivatives of p p p Δ p \Delta p Δ p
∥ ∇ 2 p ∥ H 1 ≤ C 3 ( ∥ Δ p ∥ H 1 + ∥ ∇ p ∥ H 1 + ∥ p ∥ H 3 ) . \lVert\nabla^{2}p \rVert_{H^{1}}\le C_{3}\left(\lVert\Delta p \rVert_{H^{1}}+\lVert\nabla p \rVert_{H^{1}}+\lVert p \rVert_{H^{3}}\right). ∥ ∇ 2 p ∥ H 1 ≤ C 3 ( ∥ Δ p ∥ H 1 + ∥ ∇ p ∥ H 1 + ∥ p ∥ H 3 ) . Together with p = ρ γ p=\rho^{\gamma} p = ρ γ (126) .
For the electric potential, Proposition 2.3 and Proposition 2.4 imply
∥ Z α ∇ ϕ ∥ L ∞ 2 ≤ C ( ∥ ∇ Z α ∇ ϕ ∥ H 1 + ∥ Z α ∇ ϕ ∥ H 1 ) ∥ Z α ∇ ϕ ∥ H 2 ≤ C 3 P ( E m ( t ) ) , ∣ α ∣ ≤ 1 , (134) \lVert Z^{\alpha}\nabla\phi\rVert_{L^{\infty}}^{2}
\le C\left(\lVert\nabla Z^{\alpha}\nabla\phi\rVert_{H^{1}}+\lVert Z^{\alpha}\nabla\phi\rVert_{H^{1}}\right)\lVert Z^{\alpha}\nabla\phi\rVert_{H^{2}}
\le C_{3}P\left(\mathcal{E}_{m}(t)\right), \qquad|\alpha|\le1, \tag*{(134)} ∥ Z α ∇ ϕ ∥ L ∞ 2 ≤ C ( ∥ ∇ Z α ∇ ϕ ∥ H 1 + ∥ Z α ∇ ϕ ∥ H 1 ) ∥ Z α ∇ ϕ ∥ H 2 ≤ C 3 P ( E m ( t ) ) , ∣ α ∣ ≤ 1 , (134) which gives (127) .
We now estimate div u \operatorname{div}u div u (42) ,
div u = − ( ln ρ ) t − u ⋅ ∇ ln ρ . (135) \operatorname{div}u=-(\ln\rho)_{t}-u\cdot\nabla\ln\rho. \tag*{(135)} div u = − ( ln ρ ) t − u ⋅ ∇ ln ρ . (135) Using Proposition 2.2, (125) , and (126) , we obtain
∥ div u ( t ) ∥ H 1 , ∞ 2 ≤ C ( ∥ ( ln ρ ) t ( t ) ∥ H 1 , ∞ 2 + ∥ u ( t ) ⋅ ∇ ln ρ ( t ) ∥ H 1 , ∞ 2 ) ≤ C 3 [ P ( Λ m 1 ( t ) ) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) ] . (136) \lVert\operatorname{div}u(t) \rVert_{H^{1,\infty}}^{2}
\le C\left(\lVert(\ln\rho)_{t}(t) \rVert_{H^{1,\infty}}^{2}+\lVert u(t)\cdot\nabla\ln\rho(t) \rVert_{H^{1,\infty}}^{2}\right)
\le C_{3}\left[P\left(\Lambda_{m}^{1}(t)\right)+P\left(\lVert\Delta p(t) \rVert_{H^{1}}^{2}\right)\right]. \tag*{(136)} ∥ div u ( t ) ∥ H 1 , ∞ 2 ≤ C ( ∥( ln ρ ) t ( t ) ∥ H 1 , ∞ 2 + ∥ u ( t ) ⋅ ∇ ln ρ ( t ) ∥ H 1 , ∞ 2 ) ≤ C 3 [ P ( Λ m 1 ( t ) ) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) ] . (136) and hence (128) . Differentiating (135) once and using the preceding estimates gives
∥ ∇ div u ( t ) ∥ L ∞ 2 ≤ C 3 P ( Q ( t ) ) , \left\|\nabla\operatorname{div}u(t)\right\|_{L^\infty}^{2}\le C_{3}P(Q(t)), ∥ ∇ div u ( t ) ∥ L ∞ 2 ≤ C 3 P ( Q ( t )) , which is (129) .
It remains to estimate ∇ div u \nabla\operatorname{div}u ∇ div u H 1 , ∞ \mathcal{H}^{1,\infty} H 1 , ∞ (135) , we have
∇ div u = − ∇ ( ln ρ ) t − ( ∇ u ) ∇ ln ρ − u ⋅ ∇ 2 ln ρ . \nabla\operatorname{div}u=-\nabla(\ln\rho)_{t}-(\nabla u)\nabla\ln\rho-u\cdot\nabla^{2}\ln\rho. ∇ div u = − ∇ ( ln ρ ) t − ( ∇ u ) ∇ ln ρ − u ⋅ ∇ 2 ln ρ . Applying Z α Z^\alpha Z α ∣ α ∣ ≤ 1 |\alpha|\le1 ∣ α ∣ ≤ 1
∥ ∇ div u ( t ) ∥ H 1 , ∞ 2 ≤ C 4 [ 1 + P ( Q ( t ) ) ] [ P ( Λ m 1 ( t ) ) + ∥ ∇ 2 p ( t ) ∥ H 1 , ∞ 2 ] ≤ C 4 [ 1 + P ( Q ( t ) ) ] { C δ P ( Λ m 1 ( t ) ) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) } + C 4 δ [ 1 + P ( Q ( t ) ) ] ∥ Δ p ( t ) ∥ H 2 2 . (137) \begin{aligned}
\left\|\nabla\operatorname{div}u(t)\right\|_{\mathcal{H}^{1,\infty}}^{2}
&\le C_{4}[1+P(Q(t))]\left[P(\Lambda_{m}^{1}(t))+\left\|\nabla^{2}p(t)\right\|_{\mathcal{H}^{1,\infty}}^{2}\right]\\
&\le C_{4}[1+P(Q(t))]\left\{C_{\delta}P(\Lambda_{m}^{1}(t))+P\left(\left\|\Delta p(t)\right\|_{\mathcal{H}^{1}}^{2}\right)\right\}\\
&\quad+C_{4}\delta[1+P(Q(t))]\left\|\Delta p(t)\right\|_{\mathcal{H}^{2}}^{2}. \tag*{(137)}
\end{aligned} ∥ ∇ div u ( t ) ∥ H 1 , ∞ 2 ≤ C 4 [ 1 + P ( Q ( t ))] [ P ( Λ m 1 ( t )) + ∇ 2 p ( t ) H 1 , ∞ 2 ] ≤ C 4 [ 1 + P ( Q ( t ))] { C δ P ( Λ m 1 ( t )) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) } + C 4 δ [ 1 + P ( Q ( t ))] ∥ Δ p ( t ) ∥ H 2 2 . (137) This proves (130) . Finally, (131) follows from (125) –(130) . This completes the proof.
Lemma 3.11. Let m ≥ 6 m\ge6 m ≥ 6
∥ ∇ u ( t ) ∥ H 1 , ∞ 2 ≤ C m + 2 ( ∥ ( u 0 , ∇ u 0 ) ∥ H 1 , ∞ 2 + P ( Λ m 1 ( t ) ) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) ) + ε 2 t ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H 4 2 d τ + t ∫ 0 t [ 1 + P ( Λ m 1 ( τ ) ) + P ( Q ( τ ) ) ] × [ 1 + ε 2 ∥ Δ p ( τ ) ∥ H 2 2 ] d τ . (138) \begin{aligned}
\left\|\nabla u(t)\right\|_{\mathcal{H}^{1,\infty}}^{2}
&\le C_{m+2}\left(\left\|(u_{0},\nabla u_{0})\right\|_{\mathcal{H}^{1,\infty}}^{2}+P(\Lambda_{m}^{1}(t))+P\left(\left\|\Delta p(t)\right\|_{\mathcal{H}^{1}}^{2}\right)\right)\\
&\quad+\varepsilon^{2}t\int_{0}^{t}\left\|\nabla^{2}u(\tau)\right\|_{\mathcal{H}^{4}}^{2}\,d\tau
+t\int_{0}^{t}\left[1+P(\Lambda_{m}^{1}(\tau))+P(Q(\tau))\right]\\
&\qquad\times\left[1+\varepsilon^{2}\left\|\Delta p(\tau)\right\|_{\mathcal{H}^{2}}^{2}\right]\,d\tau. \tag*{(138)}
\end{aligned} ∥ ∇ u ( t ) ∥ H 1 , ∞ 2 ≤ C m + 2 ( ∥ ( u 0 , ∇ u 0 ) ∥ H 1 , ∞ 2 + P ( Λ m 1 ( t )) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) ) + ε 2 t ∫ 0 t ∇ 2 u ( τ ) H 4 2 d τ + t ∫ 0 t [ 1 + P ( Λ m 1 ( τ )) + P ( Q ( τ )) ] × [ 1 + ε 2 ∥ Δ p ( τ ) ∥ H 2 2 ] d τ . (138) Proof. The argument follows the L ∞ L^\infty L ∞ [34 ] , with additional electric terms treated by the Poisson estimates. Let χ 0 \chi_{0} χ 0 χ \chi χ ∣ α ∣ ≤ 1 |\alpha|\le1 ∣ α ∣ ≤ 1
∥ χ 0 Z α ∇ u ( t ) ∥ L ∞ 2 + ∥ χ 0 ∇ u ( t ) ∥ L ∞ 2 ≤ C ∥ u ( t ) ∥ H m 2 ≤ C Λ m 1 ( t ) . (139) \left\|\chi_{0}Z^{\alpha}\nabla u(t)\right\|_{L^\infty}^{2}+\left\|\chi_{0}\nabla u(t)\right\|_{L^\infty}^{2}\le C\left\|u(t)\right\|_{\mathcal{H}^{m}}^{2}\le C\Lambda_{m}^{1}(t). \tag*{(139)} ∥ χ 0 Z α ∇ u ( t ) ∥ L ∞ 2 + ∥ χ 0 ∇ u ( t ) ∥ L ∞ 2 ≤ C ∥ u ( t ) ∥ H m 2 ≤ C Λ m 1 ( t ) . (139) Thus it remains to estimate χ Z α ∇ u \chi Z^{\alpha}\nabla u χ Z α ∇ u χ ∇ u \chi\nabla u χ ∇ u
In a boundary coordinate patch, write
x = Ψ n ( y , z ) = ( y ψ ( y ) ) − z n ( y ) , y = ( y 1 , y 2 ) , z ≥ 0 , x=\Psi^{n}(y,z)=\begin{pmatrix}y\\\psi(y)\end{pmatrix}-zn(y),\qquad y=(y_{1},y_{2}),\quad z\ge0, x = Ψ n ( y , z ) = ( y ψ ( y ) ) − z n ( y ) , y = ( y 1 , y 2 ) , z ≥ 0 , where y = ( y 1 , y 2 ) y=(y_{1},y_{2}) y = ( y 1 , y 2 ) z z z f f f
f ~ ( t , y , z ) : = f ( t , Ψ n ( y , z ) ) . \widetilde{f}(t,y,z):=f(t,\Psi^{n}(y,z)). f ( t , y , z ) := f ( t , Ψ n ( y , z )) . Set
( η ~ , ρ ~ , u ~ ) ( t , y , z ) : = ( η , ρ , u ) ( t , Ψ n ( y , z ) ) . (\widetilde{\eta},\widetilde{\rho},\widetilde{u})(t,y,z):=(\eta,\rho,u)(t,\Psi^{n}(y,z)). ( η , ρ , u ) ( t , y , z ) := ( η , ρ , u ) ( t , Ψ n ( y , z )) . The local decomposition of the gradient gives
∥ χ ∇ u ( t ) ∥ H 1 , ∞ 2 ≤ C m + 1 ( ∥ η ~ ( t ) ∥ H 1 , ∞ 2 + ∥ div u ( t ) ∥ H 1 , ∞ 2 + ∥ u ( t ) ∥ H 2 , ∞ 2 ) . (140) \left\|\chi\nabla u(t)\right\|_{\mathcal{H}^{1,\infty}}^{2}
\le C_{m+1}\left(\left\|\widetilde{\eta}(t)\right\|_{\mathcal{H}^{1,\infty}}^{2}+\left\|\operatorname{div}u(t)\right\|_{\mathcal{H}^{1,\infty}}^{2}+\left\|u(t)\right\|_{\mathcal{H}^{2,\infty}}^{2}\right). \tag*{(140)} ∥ χ ∇ u ( t ) ∥ H 1 , ∞ 2 ≤ C m + 1 ( ∥ η ( t ) ∥ H 1 , ∞ 2 + ∥ div u ( t ) ∥ H 1 , ∞ 2 + ∥ u ( t ) ∥ H 2 , ∞ 2 ) . (140) Hence, by Lemma 3.10,
∥ χ ∇ u ( t ) ∥ H 1 , ∞ 2 ≤ C m + 2 ( ∥ η ~ ( t ) ∥ H 1 , ∞ 2 + P ( Λ m 1 ( t ) ) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) ) . (141) \left\|\chi\nabla u(t)\right\|_{\mathcal{H}^{1,\infty}}^{2}
\le C_{m+2}\left(\left\|\widetilde{\eta}(t)\right\|_{\mathcal{H}^{1,\infty}}^{2}+P(\Lambda_{m}^{1}(t))+P\left(\left\|\Delta p(t)\right\|_{\mathcal{H}^{1}}^{2}\right)\right). \tag*{(141)} ∥ χ ∇ u ( t ) ∥ H 1 , ∞ 2 ≤ C m + 2 ( ∥ η ( t ) ∥ H 1 , ∞ 2 + P ( Λ m 1 ( t )) + P ( ∥ Δ p ( t ) ∥ H 1 2 ) ) . (141) It remains to estimate η ˉ \bar{\eta} η ˉ
By (112) , after transforming to the local coordinates, we have
ρ ~ ∂ t η ˉ + ρ ~ u ~ y ⋅ ∇ y η ˉ + ρ ~ ( u ~ ⋅ n ) ∂ z η ˉ − μ ε ∂ z z η ˉ = F ~ η 0 + F ~ η E + μ ε ( Δ ~ η − ∂ z z η ˉ ) (142) \tilde{\rho}\,\partial_t\bar{\eta}+\tilde{\rho}\,\tilde{u}_y\cdot\nabla_y\bar{\eta}+\tilde{\rho}(\tilde{u}\cdot n)\partial_z\bar{\eta}-\mu\varepsilon\,\partial_{zz}\bar{\eta}=\tilde{F}_\eta^0+\tilde{F}_\eta^E+\mu\varepsilon\left(\widetilde{\Delta}\eta-\partial_{zz}\bar{\eta}\right) \tag*{(142)} ρ ~ ∂ t η ˉ + ρ ~ u ~ y ⋅ ∇ y η ˉ + ρ ~ ( u ~ ⋅ n ) ∂ z η ˉ − μ ε ∂ z z η ˉ = F ~ η 0 + F ~ η E + μ ε ( Δ η − ∂ z z η ˉ ) (142) with
η ˉ ∣ z = 0 = 0. \left.\bar{\eta}\right|_{z=0}=0. η ˉ ∣ z = 0 = 0. Since u ⋅ n = 0 u\cdot n=0 u ⋅ n = 0 ∂ Ω \partial\Omega ∂ Ω
( u ~ ⋅ n ) ( t , y , 0 ) = 0. (\tilde{u}\cdot n)(t,y,0)=0. ( u ~ ⋅ n ) ( t , y , 0 ) = 0. Let
a ( t , y ) : = ∂ z ( u ~ ⋅ n ) ( t , y , 0 ) . a(t,y):=\partial_z(\tilde{u}\cdot n)(t,y,0). a ( t , y ) := ∂ z ( u ~ ⋅ n ) ( t , y , 0 ) . Then
u ~ ⋅ n = z a ( t , y ) + [ u ~ ⋅ n − z a ( t , y ) ] . \tilde{u}\cdot n=za(t,y)+[\tilde{u}\cdot n-za(t,y)]. u ~ ⋅ n = z a ( t , y ) + [ u ~ ⋅ n − z a ( t , y )] . Therefore, η ˉ \bar{\eta} η ˉ
ρ ~ ∂ t η ˉ + ρ ~ u ~ y ⋅ ∇ y η ˉ + ρ ~ z a ( t , y ) ∂ z η ˉ − μ ε ∂ z z η ˉ = G , η ˉ ∣ z = 0 = 0 , (143) \tilde{\rho}\,\partial_t\bar{\eta}+\tilde{\rho}\,\tilde{u}_y\cdot\nabla_y\bar{\eta}+\tilde{\rho}\,za(t,y)\partial_z\bar{\eta}-\mu\varepsilon\,\partial_{zz}\bar{\eta}=G,\qquad\left.\bar{\eta}\right|_{z=0}=0, \tag*{(143)} ρ ~ ∂ t η ˉ + ρ ~ u ~ y ⋅ ∇ y η ˉ + ρ ~ z a ( t , y ) ∂ z η ˉ − μ ε ∂ z z η ˉ = G , η ˉ ∣ z = 0 = 0 , (143) where
G : = F ~ η 0 + F ~ η E + μ ε ( Δ ~ η − ∂ z z η ˉ ) − ρ ~ [ u ~ ⋅ n − z a ( t , y ) ] ∂ z η ˉ . (144) G:=\tilde{F}_\eta^0+\tilde{F}_\eta^E+\mu\varepsilon\left(\widetilde{\Delta}\eta-\partial_{zz}\bar{\eta}\right)-\tilde{\rho}[\tilde{u}\cdot n-za(t,y)]\partial_z\bar{\eta}. \tag*{(144)} G := F ~ η 0 + F ~ η E + μ ε ( Δ η − ∂ z z η ˉ ) − ρ ~ [ u ~ ⋅ n − z a ( t , y )] ∂ z η ˉ . (144) Set η ˉ 0 = η ˉ ( 0 , y , z ) \bar{\eta}_0=\bar{\eta}(0,y,z) η ˉ 0 = η ˉ ( 0 , y , z ) L ∞ L^\infty L ∞ (143) , and using Proposition 2.2 , we get
∥ η ˉ ( t ) ∥ H 1 , ∞ 2 ≤ C ( ∥ η ˉ 0 ∥ H 1 , ∞ 2 + t ∫ 0 t ∥ G ( τ ) ∥ H 1 , ∞ 2 d τ + t ∫ 0 t [ 1 + P ( Λ m 1 ( τ ) ) + P ( Q ( τ ) ) ] ∥ η ˉ ( τ ) ∥ H 1 , ∞ 2 d τ ) (145) \|\bar{\eta}(t)\|_{H^1,\infty}^2\le C\left(\|\bar{\eta}_0\|_{H^1,\infty}^2+t\int_0^t\|G(\tau)\|_{H^1,\infty}^2\,d\tau+t\int_0^t[1+P(\Lambda_m^1(\tau))+P(Q(\tau))]\|\bar{\eta}(\tau)\|_{H^1,\infty}^2\,d\tau\right) \tag*{(145)} ∥ η ˉ ( t ) ∥ H 1 , ∞ 2 ≤ C ( ∥ η ˉ 0 ∥ H 1 , ∞ 2 + t ∫ 0 t ∥ G ( τ ) ∥ H 1 , ∞ 2 d τ + t ∫ 0 t [ 1 + P ( Λ m 1 ( τ )) + P ( Q ( τ ))] ∥ η ˉ ( τ ) ∥ H 1 , ∞ 2 d τ ) (145) We now estimate G G G F ~ η 0 \tilde{F}_\eta^0 F ~ η 0
∥ F ~ η 0 ∥ H 1 , ∞ 2 + ε 2 ∥ Δ ~ η − ∂ z z η ˉ ∥ H 1 , ∞ 2 ≤ C m + 2 [ P ( Q ( t ) ) + P ( Λ m 1 ( t ) ) + ε 2 ∥ ∇ 2 u ( t ) ∥ H 4 2 ] . (146) \|\tilde{F}_\eta^0\|_{H^1,\infty}^2+\varepsilon^2\|\widetilde{\Delta}\eta-\partial_{zz}\bar{\eta}\|_{H^1,\infty}^2\le C_{m+2}[P(Q(t))+P(\Lambda_m^1(t))+\varepsilon^2\|\nabla^2u(t)\|_{\mathcal{H}^4}^2]. \tag*{(146)} ∥ F ~ η 0 ∥ H 1 , ∞ 2 + ε 2 ∥ Δ η − ∂ z z η ˉ ∥ H 1 , ∞ 2 ≤ C m + 2 [ P ( Q ( t )) + P ( Λ m 1 ( t )) + ε 2 ∥ ∇ 2 u ( t ) ∥ H 4 2 ] . (146) For the electric part, by (113) , Proposition 2.2 , and the elliptic estimate for the Poisson equation,
∥ F ~ η E ( t ) ∥ H 1 , ∞ 2 ≤ C m + 2 ( ∥ ∇ ρ × ∇ φ ∥ H 1 , ∞ 2 + ∥ ρ ∇ φ ∥ H 1 , ∞ 2 ) ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] P ( E m ( t ) ) . (147) \begin{aligned}
\|\tilde{F}_\eta^E(t)\|_{H^1,\infty}^2\le C_{m+2}\left(\|\nabla\rho\times\nabla\varphi\|_{H^1,\infty}^2+\|\rho\nabla\varphi\|_{H^1,\infty}^2\right) \\
&\le C_{m+2}[1+P(Q(t))]P(\mathcal{E}_m(t)). \tag*{(147)}
\end{aligned} ∥ F ~ η E ( t ) ∥ H 1 , ∞ 2 ≤ C m + 2 ( ∥∇ ρ × ∇ φ ∥ H 1 , ∞ 2 + ∥ ρ ∇ φ ∥ H 1 , ∞ 2 ) ≤ C m + 2 [ 1 + P ( Q ( t ))] P ( E m ( t )) . (147) It remains to control the last term in (144) . Since
u ~ ⋅ n − z a ( t , y ) = ∫ 0 z ∫ 0 s ∂ z z ( u ~ ⋅ n ) ( t , y , r ) d r d s , \tilde{u}\cdot n-za(t,y)=\int_0^z\int_0^s\partial_{zz}(\tilde{u}\cdot n)(t,y,r)\,dr\,ds, u ~ ⋅ n − z a ( t , y ) = ∫ 0 z ∫ 0 s ∂ z z ( u ~ ⋅ n ) ( t , y , r ) d r d s , we have
∥ u ~ ⋅ n − z a ( t , y ) z ∥ L ∞ ≤ C ∥ ∂ z z ( u ~ ⋅ n ) ∥ L ∞ . \left\|\frac{\tilde{u}\cdot n-za(t,y)}{z}\right\|_{L^\infty}\le C\|\partial_{zz}(\tilde{u}\cdot n)\|_{L^\infty}. z u ~ ⋅ n − z a ( t , y ) L ∞ ≤ C ∥ ∂ z z ( u ~ ⋅ n ) ∥ L ∞ . Moreover, since n n n z z z
∂ z z ( u ~ ⋅ n ) = ∂ z div u − ∂ z ( Π ∂ y 1 u ) 1 − ∂ z ( Π ∂ y 2 u ) 2 . (148) \partial_{zz}(\tilde{u}\cdot n)=\partial_z\operatorname{div}u-\partial_z(\Pi\partial_{y_1}u)_1-\partial_z(\Pi\partial_{y_2}u)_2. \tag*{(148)} ∂ z z ( u ~ ⋅ n ) = ∂ z div u − ∂ z ( Π ∂ y 1 u ) 1 − ∂ z ( Π ∂ y 2 u ) 2 . (148) Thus, by Lemma 3.10 and the local decomposition of ∇ u \nabla u ∇ u
∥ ∂ z z ( u ~ ⋅ n ) ( t ) ∥ L ∞ 2 ≤ C 3 P ( Q ( t ) ) . (149) \|\partial_{zz}(\widetilde{u}\cdot n)(t)\|_{L^\infty}^{2}\le C_{3}P(Q(t)). \tag*{(149)} ∥ ∂ z z ( u ⋅ n ) ( t ) ∥ L ∞ 2 ≤ C 3 P ( Q ( t )) . (149) Consequently,
∥ ρ ~ [ u ~ ⋅ n − z a ( t , y ) ] ∂ z η ~ ∥ H 1 , ∞ 2 ≤ C 4 [ 1 + P ( Q ( t ) ) ] ( ∥ η ~ ( t ) ∥ H 1 , ∞ 2 + P ( Λ m 1 ( t ) ) ) . (150) \|\widetilde{\rho}[\widetilde{u}\cdot n-za(t,y)]\partial_z\widetilde{\eta}\|_{H^{1,\infty}}^{2}\le C_{4}[1+P(Q(t))]\left(\|\widetilde{\eta}(t)\|_{H^{1,\infty}}^{2}+P(\Lambda_m^{1}(t))\right). \tag*{(150)} ∥ ρ [ u ⋅ n − z a ( t , y )] ∂ z η ∥ H 1 , ∞ 2 ≤ C 4 [ 1 + P ( Q ( t ))] ( ∥ η ( t ) ∥ H 1 , ∞ 2 + P ( Λ m 1 ( t )) ) . (150) Combining (146) –(150) , we obtain
∥ G ( t ) ∥ H 1 , ∞ 2 ≤ C 4 [ 1 + P ( Q ( t ) ) ] [ ∥ η ~ ( t ) ∥ H 1 , ∞ 2 + P ( Λ m 1 ( t ) ) + P ( E m ( t ) ) + ε 2 ∥ ∇ 2 u ( t ) ∥ H 4 2 + ε 2 ∥ Δ p ( t ) ∥ H 2 2 ] . (151) \|G(t)\|_{H^{1,\infty}}^{2}\le C_{4}[1+P(Q(t))]\left[\|\widetilde{\eta}(t)\|_{H^{1,\infty}}^{2}+P(\Lambda_m^{1}(t))+P(\mathcal{E}_m(t))+\varepsilon^{2}\|\nabla^{2}u(t)\|_{H^{4}}^{2}+\varepsilon^{2}\|\Delta p(t)\|_{H^{2}}^{2}\right]. \tag*{(151)} ∥ G ( t ) ∥ H 1 , ∞ 2 ≤ C 4 [ 1 + P ( Q ( t ))] [ ∥ η ( t ) ∥ H 1 , ∞ 2 + P ( Λ m 1 ( t )) + P ( E m ( t )) + ε 2 ∥ ∇ 2 u ( t ) ∥ H 4 2 + ε 2 ∥Δ p ( t ) ∥ H 2 2 ] . (151) Substituting (151) into (145) and applying Gronwall’s inequality yields
∥ η ~ ( t ) ∥ H 1 , ∞ 2 ≤ C m + 2 ( ∥ η ~ 0 ∥ H 1 , ∞ 2 + ε 2 t ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H 4 2 d τ + t ∫ 0 t [ 1 + P ( Λ m 1 ( τ ) ) + P ( Q ( τ ) ) ] [ 1 + ε 2 ∥ Δ p ( τ ) ∥ H 2 2 ] d τ ) . (152) \|\widetilde{\eta}(t)\|_{H^{1,\infty}}^{2}\le C_{m+2}\left(\|\widetilde{\eta}_{0}\|_{H^{1,\infty}}^{2}+\varepsilon^{2}t\int_{0}^{t}\|\nabla^{2}u(\tau)\|_{H^{4}}^{2}\,d\tau+t\int_{0}^{t}[1+P(\Lambda_m^{1}(\tau))+P(Q(\tau))][1+\varepsilon^{2}\|\Delta p(\tau)\|_{H^{2}}^{2}]\,d\tau\right). \tag*{(152)} ∥ η ( t ) ∥ H 1 , ∞ 2 ≤ C m + 2 ( ∥ η 0 ∥ H 1 , ∞ 2 + ε 2 t ∫ 0 t ∥ ∇ 2 u ( τ ) ∥ H 4 2 d τ + t ∫ 0 t [ 1 + P ( Λ m 1 ( τ )) + P ( Q ( τ ))] [ 1 + ε 2 ∥Δ p ( τ ) ∥ H 2 2 ] d τ ) . (152) Finally, from (108) , (109) , (141) , (139) , and (152) , we obtain (138) . This completes the proof.
Lemma 3.12. Let m ≥ 6 m\ge6 m ≥ 6 0 ≤ t ≤ min { T , 1 } 0\le t\le\min\{T,1\} 0 ≤ t ≤ min { T , 1 }
sup 0 ≤ τ ≤ t ( ∥ Δ p ( τ ) ∥ H 1 2 + ε ∥ Δ p ( τ ) ∥ H 2 2 ) + ∫ 0 t ∥ Δ p ( τ ) ∥ H 2 2 d τ ≤ C m + 2 [ P ( ∥ ( p 0 , u 0 , ∇ ϕ 0 ) ∥ X m 2 ) + [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( ∥ ( p , u , ∇ ϕ ) ( τ ) ∥ X m 2 ) d τ ] . (153) \sup_{0\le\tau\le t}\left(\|\Delta p(\tau)\|_{H^{1}}^{2}+\varepsilon\|\Delta p(\tau)\|_{H^{2}}^{2}\right)+\int_{0}^{t}\|\Delta p(\tau)\|_{H^{2}}^{2}\,d\tau
\le C_{m+2}\left[P\left(\|(p_{0},u_{0},\nabla\phi_{0})\|_{X_{m}^{2}}\right)+[1+P(Q(t))]\int_{0}^{t}P\left(\|(p,u,\nabla\phi)(\tau)\|_{X_{m}^{2}}\right)\,d\tau\right]. \tag*{(153)} 0 ≤ τ ≤ t sup ( ∥Δ p ( τ ) ∥ H 1 2 + ε ∥Δ p ( τ ) ∥ H 2 2 ) + ∫ 0 t ∥Δ p ( τ ) ∥ H 2 2 d τ ≤ C m + 2 [ P ( ∥ ( p 0 , u 0 , ∇ ϕ 0 ) ∥ X m 2 ) + [ 1 + P ( Q ( t ))] ∫ 0 t P ( ∥ ( p , u , ∇ ϕ ) ( τ ) ∥ X m 2 ) d τ ] . (153) Proof. Let
q = ln ρ , u ˙ = u t + u ⋅ ∇ u . q=\ln\rho,\qquad\dot{u}=u_{t}+u\cdot\nabla u. q = ln ρ , u ˙ = u t + u ⋅ ∇ u . Taking divergence of (24) gives
− ε Δ div u + 1 2 μ + λ Δ p = − 1 2 μ + λ div ( ρ u ˙ ) + 1 2 μ + λ div ( ρ ∇ ϕ ) . (154) -\varepsilon\Delta\operatorname{div}u+\frac{1}{2\mu+\lambda}\Delta p=-\frac{1}{2\mu+\lambda}\operatorname{div}(\rho\dot{u})+\frac{1}{2\mu+\lambda}\operatorname{div}(\rho\nabla\phi). \tag*{(154)} − ε Δ div u + 2 μ + λ 1 Δ p = − 2 μ + λ 1 div ( ρ u ˙ ) + 2 μ + λ 1 div ( ρ ∇ ϕ ) . (154) On the other hand, the continuity equation gives
q t + u ⋅ ∇ q + div u = 0. q_{t}+u\cdot\nabla q+\operatorname{div}u=0. q t + u ⋅ ∇ q + div u = 0. Hence
− Δ div u = ( Δ q ) t + u ⋅ ∇ Δ q + Δ u ⋅ ∇ q + 2 ∑ i = 1 3 ∂ i u ⋅ ∇ ∂ i q . -\Delta\operatorname{div}u=(\Delta q)_{t}+u\cdot\nabla\Delta q+\Delta u\cdot\nabla q+2\sum_{i=1}^{3}\partial_{i}u\cdot\nabla\partial_{i}q. − Δ div u = ( Δ q ) t + u ⋅ ∇Δ q + Δ u ⋅ ∇ q + 2 i = 1 ∑ 3 ∂ i u ⋅ ∇ ∂ i q . Substituting this identity into (154) , we obtain
ε ( Δ q ) t + ε u ⋅ ∇ Δ q + 1 2 μ + λ Δ p = − ε Δ u ⋅ ∇ q − 2 ε ∑ i = 1 3 ∂ i u ⋅ ∇ ∂ i q − 1 2 μ + λ div ( ρ u ˙ ) + 1 2 μ + λ div ( ρ ∇ ϕ ) . (155) \varepsilon(\Delta q)_{t}+\varepsilon u\cdot\nabla\Delta q+\frac{1}{2\mu+\lambda}\Delta p
=-\varepsilon\Delta u\cdot\nabla q-2\varepsilon\sum_{i=1}^{3}\partial_{i}u\cdot\nabla\partial_{i}q-\frac{1}{2\mu+\lambda}\operatorname{div}(\rho\dot{u})+\frac{1}{2\mu+\lambda}\operatorname{div}(\rho\nabla\phi). \tag*{(155)} ε ( Δ q ) t + ε u ⋅ ∇Δ q + 2 μ + λ 1 Δ p = − ε Δ u ⋅ ∇ q − 2 ε i = 1 ∑ 3 ∂ i u ⋅ ∇ ∂ i q − 2 μ + λ 1 div ( ρ u ˙ ) + 2 μ + λ 1 div ( ρ ∇ ϕ ) . (155) We follow the argument of [34 Lemma 3.16] , with the additional electric term estimated by the preceding Poisson estimates. Applying Z α Z^\alpha Z α ∣ α ∣ ≤ 2 |\alpha| \le2 ∣ α ∣ ≤ 2 (155) , multiplying by Z α Δ q Z^\alpha\Delta q Z α Δ q Ω \Omega Ω
ε 2 d d t ∥ Z α Δ q ∥ L 2 2 + 1 2 μ + λ ∫ Ω Z α Δ p Z α Δ q d x = − ε ∫ Ω Z α ( u ⋅ ∇ Δ q ) Z α Δ q d x − ε ∫ Ω Z α ( Δ u ⋅ ∇ q ) Z α Δ q d x − 2 ε ∑ i = 1 3 ∫ Ω Z α ( ∂ i u ⋅ ∇ ∂ i q ) Z α Δ q d x − 1 2 μ + λ ∫ Ω Z α div ( ρ u ˙ ) Z α Δ q d x + 1 2 μ + λ ∫ Ω Z α div ( ρ ∇ ϕ ) Z α Δ q d x . (156) \begin{aligned}
\frac{\varepsilon}{2}\frac{d}{dt}\|Z^\alpha\Delta q\|_{L^2}^2
&+\frac{1}{2\mu+\lambda}\int_{\Omega} Z^\alpha\Delta p Z^\alpha\Delta q\,dx \\
&=-\varepsilon\int_{\Omega} Z^\alpha(u\cdot\nabla\Delta q)Z^\alpha\Delta q\,dx
-\varepsilon\int_{\Omega} Z^\alpha(\Delta u\cdot\nabla q)Z^\alpha\Delta q\,dx \\
&\quad-2\varepsilon\sum_{i=1}^{3}\int_{\Omega} Z^\alpha(\partial_i u\cdot\nabla\partial_i q)Z^\alpha\Delta q\,dx
-\frac{1}{2\mu+\lambda}\int_{\Omega} Z^\alpha\operatorname{div}(\rho\dot{u})Z^\alpha\Delta q\,dx \\
&\quad+\frac{1}{2\mu+\lambda}\int_{\Omega} Z^\alpha\operatorname{div}(\rho\nabla\phi)Z^\alpha\Delta q\,dx. \tag*{(156)}
\end{aligned} 2 ε d t d ∥ Z α Δ q ∥ L 2 2 + 2 μ + λ 1 ∫ Ω Z α Δ p Z α Δ q d x = − ε ∫ Ω Z α ( u ⋅ ∇Δ q ) Z α Δ q d x − ε ∫ Ω Z α ( Δ u ⋅ ∇ q ) Z α Δ q d x − 2 ε i = 1 ∑ 3 ∫ Ω Z α ( ∂ i u ⋅ ∇ ∂ i q ) Z α Δ q d x − 2 μ + λ 1 ∫ Ω Z α div ( ρ u ˙ ) Z α Δ q d x + 2 μ + λ 1 ∫ Ω Z α div ( ρ ∇ ϕ ) Z α Δ q d x . (156) We first estimate the electric term. As in (40) , the Poisson equation yields
div ( ρ ∇ ϕ ) = ∇ ρ ⋅ ∇ ϕ + ρ ( ρ − 1 ) . \operatorname{div}(\rho\nabla\phi)=\nabla\rho\cdot\nabla\phi+\rho(\rho-1). div ( ρ ∇ ϕ ) = ∇ ρ ⋅ ∇ ϕ + ρ ( ρ − 1 ) . By Proposition 2.2 and Proposition 2.4,
∫ 0 t ∥ div ( ρ ∇ ϕ ) ∥ H 2 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( E m ( τ ) ) d τ . \int_{0}^{t}\|\operatorname{div}(\rho\nabla\phi)\|_{H^2}^{2}\,d\tau
\le C_{m+2}[1+P(Q(t))]\int_{0}^{t}P(\mathcal{E}_m(\tau))\,d\tau. ∫ 0 t ∥ div ( ρ ∇ ϕ ) ∥ H 2 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t P ( E m ( τ )) d τ . Therefore, Young’s inequality gives
∑ ∣ α ∣ ≤ 2 ∫ 0 t ∣ ∫ Ω Z α div ( ρ ∇ ϕ ) Z α Δ q d x ∣ d τ ≤ δ ∫ 0 t ∥ Δ q ∥ H 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( E m ( τ ) ) d τ . (157) \sum_{|\alpha|\le2}\int_{0}^{t}\left|\int_{\Omega} Z^\alpha\operatorname{div}(\rho\nabla\phi)Z^\alpha\Delta q\,dx\right|\,d\tau
\le\delta\int_{0}^{t}\|\Delta q\|_{H^2}^{2}\,d\tau+C_{\delta}C_{m+2}[1+P(Q(t))]\int_{0}^{t}P(\mathcal{E}_m(\tau))\,d\tau. \tag*{(157)} ∣ α ∣ ≤ 2 ∑ ∫ 0 t ∫ Ω Z α div ( ρ ∇ ϕ ) Z α Δ q d x d τ ≤ δ ∫ 0 t ∥Δ q ∥ H 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t P ( E m ( τ )) d τ . (157) Next, since p = e γ q p=e^{\gamma q} p = e γ q
Δ p = γ p Δ q + γ 2 p ∣ ∇ q ∣ 2 , Z α Δ p = γ p Z α Δ q + γ [ Z α , p ] Δ q + γ 2 Z α ( p ∣ ∇ q ∣ 2 ) . \Delta p=\gamma p\Delta q+\gamma^2p|\nabla q|^2,\qquad
Z^\alpha\Delta p=\gamma pZ^\alpha\Delta q+\gamma[Z^\alpha,p]\Delta q+\gamma^2Z^\alpha(p|\nabla q|^2). Δ p = γ p Δ q + γ 2 p ∣∇ q ∣ 2 , Z α Δ p = γ p Z α Δ q + γ [ Z α , p ] Δ q + γ 2 Z α ( p ∣∇ q ∣ 2 ) . Using the positive lower bound of p p p
1 2 μ + λ ∑ ∣ α ∣ ≤ 2 ∫ 0 t ∫ Ω Z α Δ p Z α Δ q d x d τ ≥ c ∫ 0 t ∥ Δ q ∥ H 2 2 d τ − C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t ( P ( E m ( τ ) ) + ∥ Δ q ∥ H 1 2 ) d τ . (158) \begin{aligned}
\frac{1}{2\mu+\lambda}\sum_{|\alpha|\le2}\int_{0}^{t}\int_{\Omega}Z^\alpha\Delta pZ^\alpha\Delta q\,dx\,d\tau
\ge{}&c\int_{0}^{t}\|\Delta q\|_{H^2}^{2}\,d\tau\\
&-C_{m+2}[1+P(Q(t))]\int_{0}^{t}\left(P(\mathcal{E}_m(\tau))+\|\Delta q\|_{H^1}^{2}\right)\,d\tau. \tag*{(158)}
\end{aligned} 2 μ + λ 1 ∣ α ∣ ≤ 2 ∑ ∫ 0 t ∫ Ω Z α Δ p Z α Δ q d x d τ ≥ c ∫ 0 t ∥Δ q ∥ H 2 2 d τ − C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t ( P ( E m ( τ )) + ∥Δ q ∥ H 1 2 ) d τ . (158) The remaining terms are estimated as in [34 (3.187)–(3.192)] . We give the estimates here for completeness. By the momentum equation, (129) , and Lemma 3.8 at the lower orders, we have
ε 2 ∥ Δ u ∥ L ∞ 2 ≤ C [ 1 + P ( Q ( t ) ) ] , \varepsilon^2\|\Delta u\|_{L^\infty}^{2}\le C[1+P(Q(t))], ε 2 ∥Δ u ∥ L ∞ 2 ≤ C [ 1 + P ( Q ( t ))] , ε 2 ∫ 0 t ∥ Δ u ∥ H 2 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( E m ( τ ) ) d τ . \varepsilon^2\int_{0}^{t}\|\Delta u\|_{H^2}^{2}\,d\tau
\le C_{m+2}[1+P(Q(t))]\int_{0}^{t}P(\mathcal{E}_m(\tau))\,d\tau. ε 2 ∫ 0 t ∥Δ u ∥ H 2 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t P ( E m ( τ )) d τ . The electric force ρ ∇ ϕ \rho\nabla\phi ρ ∇ ϕ (97) . Together with Proposition 2.3, the local decomposition of the Laplacian, and Young’s inequality, this gives
ε ∑ ∣ α ∣ ≤ 2 ∫ 0 t ∣ ∫ Ω Z α ( Δ u ⋅ ∇ q ) Z α Δ q d x ∣ d τ + 2 ε ∑ ∣ α ∣ ≤ 2 ∑ i = 1 3 ∫ 0 t ∣ ∫ Ω Z α ( ∂ i u ⋅ ∇ ∂ i q ) Z α Δ q d x ∣ d τ ≤ δ ∫ 0 t ∥ Δ q ∥ H 2 2 d τ + C δ ε 2 ∫ 0 t ∥ ∇ 2 u ∥ H 3 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t ( P ( E m ( τ ) ) + ε 2 ∥ Δ q ∥ H 2 2 + P ( ∥ Δ q ∥ H 1 2 ) ) d τ . (159) \begin{aligned}
\varepsilon\sum_{|\alpha|\leq2}\int_{0}^{t}\left|\int_{\Omega} Z^{\alpha}(\Delta u\cdot\nabla q)Z^{\alpha}\Delta q\,dx\right|\,d\tau\\
&+2\varepsilon\sum_{|\alpha|\leq2}\sum_{i=1}^{3}\int_{0}^{t}\left|\int_{\Omega} Z^{\alpha}(\partial_i u\cdot\nabla\partial_i q)Z^{\alpha}\Delta q\,dx\right|\,d\tau\\
&\leq\delta\int_{0}^{t}\|\Delta q\|_{\mathcal{H}^{2}}^{2}\,d\tau+C_{\delta}\varepsilon^{2}\int_{0}^{t}\|\nabla^{2}u\|_{\mathcal{H}^{3}}^{2}\,d\tau\\
&+C_{\delta}C_{m+2}[1+P(Q(t))]\int_{0}^{t}\left(P(\mathcal{E}_{m}(\tau))+\varepsilon^{2}\|\Delta q\|_{\mathcal{H}^{2}}^{2}+P(\|\Delta q\|_{\mathcal{H}^{1}}^{2})\right)\,d\tau. \tag*{(159)}
\end{aligned} ε ∣ α ∣ ≤ 2 ∑ ∫ 0 t ∫ Ω Z α ( Δ u ⋅ ∇ q ) Z α Δ q d x d τ + 2 ε ∣ α ∣ ≤ 2 ∑ i = 1 ∑ 3 ∫ 0 t ∫ Ω Z α ( ∂ i u ⋅ ∇ ∂ i q ) Z α Δ q d x d τ ≤ δ ∫ 0 t ∥Δ q ∥ H 2 2 d τ + C δ ε 2 ∫ 0 t ∥ ∇ 2 u ∥ H 3 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t ( P ( E m ( τ )) + ε 2 ∥Δ q ∥ H 2 2 + P ( ∥Δ q ∥ H 1 2 ) ) d τ . (159) For the transport term, integration by parts, u ⋅ n = 0 u\cdot n=0 u ⋅ n = 0 ( u ⋅ N ) / φ ( z ) (u\cdot N)/\varphi(z) ( u ⋅ N ) / φ ( z )
ε ∑ ∣ α ∣ ≤ 2 ∫ 0 t ∣ ∫ Ω Z α ( u ⋅ ∇ Δ q ) Z α Δ q d x ∣ d τ ≤ δ ∫ 0 t ∥ Δ q ∥ H 2 2 d τ + C δ ε 2 ∫ 0 t ∥ ∇ 2 u ∥ H 3 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t ( P ( E m ( τ ) ) + ε 2 ∥ Δ q ∥ H 2 2 ) d τ . (160) \begin{aligned}
\varepsilon\sum_{|\alpha|\leq2}\int_{0}^{t}\left|\int_{\Omega} Z^{\alpha}(u\cdot\nabla\Delta q)Z^{\alpha}\Delta q\,dx\right|\,d\tau\\
&\leq\delta\int_{0}^{t}\|\Delta q\|_{\mathcal{H}^{2}}^{2}\,d\tau+C_{\delta}\varepsilon^{2}\int_{0}^{t}\|\nabla^{2}u\|_{\mathcal{H}^{3}}^{2}\,d\tau\\
&+C_{\delta}C_{m+2}[1+P(Q(t))]\int_{0}^{t}\left(P(\mathcal{E}_{m}(\tau))+\varepsilon^{2}\|\Delta q\|_{\mathcal{H}^{2}}^{2}\right)\,d\tau. \tag*{(160)}
\end{aligned} ε ∣ α ∣ ≤ 2 ∑ ∫ 0 t ∫ Ω Z α ( u ⋅ ∇Δ q ) Z α Δ q d x d τ ≤ δ ∫ 0 t ∥Δ q ∥ H 2 2 d τ + C δ ε 2 ∫ 0 t ∥ ∇ 2 u ∥ H 3 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t ( P ( E m ( τ )) + ε 2 ∥Δ q ∥ H 2 2 ) d τ . (160) For the material derivative term, we use
div ( ρ u ˙ ) = ρ ( div u ) t + ρ u ⋅ ∇ div u + ∇ ρ ⋅ u ˙ + ρ ∑ i , j = 1 3 ∂ i u j ∂ j u i . \operatorname{div}(\rho\dot{u})=\rho(\operatorname{div}u)_{t}+\rho u\cdot\nabla\operatorname{div}u+\nabla\rho\cdot\dot{u}+\rho\sum_{i,j=1}^{3}\partial_i u_j\partial_j u_i. div ( ρ u ˙ ) = ρ ( div u ) t + ρ u ⋅ ∇ div u + ∇ ρ ⋅ u ˙ + ρ i , j = 1 ∑ 3 ∂ i u j ∂ j u i . Using (3.19), the Hardy inequality, and Proposition 2.2, we obtain
∫ 0 t ∥ div ( ρ u ˙ ) ∥ H 2 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( E m ( τ ) ) d τ . \int_{0}^{t}\|\operatorname{div}(\rho\dot{u})\|_{\mathcal{H}^{2}}^{2}\,d\tau\leq C_{m+2}[1+P(Q(t))]\int_{0}^{t}P(\mathcal{E}_{m}(\tau))\,d\tau. ∫ 0 t ∥ div ( ρ u ˙ ) ∥ H 2 2 d τ ≤ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t P ( E m ( τ )) d τ . Consequently,
∑ ∣ α ∣ ≤ 2 ∫ 0 t ∣ ∫ Ω Z α div ( ρ u ˙ ) Z α Δ q d x ∣ d τ ≤ δ ∫ 0 t ∥ Δ q ∥ H 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( E m ( τ ) ) d τ . (161) \begin{aligned}
\sum_{|\alpha|\leq2}\int_{0}^{t}\left|\int_{\Omega}Z^{\alpha}\operatorname{div}(\rho\dot{u})Z^{\alpha}\Delta q\,dx\right|\,d\tau\\
&\leq\delta\int_{0}^{t}\|\Delta q\|_{\mathcal{H}^{2}}^{2}\,d\tau+C_{\delta}C_{m+2}[1+P(Q(t))]\int_{0}^{t}P(\mathcal{E}_{m}(\tau))\,d\tau. \tag*{(161)}
\end{aligned} ∣ α ∣ ≤ 2 ∑ ∫ 0 t ∫ Ω Z α div ( ρ u ˙ ) Z α Δ q d x d τ ≤ δ ∫ 0 t ∥Δ q ∥ H 2 2 d τ + C δ C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t P ( E m ( τ )) d τ . (161) Combining (3.133)–(3.138) and choosing δ > 0 \delta>0 δ > 0
ε sup 0 ≤ τ ≤ t ∥ Δ q ( τ ) ∥ H 2 2 + c ∫ 0 t ∥ Δ q ∥ H 2 2 d τ ≤ C ε ∥ Δ q 0 ∥ H 2 2 + C ε 2 ∫ 0 t ∥ ∇ 2 u ∥ H 3 2 d τ + C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t ( ε ∥ Δ q ∥ H 2 2 + P ( ∥ Δ q ∥ H 1 ) + P ( E m ( τ ) ) ) d τ . (162) \begin{aligned}
\varepsilon\sup_{0\leq\tau\leq t}\|\Delta q(\tau)\|_{\mathcal{H}^{2}}^{2}+c\int_{0}^{t}\|\Delta q\|_{\mathcal{H}^{2}}^{2}\,d\tau\\
&\leq C\varepsilon\|\Delta q_{0}\|_{\mathcal{H}^{2}}^{2}+C\varepsilon^{2}\int_{0}^{t}\|\nabla^{2}u\|_{\mathcal{H}^{3}}^{2}\,d\tau\\
&+C_{m+2}[1+P(Q(t))]\int_{0}^{t}\left(\varepsilon\|\Delta q\|_{\mathcal{H}^{2}}^{2}+P(\|\Delta q\|_{\mathcal{H}^{1}})+P(\mathcal{E}_{m}(\tau))\right)\,d\tau. \tag*{(162)}
\end{aligned} ε 0 ≤ τ ≤ t sup ∥Δ q ( τ ) ∥ H 2 2 + c ∫ 0 t ∥Δ q ∥ H 2 2 d τ ≤ C ε ∥Δ q 0 ∥ H 2 2 + C ε 2 ∫ 0 t ∥ ∇ 2 u ∥ H 3 2 d τ + C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t ( ε ∥Δ q ∥ H 2 2 + P ( ∥Δ q ∥ H 1 ) + P ( E m ( τ )) ) d τ . (162) On the other hand, since ∂ t \partial_t ∂ t
sup 0 ≤ τ ≤ t ∥ Δ q ( τ ) ∥ H 1 2 ≤ ∥ Δ q 0 ∥ H 1 2 + 2 ∫ 0 t ∥ Δ q ∥ H 1 ∥ ∂ t Δ q ∥ H 1 d τ ≤ ∥ Δ q 0 ∥ H 1 2 + δ ∫ 0 t ∥ Δ q ∥ H 2 2 d τ + C δ ∫ 0 t ∥ Δ q ∥ H 1 2 d τ . (163) \begin{aligned}
\sup_{0\leq\tau\leq t}\|\Delta q(\tau)\|_{\mathcal{H}^{1}}^{2}\leq\|\Delta q_{0}\|_{\mathcal{H}^{1}}^{2}+2\int_{0}^{t}\|\Delta q\|_{\mathcal{H}^{1}}\|\partial_t\Delta q\|_{\mathcal{H}^{1}}\,d\tau\\
&\leq\|\Delta q_{0}\|_{\mathcal{H}^{1}}^{2}+\delta\int_{0}^{t}\|\Delta q\|_{\mathcal{H}^{2}}^{2}\,d\tau+C_{\delta}\int_{0}^{t}\|\Delta q\|_{\mathcal{H}^{1}}^{2}\,d\tau. \tag*{(163)}
\end{aligned} 0 ≤ τ ≤ t sup ∥Δ q ( τ ) ∥ H 1 2 ≤ ∥Δ q 0 ∥ H 1 2 + 2 ∫ 0 t ∥Δ q ∥ H 1 ∥ ∂ t Δ q ∥ H 1 d τ ≤ ∥Δ q 0 ∥ H 1 2 + δ ∫ 0 t ∥Δ q ∥ H 2 2 d τ + C δ ∫ 0 t ∥Δ q ∥ H 1 2 d τ . (163) Taking δ < c / 2 \delta< c/2 δ < c /2 (163) to (162) , and using (124) , we get
sup 0 ≤ τ ≤ t ( ∥ Δ q ( τ ) ∥ H 1 2 + ε ∥ Δ q ( τ ) ∥ H 2 2 ) + ∫ 0 t ∥ Δ q ∥ H 2 2 d τ ≤ C ( ∥ Δ q 0 ∥ H 1 2 + ε ∥ Δ q 0 ∥ H 2 2 + E m ( 0 ) ) (164) \sup_{0\leq\tau\leq t}\left(\|\Delta q(\tau)\|_{H^1}^2+\varepsilon\|\Delta q(\tau)\|_{H^2}^2\right)+\int_0^t\|\Delta q\|_{H^2}^2\,\mathrm{d}\tau
\leq C\left(\|\Delta q_0\|_{H^1}^2+\varepsilon\|\Delta q_0\|_{H^2}^2+\mathcal{E}_m(0)\right) \tag*{(164)} 0 ≤ τ ≤ t sup ( ∥Δ q ( τ ) ∥ H 1 2 + ε ∥Δ q ( τ ) ∥ H 2 2 ) + ∫ 0 t ∥Δ q ∥ H 2 2 d τ ≤ C ( ∥Δ q 0 ∥ H 1 2 + ε ∥Δ q 0 ∥ H 2 2 + E m ( 0 ) ) (164) + C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t ( ε ∥ Δ q ∥ H 2 2 + P ( ∥ Δ q ∥ H 1 2 ) + P ( E m ( τ ) ) ) d τ . {}+C_{m+2}[1+P(Q(t))]\int_0^t\left(\varepsilon\|\Delta q\|_{H^2}^2+P(\|\Delta q\|_{H^1}^2)+P(\mathcal{E}_m(\tau))\right)\,\mathrm{d}\tau. + C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t ( ε ∥Δ q ∥ H 2 2 + P ( ∥Δ q ∥ H 1 2 ) + P ( E m ( τ )) ) d τ . It remains to return from q q q p p p p = e γ q p=e^{\gamma q} p = e γ q [34 (3.148)] ,
∥ Δ q ∥ H 1 2 ≤ C [ P ( ∥ Δ p ∥ H 1 2 ) + P ( E m ) ] , \|\Delta q\|_{H^1}^2\leq C\left[P\left(\|\Delta p\|_{H^1}^2\right)+P(\mathcal{E}_m)\right], ∥Δ q ∥ H 1 2 ≤ C [ P ( ∥Δ p ∥ H 1 2 ) + P ( E m ) ] , ∥ Δ p ∥ H 1 2 ≤ C [ P ( ∥ Δ q ∥ H 1 2 ) + P ( E m ) ] , \|\Delta p\|_{H^1}^2\leq C\left[P\left(\|\Delta q\|_{H^1}^2\right)+P(\mathcal{E}_m)\right], ∥Δ p ∥ H 1 2 ≤ C [ P ( ∥Δ q ∥ H 1 2 ) + P ( E m ) ] , ∥ Δ q ∥ H 2 2 ≤ C [ ∥ Δ p ∥ H 2 2 + P ( ∥ Δ q ∥ H 1 2 ) + P ( E m ) ] , \|\Delta q\|_{H^2}^2\leq C\left[\|\Delta p\|_{H^2}^2+P\left(\|\Delta q\|_{H^1}^2\right)+P(\mathcal{E}_m)\right], ∥Δ q ∥ H 2 2 ≤ C [ ∥Δ p ∥ H 2 2 + P ( ∥Δ q ∥ H 1 2 ) + P ( E m ) ] , ∥ Δ p ∥ H 2 2 ≤ C [ ∥ Δ q ∥ H 2 2 + P ( ∥ Δ q ∥ H 1 2 ) + P ( E m ) ] . (165) \|\Delta p\|_{H^2}^2\leq C\left[\|\Delta q\|_{H^2}^2+P\left(\|\Delta q\|_{H^1}^2\right)+P(\mathcal{E}_m)\right]. \tag*{(165)} ∥Δ p ∥ H 2 2 ≤ C [ ∥Δ q ∥ H 2 2 + P ( ∥Δ q ∥ H 1 2 ) + P ( E m ) ] . (165) Thus the initial terms in (164) are bounded by C P ( ∥ ( p 0 , u 0 , ∇ ϕ 0 ) ∥ X m 2 ) CP(\|(p_0,u_0,\nabla\phi_0)\|_{\mathcal{X}_m}^2) C P ( ∥ ( p 0 , u 0 , ∇ ϕ 0 ) ∥ X m 2 )
C m + 2 [ 1 + P ( Q ( t ) ) ] ∫ 0 t P ( ∥ ( p , u , ∇ ϕ ) ( τ ) ∥ X m 2 ) d τ . C_{m+2}[1+P(Q(t))]\int_0^t P(\|(p,u,\nabla\phi)(\tau)\|_{\mathcal{X}_m}^2)\,\mathrm{d}\tau. C m + 2 [ 1 + P ( Q ( t ))] ∫ 0 t P ( ∥ ( p , u , ∇ ϕ ) ( τ ) ∥ X m 2 ) d τ . Finally, applying (165) to the endpoint norms and the time integral, using (124) , and enlarging P P P (153) . This completes the proof.
Proofs of Theorem 1.1 and Theorem 1.2
With the uniform a priori estimates established in Section 3, the proofs of the local existence of solutions on a time interval [ 0 , T 0 ] [0,T_0] [ 0 , T 0 ] ε \varepsilon ε [22 ] . Indeed, one may first mollify the initial data so that the classical local well-posedness theory for the viscous problem applies. The uniform estimates obtained in Section 3, together with a continuous induction argument, yield a lifespan independent of ε \varepsilon ε ε → 0 \varepsilon\to0 ε → 0
Acknowledgements
This research is supported by the National Natural Science Foundation of China (No. 12471198, No. 12431018).
Data availability
Data sharing is not applicable to this article as no data sets were generated or analysed during the current study.
