Introduction
The motion of a conducting fluid through a magnetic field induces electric currents, which in turn generate secondary magnetic fields. This magneto-induction mechanism is described by the magnetic induction equation and underlies a wide range of phenomena in geophysics, astrophysics, space physics, and industrial applications; see, for example, [1 , 2 , 3 , 4 , 5 ] . In many of these settings, direct measurements of the fluid velocity are difficult or impractical because of the inaccessibility of the flow, the opacity of the medium, or the extreme physical conditions. Magnetic measurements therefore provide a potentially valuable indirect means of recovering the underlying fluid motion. This motivates the inverse problem of reconstructing a velocity field from magnetic field observations, which has received considerable attention in the literature; see, e.g., [6 , 7 , 8 , 9 ] .
The evolution of electromagnetic fields in a conducting medium is governed by Maxwell’s equations. Under the magnetohydrodynamic approximation, neglecting displacement currents and free-charge accumulation, and using Ohm’s law
J = σ ( E + v × B ) , \mathbf J = \sigma(\mathbf E + \mathbf v \times\mathbf B), J = σ ( E + v × B ) , one obtains the magnetic induction equation
∂ t B − η Δ B = ∇ × ( v × B ) , ∇ ⋅ B = 0 , η = 1 μ 0 σ , (1) \partial_t\mathbf B - \eta\Delta\mathbf B = \nabla\times(\mathbf v \times\mathbf B), \qquad\nabla\cdot\mathbf B = 0, \qquad\eta= \frac{1}{\mu_0\sigma}, \tag*{(1)} ∂ t B − η Δ B = ∇ × ( v × B ) , ∇ ⋅ B = 0 , η = μ 0 σ 1 , (1) where v \mathbf v v B \mathbf B B σ > 0 \sigma> 0 σ > 0 μ 0 > 0 \mu_0 > 0 μ 0 > 0 η > 0 \eta> 0 η > 0 [6 , 7 ] . Throughout this work, we set η = 1 \eta= 1 η = 1
We consider a connected bounded domain Ω ⊂ R 3 \Omega\subset\mathbb{R}^3 Ω ⊂ R 3 ∂ Ω \partial\Omega ∂ Ω T > 0 T > 0 T > 0
B ⋅ n = g 1 , n × ( ∇ × B ) = g 2 on ( 0 , T ) × ∂ Ω , (2) \mathbf B \cdot\mathbf n = g_1, \qquad\mathbf n \times(\nabla\times\mathbf B) = g_2 \quad\text{on } (0,T) \times\partial\Omega, \tag*{(2)} B ⋅ n = g 1 , n × ( ∇ × B ) = g 2 on ( 0 , T ) × ∂ Ω , (2) where n \mathbf n n g 1 g_1 g 1 g 2 g_2 g 2
∇ × B = μ 0 J , \nabla\times\mathbf B = \mu_0\mathbf J, ∇ × B = μ 0 J , the latter condition is equivalent to prescribing the tangential component of the current density, up to the factor μ 0 \mu_{0} μ 0 [10 , 11 ] . In the Hodge-theoretic formulation, the homogeneous conditions
B ⋅ n = 0 , n × ( ∇ × B ) = 0 on ∂ Ω \mathbf{B} \cdot\mathbf{n} = 0,\qquad\mathbf{n} \times(\nabla\times\mathbf{B}) = 0\quad\text{on }\partial\Omega B ⋅ n = 0 , n × ( ∇ × B ) = 0 on ∂ Ω are the magnetic (absolute) boundary conditions for the Hodge Laplacian on divergence-free vector fields. Their physical interpretation depends on the electromagnetic model and the assumptions imposed on the wall; they should therefore not be identified indiscriminately with every formulation of a perfectly conducting boundary. For the latter distinction and the associated MHD boundary-value formulations, see [4 , 12 ] . The boundary data must also satisfy the compatibility conditions imposed by the divergence constraint and the evolution equation.
The inverse problem studied here is to determine whether the velocity field v \mathbf{v} v L \mathcal{L} L B \mathbf{B} B
L ( B ) . \mathcal{L}(\mathbf{B}). L ( B ) . Depending on the observation configuration, L \mathcal{L} L
Velocity reconstruction from electromagnetic measurements has been investigated through analytical, numerical, and experimental approaches. The reconstruction problem using magnetic-field and electric-potential measurements was formulated in [6 ] . Non-uniqueness of the reconstruction problem was subsequently analyzed in [7 ] , where a regularization based on minimizing the kinetic energy of the velocity field was proposed as a selection criterion. A contactless method for velocity reconstruction in electrically conducting fluids was developed in [13 ] , and theoretical and experimental aspects of inverse problems in magnetohydrodynamics were further investigated in [14 ] . A finite-element-based Bayesian approach to reconstructing the longitudinal velocity component in electromagnetic flow tomography was proposed in [8 ] , but no uniqueness or stability results are proved there. It is worth mentioning that recently, in [15 ] , a Carleman estimate was established for the full MHD system; however, that work recovers a source term that is independent of the magnetic field or the velocity of the fluid.
More recently, [9 ] established uniqueness criteria for the linearized induction equation in whole-space and periodic settings with constant background magnetic field.
Contribution
We prove stability and uniqueness for velocity reconstruction in a bounded domain from partial parabolic observations of the magnetic field. Magnetic observations alone are not sufficient for uniqueness; prescribing the velocity on the inflow boundary together with the magnetic boundary conditions restores uniqueness under suitable conditions on the space-dependent background field and bounded-domain transport geometry. The problem has a hybrid parabolic-hyperbolic structure: magnetic diffusion is parabolic, while velocity transport along the background field is hyperbolic. This motivates a two-stage reconstruction and yields a hybrid stability estimate combining a Lipschitz contribution from the inflow trace with a weaker modulus from the parabolic observation. Unlike previous work in unbounded or periodic settings with constant background fields, we allow a space-dependent background field.
Organization of the paper
This paper is organized as follows. In Section 2, we present the linearized model around a background magnetic field, the existence result for the forward problem, and the statement of the inverse problem. In Section 3, we present the geometric and functional tools needed to announce and prove the main results for the inverse problem. In Section 4, we establish a well-posedness result for a curl–div system that is required in the proof of the main results. In Section 5, we state the main results for the inverse problem. In Section 6, we prove the main results, and in Section 7, we present the numerical validation of both the stability and uniqueness results.
The model
The nonlinearity of the induction equation makes direct analysis of the associated inverse problem difficult. Following the linearization strategy commonly used in inverse problems for magnetohydrodynamics [6 , 7 ] , we decompose the magnetic field as
B ( t , x ) = F ( x ) + b ( t , x ) , (3) B(t,x)=F(x)+b(t,x), \tag*{(3)} B ( t , x ) = F ( x ) + b ( t , x ) , (3) where F F F b b b
F ∈ [ C 2 ( Ω ‾ ) ] 3 , ∇ ⋅ F = 0 , in Ω . (4) F\in[C^{2}(\overline{\Omega})]^{3}, \qquad\nabla\cdot F=0, \qquad\text{in }\Omega. \tag*{(4)} F ∈ [ C 2 ( Ω ) ] 3 , ∇ ⋅ F = 0 , in Ω. (4) Since B B B F F F
∇ ⋅ b = ∇ ⋅ B − ∇ ⋅ F = 0. \nabla\cdot b=\nabla\cdot B-\nabla\cdot F=0. ∇ ⋅ b = ∇ ⋅ B − ∇ ⋅ F = 0. Recall from the introduction that the total magnetic field satisfies the boundary conditions
B ⋅ n = g 1 , n × ( ∇ × B ) = g 2 , on ( 0 , T ) × ∂ Ω . (5) B\cdot n=g_{1}, \qquad n\times(\nabla\times B)=g_{2}, \qquad\text{on }(0,T)\times\partial\Omega. \tag*{(5)} B ⋅ n = g 1 , n × ( ∇ × B ) = g 2 , on ( 0 , T ) × ∂ Ω. (5) Because the decomposition (3) separates the known background field from the unknown perturbation, we impose the boundary condition on the total field B B B F F F
g 1 : = F ⋅ n , g 2 : = n × ( ∇ × F ) . (6) g_{1}:=F\cdot n, \qquad g_{2}:=n\times(\nabla\times F). \tag*{(6)} g 1 := F ⋅ n , g 2 := n × ( ∇ × F ) . (6) Consequently the perturbation satisfies homogeneous absolute boundary conditions
b ⋅ n = 0 , n × ( ∇ × b ) = 0 , on ( 0 , T ) × ∂ Ω . b\cdot n=0, \qquad n\times(\nabla\times b)=0, \qquad\text{on }(0,T)\times\partial\Omega. b ⋅ n = 0 , n × ( ∇ × b ) = 0 , on ( 0 , T ) × ∂ Ω. This choice is natural: the background field already satisfies the perfectly-conducting-wall or symmetry boundary conditions, and the perturbation is the difference between the total field and this known background. Hence the boundary data of the total field are precisely those of the background, so the perturbation inherits boundary data. Physically, the linearized perturbation does not alter the normal magnetic flux or the tangential current density at the boundary.
Substituting (3) into the induction equation, using that F F F (4) , and dropping the bilinear term ∇ × ( v × b ) \nabla\times(v\times b) ∇ × ( v × b )
∂ t b − Δ b = Δ F + ∇ × ( v × F ) , in ( 0 , T ) × Ω , ∇ ⋅ b = 0 , in ( 0 , T ) × Ω , b ⋅ n = 0 , n × ( ∇ × b ) = 0 , on ( 0 , T ) × ∂ Ω , b ∣ t = 0 = b 0 , in Ω , (7) \begin{aligned}
\partial_t b-\Delta b&=\Delta F+\nabla\times(v\times F), &&\text{in }(0,T)\times\Omega,\\
\nabla\cdot b&=0, &&\text{in }(0,T)\times\Omega,\\
b\cdot n&=0,\qquad n\times(\nabla\times b)=0, &&\text{on }(0,T)\times\partial\Omega,\\
b|_{t=0}&=b_0, &&\text{in }\Omega, \tag*{(7)}
\end{aligned} ∂ t b − Δ b ∇ ⋅ b b ⋅ n b ∣ t = 0 = Δ F + ∇ × ( v × F ) , = 0 , = 0 , n × ( ∇ × b ) = 0 , = b 0 , in ( 0 , T ) × Ω , in ( 0 , T ) × Ω , on ( 0 , T ) × ∂ Ω , in Ω , (7) where b 0 = B 0 − F b_0=B_0-F b 0 = B 0 − F
The Hodge Laplacian
It has been noted in several works, see, for example, [16 , 17 , 18 ] , that, in Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3
− Δ H u : = − Δ u = ∇ × ( ∇ × u ) − ∇ ( ∇ ⋅ u ) , (8) -\Delta_Hu:=-\Delta u=\nabla\times(\nabla\times u)-\nabla(\nabla\cdot u), \tag*{(8)} − Δ H u := − Δ u = ∇ × ( ∇ × u ) − ∇ ( ∇ ⋅ u ) , (8) with the boundary conditions
u ⋅ n = 0 , n × ( ∇ × u ) = 0 on ∂ Ω , (9) u\cdot n=0,\qquad n\times(\nabla\times u)=0 \qquad\text{on }\partial\Omega, \tag*{(9)} u ⋅ n = 0 , n × ( ∇ × u ) = 0 on ∂ Ω , (9) is called the Hodge Laplacian .
We introduce the solenoidal Hilbert space
H : = { u ∈ [ L 2 ( Ω ) ] 3 : ∇ ⋅ u = 0 in Ω , u ⋅ n = 0 on ∂ Ω } . (10) \mathbb{H} := \left\{ u \in[L^{2}(\Omega)]^{3} : \nabla\cdot u = 0 \text{ in } \Omega,\quad u \cdot n = 0 \text{ on } \partial\Omega\right\}. \tag*{(10)} H := { u ∈ [ L 2 ( Ω ) ] 3 : ∇ ⋅ u = 0 in Ω , u ⋅ n = 0 on ∂ Ω } . (10) Let
P : [ L 2 ( Ω ) ] 3 ⟶ H \mathbb{P} : [L^{2}(\Omega)]^{3} \longrightarrow\mathbb{H} P : [ L 2 ( Ω ) ] 3 ⟶ H denote the Helmholtz–Leray orthogonal projection.
A key property of the Hodge realization is its compatibility with P \mathbb{P} P R 3 \mathbb{R}^{3} R 3 u u u
P u ∈ D ( Δ H ) , − P Δ u = − Δ H P u = ∇ × ( ∇ × u ) , (11) \mathbb{P}u \in D(\Delta_{\mathrm{H}}), \qquad-\mathbb{P}\Delta u = -\Delta_{\mathrm{H}}\mathbb{P}u = \nabla\times(\nabla\times u), \tag*{(11)} P u ∈ D ( Δ H ) , − P Δ u = − Δ H P u = ∇ × ( ∇ × u ) , (11) and P u \mathbb{P}u P u
( P u ) ⋅ n = 0 , n × ( ∇ × P u ) = 0 on ∂ Ω . (12) (\mathbb{P}u) \cdot n = 0, \qquad n \times(\nabla\times\mathbb{P}u) = 0 \qquad\text{on } \partial\Omega. \tag*{(12)} ( P u ) ⋅ n = 0 , n × ( ∇ × P u ) = 0 on ∂ Ω. (12) Thus, the compatibility is a property of the operator realizations and their domains, rather than an unrestricted commutation relation P Δ = Δ P \mathbb{P}\Delta= \Delta\mathbb{P} P Δ = Δ P [ L 2 ( Ω ) ] 3 [L^{2}(\Omega)]^{3} [ L 2 ( Ω ) ] 3 [18 ] .
We consider the realization of the Hodge Laplacian on H \mathbb{H} H
D ( Δ H ) : = { u ∈ H : Δ u ∈ [ L 2 ( Ω ) ] 3 , n × ( ∇ × u ) = 0 on ∂ Ω } . (13) D(\Delta_{\mathrm{H}}) := \left\{ u \in\mathbb{H} : \Delta u \in[L^{2}(\Omega)]^{3},\quad n \times(\nabla\times u) = 0 \text{ on } \partial\Omega\right\}. \tag*{(13)} D ( Δ H ) := { u ∈ H : Δ u ∈ [ L 2 ( Ω ) ] 3 , n × ( ∇ × u ) = 0 on ∂ Ω } . (13) Since ∇ ⋅ u = 0 \nabla\cdot u = 0 ∇ ⋅ u = 0 u ∈ D ( Δ H ) u \in D(\Delta_{\mathrm{H}}) u ∈ D ( Δ H )
− Δ H u = − Δ u = ∇ × ( ∇ × u ) . (14) -\Delta_{\mathrm{H}}u = -\Delta u = \nabla\times(\nabla\times u). \tag*{(14)} − Δ H u = − Δ u = ∇ × ( ∇ × u ) . (14) Hence the divergence-free constraint and the boundary conditions are incorporated directly into the Hodge–Laplacian realization. The evolution problem can therefore be formulated directly on H \mathbb{H} H
Well-posedness
For the linearized induction system (7) , we set
h ( t ) = Δ F + ∇ × ( v ( t ) × F ) , t ∈ ( 0 , T ) . (15) h(t) = \Delta F + \nabla\times(v(t) \times F), \qquad t \in(0,T). \tag*{(15)} h ( t ) = Δ F + ∇ × ( v ( t ) × F ) , t ∈ ( 0 , T ) . (15) The Hodge–Laplacian realization Δ H \Delta_{\mathrm{H}} Δ H H \mathbb{H} H Δ H \Delta_{\mathrm{H}} Δ H C 0 C_{0} C 0 H \mathbb{H} H [19 Theorem 7.1] . We have the following result [20 Theorem 3.2]
Theorem 2.1. Let T > 0 T > 0 T > 0 Ω \Omega Ω
h ∈ L 2 ( 0 , T ; H ) , b 0 ∈ D ( Δ H 1 2 ) . h \in L^{2}(0,T;\mathbb{H}), \qquad b_{0} \in D(\Delta_{\mathrm{H}}^{\frac{1}{2}}). h ∈ L 2 ( 0 , T ; H ) , b 0 ∈ D ( Δ H 2 1 ) . Then the problem (7) admits a unique strong solution
b ∈ L 2 ( [ 0 , T ] ; D ( Δ H ) ) ∩ H 1 ( 0 , T ; H ) . b \in L^{2}([0,T];D(\Delta_{\mathrm{H}})) \cap H^{1}(0,T;\mathbb{H}). b ∈ L 2 ([ 0 , T ] ; D ( Δ H )) ∩ H 1 ( 0 , T ; H ) . We therefore define the state space
Y : = { b ∈ L 2 ( [ 0 , T ] ; D ( Δ H ) ) ∩ H 1 ( 0 , T ; H ) } . (16) \mathbb{Y} := \left\{ b \in L^{2}([0,T];D(\Delta_{\mathrm{H}})) \cap H^{1}(0,T;\mathbb{H}) \right\}. \tag*{(16)} Y := { b ∈ L 2 ([ 0 , T ] ; D ( Δ H )) ∩ H 1 ( 0 , T ; H ) } . (16) Remark 2.2. By definition of the source term h h h (15) , the source term is required to belong to H \mathbb{H} H
∇ ⋅ h ( t ) = ∇ ⋅ ( Δ F + ∇ × ( v ( t ) × F ) ) = ( Δ ( ∇ ⋅ F ) + ∇ ⋅ ( ∇ × ( v ( t ) × F ) ) ) = 0 , a.e. on ( 0 , T ) , \nabla\cdot h(t) = \nabla\cdot\left(\Delta F + \nabla\times(v(t) \times F)\right) = \left(\Delta(\nabla\cdot F) + \nabla\cdot\left(\nabla\times(v(t) \times F)\right)\right) = 0,\qquad\text{a.e. on } (0,T), ∇ ⋅ h ( t ) = ∇ ⋅ ( Δ F + ∇ × ( v ( t ) × F ) ) = ( Δ ( ∇ ⋅ F ) + ∇ ⋅ ( ∇ × ( v ( t ) × F ) ) ) = 0 , a.e. on ( 0 , T ) , holds identically, the remaining condition h ⋅ n = 0 h \cdot n = 0 h ⋅ n = 0 ∂ Ω \partial\Omega ∂ Ω
n ⋅ ∇ × ( v ( t ) × F ) = − n ⋅ Δ F on ∂ Ω for a.e. on ( 0 , T ) . n \cdot\nabla\times(v(t) \times F) = -n \cdot\Delta F \qquad\text{on } \partial\Omega\text{ for a.e. on } (0,T). n ⋅ ∇ × ( v ( t ) × F ) = − n ⋅ Δ F on ∂ Ω for a.e. on ( 0 , T ) . Statement of the inverse problem
We can now formulate the main inverse problem. Assume that v v v
v ( t , x ) = α ( t ) u ( x ) , v(t,x)=\alpha(t)u(x), v ( t , x ) = α ( t ) u ( x ) , where α ∈ L 2 ( 0 , T ) \alpha\in L^{2}(0,T) α ∈ L 2 ( 0 , T ) ( 0 , T ) (0,T) ( 0 , T )
Problem 2.3 (Inverse Problem: Recovering v v v L ( y ) \mathcal{L}(y) L ( y ) u u u Ω \Omega Ω L ( y ) \mathcal{L}(y) L ( y ) u u u
First, observe that uniqueness of source recovery in Problem 2.3 cannot hold for arbitrary v v v z ∈ H z \in\mathbb{H} z ∈ H
u ⟼ ∇ × ( u × F ) , (17) u \longmapsto\nabla\times(u \times F), \tag*{(17)} u ⟼ ∇ × ( u × F ) , (17) possesses a nontrivial kernel. Specifically, let c ( x ) c(x) c ( x ) ∇ ⋅ ( c F ) = 0 \nabla\cdot(cF)=0 ∇ ⋅ ( c F ) = 0
∇ ⋅ ( c F ) = F ⋅ ∇ c = F 1 ∂ 1 c + F 2 ∂ 2 c + F 3 ∂ 3 c = 0 , in Ω . \nabla\cdot(cF)=F\cdot\nabla c=F_{1}\partial_{1}c+F_{2}\partial_{2}c+F_{3}\partial_{3}c=0,\qquad\text{in }\Omega. ∇ ⋅ ( c F ) = F ⋅ ∇ c = F 1 ∂ 1 c + F 2 ∂ 2 c + F 3 ∂ 3 c = 0 , in Ω. The existence of such a function c c c
( u + c F ) × F = u × F , t ∈ ( 0 , T ) , (u+cF)\times F=u\times F,\qquad t\in(0,T), ( u + c F ) × F = u × F , t ∈ ( 0 , T ) , and consequently,
∇ × ( ( u + c F ) × F ) = ∇ × ( u × F ) . (18) \nabla\times((u+cF)\times F)=\nabla\times(u\times F). \tag*{(18)} ∇ × (( u + c F ) × F ) = ∇ × ( u × F ) . (18) Thus, any component of v v v F F F L \mathcal{L} L
Using the vector identity
∇ × ( A × B ) = A ( ∇ ⋅ B ) − B ( ∇ ⋅ A ) + ( B ⋅ ∇ ) A − ( A ⋅ ∇ ) B , \nabla\times(A\times B)=A(\nabla\cdot B)-B(\nabla\cdot A)+(B\cdot\nabla)A-(A\cdot\nabla)B, ∇ × ( A × B ) = A ( ∇ ⋅ B ) − B ( ∇ ⋅ A ) + ( B ⋅ ∇ ) A − ( A ⋅ ∇ ) B , with A = u A=u A = u B = F B=F B = F ∇ ⋅ u = 0 \nabla\cdot u=0 ∇ ⋅ u = 0 ∇ ⋅ F = 0 \nabla\cdot F=0 ∇ ⋅ F = 0
u ⟼ ∇ × ( u × F ) = ( F ⋅ ∇ ) u − ( u ⋅ ∇ ) F , t ∈ ( 0 , T ) . (19) u\longmapsto\nabla\times(u\times F)=(F\cdot\nabla)u-(u\cdot\nabla)F,\qquad t\in(0,T). \tag*{(19)} u ⟼ ∇ × ( u × F ) = ( F ⋅ ∇ ) u − ( u ⋅ ∇ ) F , t ∈ ( 0 , T ) . (19) To make this structural non-uniqueness explicit, recall that by the identity 19 the operator 17 is a first-order transport operator along the characteristics of F F F c F cF c F
To eliminate this ambiguity, we supplement the internal state observations with measurements of u u u
Geometric and Functional settings for the inverse problem
Geometric setting
Let Γ i n ⊂ ∂ Ω \Gamma_{\mathrm{in}}\subset\partial\Omega Γ in ⊂ ∂ Ω F F F
Γ i n : = { x ∈ ∂ Ω : F ( x ) ⋅ n ( x ) < 0 } , (20) \Gamma_{\mathrm{in}}:=\{x\in\partial\Omega:F(x)\cdot n(x)<0\}, \tag*{(20)} Γ in := { x ∈ ∂ Ω : F ( x ) ⋅ n ( x ) < 0 } , (20) where n n n ∂ Ω \partial\Omega ∂ Ω
Since F ∈ [ C 2 ( Ω ‾ ) ] 3 F\in[C^{2}(\overline{\Omega})]^{3} F ∈ [ C 2 ( Ω ) ] 3 F F F C 2 C^{2} C 2 R 3 \mathbb{R}^{3} R 3
X ∈ C 1 ( R × R 3 ; R 3 ) X\in C^{1}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3}) X ∈ C 1 ( R × R 3 ; R 3 ) is uniquely defined by
∂ t X ( t , x 0 ) = F ( X ( t , x 0 ) ) , X ( 0 , x 0 ) = x 0 . (21) \partial_{t}X(t,x_{0})=F(X(t,x_{0})),\qquad X(0,x_{0})=x_{0}. \tag*{(21)} ∂ t X ( t , x 0 ) = F ( X ( t , x 0 )) , X ( 0 , x 0 ) = x 0 . (21) Definition 3.1 (Ω \Omega Ω F F F ( t , x 0 ) ↦ X ( t , x 0 ) (t,x_0) \mapsto X(t,x_0) ( t , x 0 ) ↦ X ( t , x 0 ) Γ i n \Gamma_{\mathrm{in}} Γ in σ \sigma σ τ ( x 0 ) > 0 \tau(x_0)>0 τ ( x 0 ) > 0
D : = { ( t , x 0 ) : x 0 ∈ Γ i n , 0 < t < τ ( x 0 ) } . \mathcal{D}:=\{(t,x_0):x_0\in\Gamma_{\mathrm{in}},\ 0<t<\tau(x_0)\}. D := {( t , x 0 ) : x 0 ∈ Γ in , 0 < t < τ ( x 0 )} . The flow is Ω \Omega Ω T f i l l > 0 T_{\mathrm{fill}}>0 T fill > 0 N ⊂ Ω \mathcal{N}\subset\Omega N ⊂ Ω x ∈ Ω ∖ N x\in\Omega\setminus\mathcal{N} x ∈ Ω ∖ N ( x 0 , τ ) (x_0,\tau) ( x 0 , τ )
x = X ( τ , x 0 ) , x 0 ∈ Γ i n , 0 < τ ≤ T f i l l , X ( t , x 0 ) ∈ Ω ∀ 0 < t < τ . x=X(\tau,x_0),\qquad x_0\in\Gamma_{\mathrm{in}},\quad0<\tau\leq T_{\mathrm{fill}},\quad X(t,x_0)\in\Omega\ \forall\ 0<t<\tau. x = X ( τ , x 0 ) , x 0 ∈ Γ in , 0 < τ ≤ T fill , X ( t , x 0 ) ∈ Ω ∀ 0 < t < τ . Moreover, τ ( x 0 ) ≤ T f i l l \tau(x_0)\leq T_{\mathrm{fill}} τ ( x 0 ) ≤ T fill σ \sigma σ x 0 ∈ Γ i n x_0\in\Gamma_{\mathrm{in}} x 0 ∈ Γ in
Lemma 3.2 (Injectivity of the characteristic flow). The flow
X : D → Ω , ( t , x 0 ) ↦ X ( t , x 0 ) , X:\mathcal{D}\to\Omega,\qquad(t,x_0)\mapsto X(t,x_0), X : D → Ω , ( t , x 0 ) ↦ X ( t , x 0 ) , is injective.
Proof. Suppose
X ( t 1 , x 1 ) = X ( t 2 , x 2 ) X(t_1,x_1)=X(t_2,x_2) X ( t 1 , x 1 ) = X ( t 2 , x 2 ) with ( t 1 , x 1 ) , ( t 2 , x 2 ) ∈ D (t_1,x_1),(t_2,x_2)\in\mathcal{D} ( t 1 , x 1 ) , ( t 2 , x 2 ) ∈ D t 1 ≥ t 2 t_1\geq t_2 t 1 ≥ t 2 t 1 > t 2 t_1>t_2 t 1 > t 2 s = t 1 − t 2 > 0 s=t_1-t_2>0 s = t 1 − t 2 > 0
x 2 = X ( s , x 1 ) . x_2=X(s,x_1). x 2 = X ( s , x 1 ) . But
0 < s = t 1 − t 2 < t 1 < τ ( x 1 ) , 0<s=t_1-t_2<t_1<\tau(x_1), 0 < s = t 1 − t 2 < t 1 < τ ( x 1 ) , so X ( s , x 1 ) ∈ Ω X(s,x_1)\in\Omega X ( s , x 1 ) ∈ Ω x 2 ∈ Ω x_2\in\Omega x 2 ∈ Ω x 2 ∈ Γ i n ⊂ ∂ Ω x_2\in\Gamma_{\mathrm{in}}\subset\partial\Omega x 2 ∈ Γ in ⊂ ∂ Ω t 1 = t 2 t_1=t_2 t 1 = t 2
Now
X ( t 1 , x 1 ) = X ( t 1 , x 2 ) . X(t_1,x_1)=X(t_1,x_2). X ( t 1 , x 1 ) = X ( t 1 , x 2 ) . Since t 1 < τ ( x 1 ) t_1<\tau(x_1) t 1 < τ ( x 1 ) t 1 < τ ( x 2 ) t_1<\tau(x_2) t 1 < τ ( x 2 ) x 1 = x 2 x_1=x_2 x 1 = x 2 X X X D \mathcal{D} D
Functional setting
Definition 3.5 (The space X F \mathbb{X}_F X F
X F : = { u ∈ [ L 2 ( Ω ) ] 3 : ∇ ⋅ u = 0 , ∇ × ( u × F ) ∈ H } , (23) \mathbb{X}_F:=\left\{u\in[L^2(\Omega)]^3:\nabla\cdot u=0,\quad\nabla\times(u\times F)\in\mathbf{H}\right\}, \tag*{(23)} X F := { u ∈ [ L 2 ( Ω ) ] 3 : ∇ ⋅ u = 0 , ∇ × ( u × F ) ∈ H } , (23) equipped with the norm
∥ u ∥ X F 2 : = ∥ u ∥ H 2 + ∥ ∇ × ( u × F ) ∥ H 2 . \lVert u\rVert_{\mathbb{X}_F}^2:=\lVert u\rVert_{\mathbf{H}}^2+\lVert\nabla\times(u\times F)\rVert_{\mathbf{H}}^2. ∥ u ∥ X F 2 := ∥ u ∥ H 2 + ∥ ∇ × ( u × F ) ∥ H 2 . Under the Ω \Omega Ω [22 ] yields a bounded trace operator
γ − : X F ⟶ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ ) , \gamma_-: X_F \longrightarrow L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|), γ − : X F ⟶ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ ) , where
L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ ) : = { g : Γ i n ⟶ R 3 : g is measurable and ∫ Γ i n ∣ g ∣ 2 τ ∣ F ⋅ n ∣ d σ < ∞ } , L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|):=\left\{g:\Gamma_{\mathrm{in}}\longrightarrow\mathbb{R}^3:g\text{ is measurable and }\int_{\Gamma_{\mathrm{in}}}|g|^2\tau|F\cdot n|\,\mathrm{d}\sigma<\infty\right\}, L 2 ( Γ in ; τ ∣ F ⋅ n ∣ ) := { g : Γ in ⟶ R 3 : g is measurable and ∫ Γ in ∣ g ∣ 2 τ ∣ F ⋅ n ∣ d σ < ∞ } , equipped with the norm
∥ g ∥ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ ) 2 : = ∫ Γ i n ∣ g ∣ 2 τ ∣ F ⋅ n ∣ d σ . \|g\|^2_{L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|)}:=\int_{\Gamma_{\mathrm{in}}}|g|^2\tau|F\cdot n|\,\mathrm{d}\sigma. ∥ g ∥ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ ) 2 := ∫ Γ in ∣ g ∣ 2 τ ∣ F ⋅ n ∣ d σ . We have:
Proposition 3.6. Assume that F F F X X X Ω \Omega Ω C > 0 C>0 C > 0
∫ Γ i n ∣ γ − u ∣ 2 τ ∣ F ⋅ n ∣ d σ ≤ C ∥ u ∥ X F 2 , u ∈ X F . (24) \int_{\Gamma_{\mathrm{in}}}|\gamma_-u|^2\tau|F\cdot n|\,\mathrm{d}\sigma\leq C\|u\|^2_{X_F},\qquad u\in X_F. \tag*{(24)} ∫ Γ in ∣ γ − u ∣ 2 τ ∣ F ⋅ n ∣ d σ ≤ C ∥ u ∥ X F 2 , u ∈ X F . (24) Proof. Fix u ∈ X F u\in X_F u ∈ X F
w : = ∇ × ( u × F ) , f : = w + ( u ⋅ ∇ ) F . w:=\nabla\times(u\times F),\qquad f:=w+(u\cdot\nabla)F. w := ∇ × ( u × F ) , f := w + ( u ⋅ ∇ ) F . The vector identity
∇ × ( u × F ) = u ∇ ⋅ F − F ∇ ⋅ u + ( F ⋅ ∇ ) u − ( u ⋅ ∇ ) F \nabla\times(u\times F)=u\nabla\cdot F-F\nabla\cdot u+(F\cdot\nabla)u-(u\cdot\nabla)F ∇ × ( u × F ) = u ∇ ⋅ F − F ∇ ⋅ u + ( F ⋅ ∇ ) u − ( u ⋅ ∇ ) F and the divergence-free assumptions imply, in distributions, that
( F ⋅ ∇ ) u = f . (F\cdot\nabla)u=f. ( F ⋅ ∇ ) u = f . Moreover,
∥ f ∥ L 2 ( Ω ) ≤ ∥ w ∥ L 2 ( Ω ) + ∥ D F ∥ L ∞ ( Ω ) ∥ u ∥ L 2 ( Ω ) . \|f\|_{L^2(\Omega)}\leq\|w\|_{L^2(\Omega)}+\|DF\|_{L^\infty(\Omega)}\|u\|_{L^2(\Omega)}. ∥ f ∥ L 2 ( Ω ) ≤ ∥ w ∥ L 2 ( Ω ) + ∥ D F ∥ L ∞ ( Ω ) ∥ u ∥ L 2 ( Ω ) . The change-of-variables formula transforms this distributional identity into
∂ t ( u ∘ X ) = f ∘ X \partial_t(u\circ X)=f\circ X ∂ t ( u ∘ X ) = f ∘ X in flow coordinates. This can be verified by testing locally in flow-coordinate charts. Since both pullbacks are square-integrable in the flow-coordinate measure, Fubini’s theorem implies that t ↦ u ( X ( t , x 0 ) ) t\mapsto u(X(t,x_0)) t ↦ u ( X ( t , x 0 )) H 1 ( 0 , τ ( x 0 ) ) H^1(0,\tau(x_0)) H 1 ( 0 , τ ( x 0 )) x 0 ∈ Γ i n x_0\in\Gamma_{\mathrm{in}} x 0 ∈ Γ in
For z ∈ H 1 ( 0 , τ ; R d ) z\in H^1(0,\tau;\mathbb{R}^d) z ∈ H 1 ( 0 , τ ; R d )
τ ∣ z ( 0 ) ∣ 2 ≤ 2 ∫ 0 τ ∣ z ( t ) ∣ 2 d t + τ 2 ∫ 0 τ ∣ z ′ ( t ) ∣ 2 d t . \tau|z(0)|^2\leq2\int_0^\tau|z(t)|^2\,\mathrm{d}t+\tau^2\int_0^\tau|z'(t)|^2\,\mathrm{d}t. τ ∣ z ( 0 ) ∣ 2 ≤ 2 ∫ 0 τ ∣ z ( t ) ∣ 2 d t + τ 2 ∫ 0 τ ∣ z ′ ( t ) ∣ 2 d t . Apply this inequality along almost every flow line, multiply by ∣ F ( x 0 ) ⋅ n ( x 0 ) ∣ |F(x_0)\cdot n(x_0)| ∣ F ( x 0 ) ⋅ n ( x 0 ) ∣ Γ i n \Gamma_{\mathrm{in}} Γ in τ ≤ T f i l l \tau\leq T_{\mathrm{fill}} τ ≤ T fill
∫ Γ i n ∣ γ − u ∣ 2 τ ∣ F ⋅ n ∣ d σ ≤ 2 ∥ u ∥ L 2 ( Ω ) 2 + T f i l l 2 ∥ f ∥ L 2 ( Ω ) 2 . \int_{\Gamma_{\mathrm{in}}}|\gamma_-u|^2\tau|F\cdot n|\,\mathrm{d}\sigma\leq2\|u\|^2_{L^2(\Omega)}+T_{\mathrm{fill}}^2\|f\|^2_{L^2(\Omega)}. ∫ Γ in ∣ γ − u ∣ 2 τ ∣ F ⋅ n ∣ d σ ≤ 2∥ u ∥ L 2 ( Ω ) 2 + T fill 2 ∥ f ∥ L 2 ( Ω ) 2 . Using the bound for f f f ( a + b ) 2 ≤ 2 a 2 + 2 b 2 (a+b)^2\leq2a^2+2b^2 ( a + b ) 2 ≤ 2 a 2 + 2 b 2
∫ Γ i n ∣ γ − u ∣ 2 τ ∣ F ⋅ n ∣ d σ ≤ C ( ∥ u ∥ L 2 ( Ω ) 2 + ∥ w ∥ L 2 ( Ω ) 2 ) , \int_{\Gamma_{\mathrm{in}}}|\gamma_-u|^2\tau|F\cdot n|\,\mathrm{d}\sigma\leq C\left(\|u\|^2_{L^2(\Omega)}+\|w\|^2_{L^2(\Omega)}\right), ∫ Γ in ∣ γ − u ∣ 2 τ ∣ F ⋅ n ∣ d σ ≤ C ( ∥ u ∥ L 2 ( Ω ) 2 + ∥ w ∥ L 2 ( Ω ) 2 ) , where one may take
C = max { 2 + 2 T f i l l 2 ∥ D F ∥ L ∞ 2 , 2 T f i l l 2 } , C=\max\left\{2+2T_{\mathrm{fill}}^2\|DF\|^2_{L^\infty},\,2T_{\mathrm{fill}}^2\right\}, C = max { 2 + 2 T fill 2 ∥ D F ∥ L ∞ 2 , 2 T fill 2 } , which is exactly the estimate (24).
Therefore, for any g ∈ [ L 2 ( Γ i n ) ] d \boldsymbol{g} \in[L^{2}(\Gamma_{\mathrm{in}})]^{d} g ∈ [ L 2 ( Γ in ) ] d
∥ g ∥ [ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ ) ] d 2 = ∫ Γ i n ∣ g ∣ 2 τ ∣ F ⋅ n ∣ d σ ≤ T f i l l M ∫ Γ i n ∣ g ∣ 2 d σ = T f i l l M ∥ g ∥ [ L 2 ( Γ i n ) ] d 2 . \|\boldsymbol{g}\|_{[L^{2}(\Gamma_{\mathrm{in}};\tau|F\cdot n|)]^{d}}^{2}
=\int_{\Gamma_{\mathrm{in}}}|\boldsymbol{g}|^{2}\tau|F\cdot n|\,\mathrm{d}\sigma
\le T_{\mathrm{fill}}M\int_{\Gamma_{\mathrm{in}}}|\boldsymbol{g}|^{2}\,\mathrm{d}\sigma
=T_{\mathrm{fill}}M\|\boldsymbol{g}\|_{[L^{2}(\Gamma_{\mathrm{in}})]^{d}}^{2}. ∥ g ∥ [ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ ) ] d 2 = ∫ Γ in ∣ g ∣ 2 τ ∣ F ⋅ n ∣ d σ ≤ T fill M ∫ Γ in ∣ g ∣ 2 d σ = T fill M ∥ g ∥ [ L 2 ( Γ in ) ] d 2 . Thus, the injection of [ L 2 ( Γ i n ) ] d [L^{2}(\Gamma_{\mathrm{in}})]^{d} [ L 2 ( Γ in ) ] d [ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ ) ] d [L^{2}(\Gamma_{\mathrm{in}};\tau|F\cdot n|)]^{d} [ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ ) ] d τ ∣ F ( x ) ⋅ n ( x ) ∣ \tau|F(x)\cdot n(x)| τ ∣ F ( x ) ⋅ n ( x ) ∣ [ C ∞ ( Γ i n ) ] d [C^{\infty}(\Gamma_{\mathrm{in}})]^{d} [ C ∞ ( Γ in ) ] d [ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ ) ] d [L^{2}(\Gamma_{\mathrm{in}};\tau|F\cdot n|)]^{d} [ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ ) ] d [24 Proposition A.3] .
In particular, since H 1 ( Γ i n ) ↪ L 2 ( Γ i n ) H^{1}(\Gamma_{\mathrm{in}})\hookrightarrow L^{2}(\Gamma_{\mathrm{in}}) H 1 ( Γ in ) ↪ L 2 ( Γ in )
[ H 1 ( Γ i n ) ] d ↪ [ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ ) ] d . [H^{1}(\Gamma_{\mathrm{in}})]^{d}\hookrightarrow[L^{2}(\Gamma_{\mathrm{in}};\tau|F\cdot n|)]^{d}. [ H 1 ( Γ in ) ] d ↪ [ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ ) ] d . Thus every g ∈ [ H 1 ( Γ i n ) ] d \boldsymbol{g}\in[H^{1}(\Gamma_{\mathrm{in}})]^{d} g ∈ [ H 1 ( Γ in ) ] d L 2 L^{2} L 2 τ ∣ F ⋅ n ∣ \tau|F\cdot n| τ ∣ F ⋅ n ∣
The curl–div system
In this section, we establish a well-posedness result for the stationary curl–div system
{ ∇ × ( u × F ) = z , in Ω , ∇ ⋅ u = 0 , in Ω , u = g , on Γ i n . (25) \begin{cases}
\nabla\times(\boldsymbol{u}\times F)=\boldsymbol{z}, & \text{in }\Omega,\\
\nabla\cdot\boldsymbol{u}=0, & \text{in }\Omega,\\
\boldsymbol{u}=\boldsymbol{g}, & \text{on }\Gamma_{\mathrm{in}}. \tag*{(25)}
\end{cases} ⎩ ⎨ ⎧ ∇ × ( u × F ) = z , ∇ ⋅ u = 0 , u = g , in Ω , in Ω , on Γ in . (25) This system is overdetermined: it consists of four scalar equations for the three components of u \boldsymbol{u} u g \boldsymbol{g} g
For g ∈ [ H 1 ( Γ i n ) ] d \boldsymbol{g}\in[H^{1}(\Gamma_{\mathrm{in}})]^{d} g ∈ [ H 1 ( Γ in ) ] d
C ( g ) : = ( F ⋅ n ) ∇ Γ i n ⋅ g + ( D F g − D Γ g F Γ ) ⋅ n on Γ i n , (26) \mathcal{C}(\boldsymbol{g}):=(F\cdot n)\nabla_{\Gamma_{\mathrm{in}}}\cdot\boldsymbol{g}
+\left(DF\boldsymbol{g}-D_{\Gamma}\boldsymbol{g}F_{\Gamma}\right)\cdot n
\quad\text{on }\Gamma_{\mathrm{in}}, \tag*{(26)} C ( g ) := ( F ⋅ n ) ∇ Γ in ⋅ g + ( D F g − D Γ g F Γ ) ⋅ n on Γ in , (26) where
F Γ : = F − ( F ⋅ n ) n F_{\Gamma}:=F-(F\cdot n)n F Γ := F − ( F ⋅ n ) n denotes the tangential component of F F F Γ i n \Gamma_{\mathrm{in}} Γ in ∇ Γ i n ⋅ \nabla_{\Gamma_{\mathrm{in}}}\cdot ∇ Γ in ⋅ Γ i n \Gamma_{\mathrm{in}} Γ in D Γ g D_{\Gamma}\boldsymbol{g} D Γ g g \boldsymbol{g} g
Proposition 4.1. Assume that Ω \Omega Ω F F F ( g , z ) ∈ [ H 1 ( Γ i n ) ] d × H (\boldsymbol{g},\boldsymbol{z})\in[H^{1}(\Gamma_{\mathrm{in}})]^{d}\times\mathbb{H} ( g , z ) ∈ [ H 1 ( Γ in ) ] d × H X X X Ω \Omega Ω (25) admits a unique solution u \boldsymbol{u} u
C ( g ) = 0 on Γ i n . (27) \mathcal{C}(\boldsymbol{g})=0\quad\text{on }\Gamma_{\mathrm{in}}. \tag*{(27)} C ( g ) = 0 on Γ in . (27) Furthermore, the solution satisfies the stability estimate
∥ u ∥ X F 2 ≤ C ( ∥ z ∥ H 2 + ∥ g ∥ [ H 1 ( Γ i n ) ] d 2 ) , (28) \|\boldsymbol{u}\|_{X_{F}}^{2}\le C\left(\|\boldsymbol{z}\|_{\mathbb{H}}^{2}
+\|\boldsymbol{g}\|_{[H^{1}(\Gamma_{\mathrm{in}})]^{d}}^{2}\right), \tag*{(28)} ∥ u ∥ X F 2 ≤ C ( ∥ z ∥ H 2 + ∥ g ∥ [ H 1 ( Γ in ) ] d 2 ) , (28) where C C C Ω \Omega Ω F F F
Proof. See Appendix 8.
Remark 4.2. System (25) is a first-order transport system and can be recast in the form (19) . Consequently, a natural route to well-posedness is through the characteristic flow associated with F F F ∇ ⋅ F = 0 \nabla\cdot F=0 ∇ ⋅ F = 0
To make this precise, define
Γ o u t : = { x ∈ ∂ Ω : F ( x ) ⋅ n ( x ) > 0 } , \Gamma_{\mathrm{out}}:=\{x\in\partial\Omega:F(x)\cdot n(x)>0\}, Γ out := { x ∈ ∂ Ω : F ( x ) ⋅ n ( x ) > 0 } , Temporarily ignore the constraint ∇ ⋅ u = 0 \nabla\cdot\boldsymbol{u}=0 ∇ ⋅ u = 0
W : = { u ∈ [ H 1 ( Ω ) ] 3 : u = 0 on Γ i n } W:=\{\boldsymbol{u}\in[H^{1}(\Omega)]^{3}:\boldsymbol{u}=0\text{ on }\Gamma_{\mathrm{in}}\} W := { u ∈ [ H 1 ( Ω ) ] 3 : u = 0 on Γ in } and consider the transport bilinear form
a ( u , w ) : = ∫ Ω ( ( F ⋅ ∇ ) u − ( D F ) u ) ⋅ w d x , a(\boldsymbol{u},\boldsymbol{w})
:=\int_{\Omega}\left((F\cdot\nabla)\boldsymbol{u}-(DF)\boldsymbol{u}\right)\cdot\boldsymbol{w}\,\mathrm{d}x, a ( u , w ) := ∫ Ω ( ( F ⋅ ∇ ) u − ( D F ) u ) ⋅ w d x , where ( D F ) i j = ∂ j F i (DF)_{ij} = \partial_j F_i ( D F ) ij = ∂ j F i w = u w = u w = u ∇ ⋅ F = 0 \nabla\cdot\mathbf{F} = 0 ∇ ⋅ F = 0
a ( u , u ) = 1 2 ∫ ∂ Ω ( F ⋅ n ) ∣ u ∣ 2 d S − ∫ Ω u T ( D F ) u d x . a(u,u) = \frac{1}{2} \int_{\partial\Omega} (\mathbf{F} \cdot\mathbf{n})|u|^2 \, \mathrm{d}S - \int_{\Omega} u^{T}(D\mathbf{F})u \, \mathrm{d}x. a ( u , u ) = 2 1 ∫ ∂ Ω ( F ⋅ n ) ∣ u ∣ 2 d S − ∫ Ω u T ( D F ) u d x . Since u = 0 u = 0 u = 0 Γ i n \Gamma_{\mathrm{in}} Γ in
a ( u , u ) = 1 2 ∫ Γ o u t ( F ⋅ n ) ∣ u ∣ 2 d S − ∫ Ω u T ( D F ) u d x . a(u,u) = \frac{1}{2} \int_{\Gamma_{\mathrm{out}}} (\mathbf{F} \cdot\mathbf{n})|u|^2 \, \mathrm{d}S - \int_{\Omega} u^{T}(D\mathbf{F})u \, \mathrm{d}x. a ( u , u ) = 2 1 ∫ Γ out ( F ⋅ n ) ∣ u ∣ 2 d S − ∫ Ω u T ( D F ) u d x . On Γ o u t \Gamma_{\mathrm{out}} Γ out F ⋅ n > 0 \mathbf{F} \cdot\mathbf{n} > 0 F ⋅ n > 0
ξ T ( D F ) ( x ) ξ ≤ − c 0 ∣ ξ ∣ 2 for all ξ ∈ R d and a.e. x ∈ Ω \xi^{T}(D\mathbf{F})(x)\,\xi\leq-c_0|\xi|^2 \qquad\text{for all } \xi\in\mathbb{R}^{d} \text{ and a.e. } x \in\Omega ξ T ( D F ) ( x ) ξ ≤ − c 0 ∣ ξ ∣ 2 for all ξ ∈ R d and a.e. x ∈ Ω would be sufficient, but it is incompatible with ∇ ⋅ F = 0 \nabla\cdot\mathbf{F} = 0 ∇ ⋅ F = 0 ξ = e i \xi= e_i ξ = e i { e 1 , … , e d } \{e_1,\ldots,e_d\} { e 1 , … , e d }
∇ ⋅ F = trace ( D F ) = ∑ i = 1 d e i T ( D F ) e i ≤ − d c 0 < 0 , \nabla\cdot\mathbf{F} = \operatorname{trace}(D\mathbf{F}) = \sum_{i=1}^{d} e_i^{T}(D\mathbf{F})e_i \leq-dc_0 < 0, ∇ ⋅ F = trace ( D F ) = i = 1 ∑ d e i T ( D F ) e i ≤ − d c 0 < 0 , which contradicts ∇ ⋅ F = 0 \nabla\cdot\mathbf{F} = 0 ∇ ⋅ F = 0
Therefore, the stationary curl–div system (25) is properly analyzed by propagating the inflow data along the characteristics of F \mathbf{F} F
Remark 4.3. The curl-div system (25) is a particular case of the general system
{ ∇ × ( A u ) = z , in Ω , ∇ ⋅ u = 0 , in Ω , suitable boundary conditions , on Γ i n , \begin{cases}
\nabla\times(A u) = z, & \text{in } \Omega,\\
\nabla\cdot u = 0, & \text{in } \Omega,\\
\text{suitable boundary conditions}, & \text{on } \Gamma_{\mathrm{in}},
\end{cases} ⎩ ⎨ ⎧ ∇ × ( A u ) = z , ∇ ⋅ u = 0 , suitable boundary conditions , in Ω , in Ω , on Γ in , which has been studied under the assumption that A A A [25 ] . In the present setting,
A u = u × F , A u = u \times\mathbf{F}, A u = u × F , and hence A A A u × F = 0 u \times\mathbf{F} = 0 u × F = 0 u u u F \mathbf{F} F [25 ] do not apply directly to the present degenerate setting. A systematic characterization of well-posedness for such singular curl-div systems is beyond the scope of this work.
Main results
We now establish a global stability estimate for the inverse problem, showing that the unknown source u u u
To encompass different measurement configurations, we introduce a bounded linear observation operator
L : Y ⟶ H o b s , (29) \mathcal{L}: Y \longrightarrow\mathbb{H}_{\mathrm{obs}}, \tag*{(29)} L : Y ⟶ H obs , (29) where Y Y Y (16) and H o b s \mathbb{H}_{\mathrm{obs}} H obs L \mathcal{L} L
The framework includes, in particular, internal observations, boundary flux observations, and final-time measurements, such as
L ( b ) = b ∣ ( 0 , T ) × ω o b s , H o b s = L 2 ( 0 , T ; [ L 2 ( ω o b s ) ] d ) , (30) \mathcal{L}(b) = b|_{(0,T)\times\omega_{\mathrm{obs}}}, \qquad\mathbb{H}_{\mathrm{obs}} = L^2(0,T;[L^2(\omega_{\mathrm{obs}})]^d), \tag*{(30)} L ( b ) = b ∣ ( 0 , T ) × ω obs , H obs = L 2 ( 0 , T ; [ L 2 ( ω obs ) ] d ) , (30) or
L ( b ) = b ( T , ⋅ ) ∣ ω o b s , H o b s = [ L 2 ( ω o b s ) ] d . (31) \mathcal{L}(b) = b(T,\cdot)|_{\omega_{\mathrm{obs}}}, \qquad\mathbb{H}_{\mathrm{obs}} = [L^2(\omega_{\mathrm{obs}})]^d. \tag*{(31)} L ( b ) = b ( T , ⋅ ) ∣ ω obs , H obs = [ L 2 ( ω obs ) ] d . (31) Such observation configurations are standard in parabolic inverse source problems; see, e.g., [26 , 27 , 28 ] . Accordingly, for system (7) , we define the generalized observation operator
Λ L : X F ⟶ H o b s × [ H 1 ( Γ i n ) ] d , (32) \Lambda_{\mathcal{L}}: X_{\mathbf{F}} \longrightarrow\mathbb{H}_{\mathrm{obs}} \times[H^1(\Gamma_{\mathrm{in}})]^d, \tag*{(32)} Λ L : X F ⟶ H obs × [ H 1 ( Γ in ) ] d , (32) u ⟼ ( L ( b ) , u ∣ Γ i n ) , u \longmapsto\left(\mathcal{L}(b),u|_{\Gamma_{\mathrm{in}}}\right), u ⟼ ( L ( b ) , u ∣ Γ in ) , where b b b v = α u v = \alpha u v = α u (7) .
Stability result
We first introduce the auxiliary parabolic system for the difference fields. For i = 1 , 2 i=1,2 i = 1 , 2 z i ∈ H z_i \in\mathbb{H} z i ∈ H y i y_i y i
{ ∂ t y i − Δ y i = α ( t ) z i + Δ F , in ( 0 , T ) × Ω , ∇ ⋅ y i = 0 , in ( 0 , T ) × Ω , y i ⋅ n = 0 , n × ( ∇ × y i ) = 0 , on ( 0 , T ) × ∂ Ω , y i ∣ t = 0 = y 0 , in Ω . (33) \begin{cases}
\partial_t y_i-\Delta y_i=\alpha(t)z_i+\Delta F, & \text{in }(0,T)\times\Omega,\\
\nabla\cdot y_i=0, & \text{in }(0,T)\times\Omega,\\
y_i\cdot n=0,\quad n\times(\nabla\times y_i)=0, & \text{on }(0,T)\times\partial\Omega,\\
y_i|_{t=0}=y_0, & \text{in }\Omega. \tag*{(33)}
\end{cases} ⎩ ⎨ ⎧ ∂ t y i − Δ y i = α ( t ) z i + Δ F , ∇ ⋅ y i = 0 , y i ⋅ n = 0 , n × ( ∇ × y i ) = 0 , y i ∣ t = 0 = y 0 , in ( 0 , T ) × Ω , in ( 0 , T ) × Ω , on ( 0 , T ) × ∂ Ω , in Ω. (33) By Theorem 2.1 , for α z i + Δ F ∈ L 2 ( 0 , T ; H ) \alpha z_i+\Delta F\in L^2(0,T;\mathbb{H}) α z i + Δ F ∈ L 2 ( 0 , T ; H ) (33) belongs to Y \mathbb{Y} Y (16) . The difference y : = y 1 − y 2 y:=y_1-y_2 y := y 1 − y 2
{ ∂ t y − Δ y = α ( t ) ( z 1 − z 2 ) , in ( 0 , T ) × Ω , ∇ ⋅ y = 0 , in ( 0 , T ) × Ω , y ⋅ n = 0 , n × ( ∇ × y ) = 0 , on ( 0 , T ) × ∂ Ω , y ∣ t = 0 = 0 , in Ω . (34) \begin{cases}
\partial_t y-\Delta y=\alpha(t)(z_1-z_2), & \text{in }(0,T)\times\Omega,\\
\nabla\cdot y=0, & \text{in }(0,T)\times\Omega,\\
y\cdot n=0,\quad n\times(\nabla\times y)=0, & \text{on }(0,T)\times\partial\Omega,\\
y|_{t=0}=0, & \text{in }\Omega. \tag*{(34)}
\end{cases} ⎩ ⎨ ⎧ ∂ t y − Δ y = α ( t ) ( z 1 − z 2 ) , ∇ ⋅ y = 0 , y ⋅ n = 0 , n × ( ∇ × y ) = 0 , y ∣ t = 0 = 0 , in ( 0 , T ) × Ω , in ( 0 , T ) × Ω , on ( 0 , T ) × ∂ Ω , in Ω. (34) For the magnetic perturbations b i b_i b i v i = α u i v_i=\alpha u_i v i = α u i b 0 b_0 b 0 F F F
b 1 − b 2 = y 1 − y 2 . (35) b_1-b_2=y_1-y_2. \tag*{(35)} b 1 − b 2 = y 1 − y 2 . (35) Theorem 5.1. Assume that F F F (4) and let γ − u i ∈ [ H 1 ( Γ i n ) ] 3 \gamma_-u_i\in[H^1(\Gamma_{\mathrm{in}})]^3 γ − u i ∈ [ H 1 ( Γ in ) ] 3 i = 1 , 2 i=1,2 i = 1 , 2 Θ : [ 0 , ∞ ) → [ 0 , ∞ ) \Theta:[0,\infty)\to[0,\infty) Θ : [ 0 , ∞ ) → [ 0 , ∞ ) lim s → 0 Θ ( s ) = 0 \lim_{s\to0}\Theta(s)=0 lim s → 0 Θ ( s ) = 0
(i) α ∈ L 2 ( 0 , T ) \alpha\in L^2(0,T) α ∈ L 2 ( 0 , T ) (34) admits the stability estimate
∥ z 1 − z 2 ∥ H 2 ≤ Θ ( ∥ L ( y 1 − y 2 ) ∥ H o b s 2 ) , (36) \|z_1-z_2\|_{\mathbb{H}}^2\leq\Theta\left(\|\mathcal{L}(y_1-y_2)\|_{\mathbb{H}_{\mathrm{obs}}}^2\right), \tag*{(36)} ∥ z 1 − z 2 ∥ H 2 ≤ Θ ( ∥ L ( y 1 − y 2 ) ∥ H obs 2 ) , (36) where y i y_i y i (33) associated with z i z_i z i i = 1 , 2 i=1,2 i = 1 , 2 L \mathcal{L} L (29) .
(ii) The boundary measurement difference γ − ( u 1 − u 2 ) \gamma_-(u_1-u_2) γ − ( u 1 − u 2 )
C ( γ − ( u 1 − u 2 ) ) = 0 , on Γ i n , (37) \mathcal{C}\left(\gamma_-(u_1-u_2)\right)=0,\qquad\text{on }\Gamma_{\mathrm{in}}, \tag*{(37)} C ( γ − ( u 1 − u 2 ) ) = 0 , on Γ in , (37) where C \mathcal{C} C (26) .
(iii) The background vector field F F F (4) and the flow X X X Ω \Omega Ω Definition 3.1 .
Then the velocity difference u 1 − u 2 ∈ X F u_1-u_2\in X_F u 1 − u 2 ∈ X F
∥ u 1 − u 2 ∥ X F 2 ≤ C [ Θ ( ∥ L ( y 1 − y 2 ) ∥ H o b s 2 ) + ∥ γ − ( u 1 − u 2 ) ∥ [ H 1 ( Γ i n ) ] 3 2 ] , (38) \|u_1-u_2\|_{X_F}^2\leq C\left[\Theta\left(\|\mathcal{L}(y_1-y_2)\|_{\mathbb{H}_{\mathrm{obs}}}^2\right)+\|\gamma_-(u_1-u_2)\|_{[H^1(\Gamma_{\mathrm{in}})]^3}^2\right], \tag*{(38)} ∥ u 1 − u 2 ∥ X F 2 ≤ C [ Θ ( ∥ L ( y 1 − y 2 ) ∥ H obs 2 ) + ∥ γ − ( u 1 − u 2 ) ∥ [ H 1 ( Γ in ) ] 3 2 ] , (38) where C > 0 C>0 C > 0 u 1 u_1 u 1 u 2 u_2 u 2
Remark 5.2 (Hybrid Error Propagation). The stability estimate (38) reflects the hybrid parabolic–hyperbolic nature of the inverse problem (2.3). The error in the inflow boundary data propagates into the domain with Lipschitz stability, as expected for the underlying first-order transport equation along the characteristics of F F F L ( y 1 − y 2 ) \mathcal{L}(y_1-y_2) L ( y 1 − y 2 ) Θ \Theta Θ [29 , 30 , 31 ] for L \mathcal{L} L (30) –(31) , reflecting the ill-posedness of the associated parabolic inverse source problem (these references handle the Dirichlet boundary condition case). Thus, the overall stability combines a Lipschitz contribution from transport with a potentially weaker contribution from parabolic source recovery. Since b 1 − b 2 = y 1 − y 2 b_1-b_2=y_1-y_2 b 1 − b 2 = y 1 − y 2
Stability from the terminal observation
The injectivity result of Theorem 5.8 (ii) shows that, under the nonvanishing condition (44) , the terminal observation y ( T ) y(T) y ( T ) z z z L ( y ) = y ( T ) ∈ D ( Δ H ) \mathcal{L}(y)=y(T)\in D(\Delta_{\mathrm{H}}) L ( y ) = y ( T ) ∈ D ( Δ H ) ω o b s \omega_{\mathrm{obs}} ω obs
Theorem 5.3. Assume that there exists a constant c 0 > 0 c_0>0 c 0 > 0
∣ γ n ∣ ≥ c 0 1 + λ n , n ≥ 1 , (39) |\gamma_n|\geq\frac{c_0}{1+\lambda_n},\qquad n\geq1, \tag*{(39)} ∣ γ n ∣ ≥ 1 + λ n c 0 , n ≥ 1 , (39) where γ n \gamma_n γ n (44) . Then for every z ∈ H z\in\mathbb{H} z ∈ H y y y (33) satisfies
∥ z ∥ H 2 ≤ C ( ∥ y ( T ) ∥ H 2 + ∥ Δ H y ( T ) ∥ H 2 ) , (40) \|z\|_{\mathbb{H}}^2\leq C\left(\|y(T)\|_{\mathbb{H}}^2+\|\Delta_{\mathrm{H}}y(T)\|_{\mathbb{H}}^2\right), \tag*{(40)} ∥ z ∥ H 2 ≤ C ( ∥ y ( T ) ∥ H 2 + ∥ Δ H y ( T ) ∥ H 2 ) , (40) for some constant C > 0 C>0 C > 0 z z z
Proof. We write y ( T ) y(T) y ( T )
y ( T ) = ∑ n = 1 ∞ ∑ k = 1 m n γ n z n , k ϕ n , k . y(T)=\sum_{n=1}^{\infty}\sum_{k=1}^{m_n}\gamma_n z_{n,k}\phi_{n,k}. y ( T ) = n = 1 ∑ ∞ k = 1 ∑ m n γ n z n , k ϕ n , k . Using (39) coefficientwise, for each ( n , k ) (n,k) ( n , k )
∣ z n , k ∣ = ∣ z n , k γ n ∣ ∣ γ n ∣ ≤ 1 + λ n c 0 ∣ z n , k γ n ∣ . |z_{n,k}|=\frac{|z_{n,k}\gamma_n|}{|\gamma_n|}\leq\frac{1+\lambda_n}{c_0}|z_{n,k}\gamma_n|. ∣ z n , k ∣ = ∣ γ n ∣ ∣ z n , k γ n ∣ ≤ c 0 1 + λ n ∣ z n , k γ n ∣. Squaring and summing over n , k n,k n , k
∥ z ∥ H 2 = ∑ n = 1 ∞ ∑ k = 1 m n ∣ z n , k ∣ 2 ≤ 1 c 0 2 ∑ n = 1 ∞ ∑ k = 1 m n ( 1 + λ n ) 2 ∣ z n , k γ n ∣ 2 = 1 c 0 2 ∥ ( I − Δ H ) y ( T ) ∥ H 2 ≤ C ( ∥ y ( T ) ∥ H 2 + ∥ Δ H y ( T ) ∥ H 2 ) , \begin{aligned}
\|z\|_{\mathbb{H}}^2
&=\sum_{n=1}^{\infty}\sum_{k=1}^{m_n}|z_{n,k}|^2
\leq\frac{1}{c_0^2}\sum_{n=1}^{\infty}\sum_{k=1}^{m_n}(1+\lambda_n)^2|z_{n,k}\gamma_n|^2\\
&=\frac{1}{c_0^2}\|(I-\Delta_{\mathrm{H}})y(T)\|_{\mathbb{H}}^2
\leq C\left(\|y(T)\|_{\mathbb{H}}^2+\|\Delta_{\mathrm{H}}y(T)\|_{\mathbb{H}}^2\right),
\end{aligned} ∥ z ∥ H 2 = n = 1 ∑ ∞ k = 1 ∑ m n ∣ z n , k ∣ 2 ≤ c 0 2 1 n = 1 ∑ ∞ k = 1 ∑ m n ( 1 + λ n ) 2 ∣ z n , k γ n ∣ 2 = c 0 2 1 ∥ ( I − Δ H ) y ( T ) ∥ H 2 ≤ C ( ∥ y ( T ) ∥ H 2 + ∥ Δ H y ( T ) ∥ H 2 ) , where we used that − Δ H ϕ n , k = λ n ϕ n , k -\Delta_{\mathrm{H}}\phi_{n,k}=\lambda_n\phi_{n,k} − Δ H ϕ n , k = λ n ϕ n , k ( 1 + λ ) 2 ≤ 2 ( 1 + λ 2 ) (1+\lambda)^2\leq2(1+\lambda^2) ( 1 + λ ) 2 ≤ 2 ( 1 + λ 2 ) λ ≥ 0 \lambda\geq0 λ ≥ 0
Uniqueness Result
We now state the main uniqueness result for the inverse problem (2.3). The result follows from the injectivity of the observation of the auxiliary variable and the Ω \Omega Ω F F F
Theorem 5.5. Assume that F F F (4) . Suppose, in addition, that:
α ∈ L 2 ( 0 , T ) \alpha\in L^2(0,T) α ∈ L 2 ( 0 , T )
O : H ⟶ H o b s , O ( z ) = L ( y ( z ) ) , (41) \begin{aligned}
\mathcal{O}&:\mathbb{H}\longrightarrow\mathbb{H}_{\mathrm{obs}},\\
\mathcal{O}(z)&=\mathcal{L}(y(z)), \tag*{(41)}
\end{aligned} O O ( z ) : H ⟶ H obs , = L ( y ( z )) , (41) is injective, where y ( z ) y(z) y ( z ) (33) associated with z z z 0 0 0 L \mathcal{L} L (29) ;
(ii) The background vector field F \mathbf{F} F (4) and the flow X X X Ω \Omega Ω Definition 3.1 .
Then the observation operator L Λ \mathcal{L}_{\Lambda} L Λ (32) is injective. More precisely, if
L ( y 1 ) = L ( y 2 ) and γ − u 1 = γ − u 2 on Γ i n , (42) \mathcal{L}(\mathbf{y}_1)=\mathcal{L}(\mathbf{y}_2) \qquad\text{and}\qquad\gamma_{-}\mathbf{u}_1=\gamma_{-}\mathbf{u}_2 \quad\text{on }\Gamma_{\mathrm{in}}, \tag*{(42)} L ( y 1 ) = L ( y 2 ) and γ − u 1 = γ − u 2 on Γ in , (42) then
u 1 = u 2 . (43) \mathbf{u}_1=\mathbf{u}_2. \tag*{(43)} u 1 = u 2 . (43) By Theorem 5.5, the source u \mathbf{u} u L ( y ) \mathcal{L}(\mathbf{y}) L ( y ) γ − u \gamma_{-}\mathbf{u} γ − u
Remark 5.6 (Gauge fixing and hybrid parabolic–hyperbolic uniqueness). The measurement Λ L \Lambda_{\mathcal{L}} Λ L L \mathcal{L} L z z z α z = ∇ × ( u × F ) − Δ F \alpha z=\nabla\times(\mathbf{u}\times\mathbf{F})-\Delta\mathbf{F} α z = ∇ × ( u × F ) − Δ F u ↦ ∇ × ( u × F ) \mathbf{u}\mapsto\nabla\times(\mathbf{u}\times\mathbf{F}) u ↦ ∇ × ( u × F ) v \mathbf{v} v F \mathbf{F} F γ − ( u ) \gamma_{-}(\mathbf{u}) γ − ( u ) F \mathbf{F} F
Remark 5.7 (Applications to partial observations). The abstract observation operator L \mathcal{L} L [32 , 33 , 31 ] . Whenever the corresponding observation determines the spatial source difference z 1 − z 2 z_1-z_2 z 1 − z 2
Application of the uniqueness result
For a bounded connected domain Ω \Omega Ω − Δ H -\Delta_{\mathrm{H}} − Δ H H \mathbb{H} H [34 ] . Consequently, its spectrum is discrete. Denote its distinct eigenvalues by
0 < λ 1 ≤ λ 2 ≤ ⋯ , λ n ⟶ + ∞ , 0<\lambda_1\leq\lambda_2\leq\cdots,\qquad\lambda_n\longrightarrow+\infty, 0 < λ 1 ≤ λ 2 ≤ ⋯ , λ n ⟶ + ∞ , and let
E λ n = ker ( − Δ H − λ n I ) E_{\lambda_n}=\ker(-\Delta_{\mathrm{H}}-\lambda_n I) E λ n = ker ( − Δ H − λ n I ) be the corresponding finite-dimensional eigenspaces. For each n ≥ 1 n\geq1 n ≥ 1
{ ϕ n , k : 1 ≤ k ≤ m n } \{\phi_{n,k}:1\leq k\leq m_n\} { ϕ n , k : 1 ≤ k ≤ m n } be an orthonormal basis of E λ n E_{\lambda_n} E λ n m n = dim E λ n m_n=\dim E_{\lambda_n} m n = dim E λ n
ϕ n , k ⋅ n = 0 , ( ∇ × ϕ n , k ) × n = 0 on ∂ Ω , ∇ ⋅ ϕ n , k = 0. \phi_{n,k}\cdot\mathbf{n}=0,\qquad(\nabla\times\phi_{n,k})\times\mathbf{n}=0\quad\text{on }\partial\Omega,\qquad\nabla\cdot\phi_{n,k}=0. ϕ n , k ⋅ n = 0 , ( ∇ × ϕ n , k ) × n = 0 on ∂ Ω , ∇ ⋅ ϕ n , k = 0. For each z ∈ H z\in\mathbb{H} z ∈ H y ( z ) y(z) y ( z ) (33) and let O \mathcal{O} O (41) . We consider the following two observation configurations defined in (30) and (31) .
Theorem 5.8. Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^3 Ω ⊂ R 3 α ∈ L ∞ ( 0 , T ) \alpha\in L^\infty(0,T) α ∈ L ∞ ( 0 , T )
(i) Assume that ω o b s \omega_{\mathrm{obs}} ω obs α ≢ 0 \alpha\not\equiv0 α ≡ 0
supp α ⊂ [ 0 , T 0 ] for some T 0 < T . \operatorname{supp}\alpha\subset[0,T_0]\qquad\text{for some }T_0<T. supp α ⊂ [ 0 , T 0 ] for some T 0 < T . Then O \mathcal{O} O (41) is injective for the observation L \mathcal{L} L (30) .
(ii) Suppose that
γ i : = ∫ 0 T e − λ i ( T − s ) α ( s ) d s ≠ 0 , i ≥ 1. (44) \gamma_i:=\int_0^T e^{-\lambda_i(T-s)}\alpha(s)\,\mathrm{d}s\ne0,\qquad i\geq1. \tag*{(44)} γ i := ∫ 0 T e − λ i ( T − s ) α ( s ) d s = 0 , i ≥ 1. (44) Then O \mathcal{O} O (41) is injective for the terminal observation L \mathcal{L} L (31) :
O ( y ) = y ( T ) in H . \mathcal{O}(\mathbf{y})=\mathbf{y}(T)\quad\text{in }\mathbb{H}. O ( y ) = y ( T ) in H . Proof. Suppose that
O ( z 1 ) = O ( z 2 ) \mathcal{O}(z_1)=\mathcal{O}(z_2) O ( z 1 ) = O ( z 2 )
Set
z = z 1 − z 2 , w = y ( z 1 ) − y ( z 2 ) z=z_1-z_2,\qquad w=y(z_1)-y(z_2) z = z 1 − z 2 , w = y ( z 1 ) − y ( z 2 )
Since the two solutions have the same initial condition, their difference satisfies
w ′ ( t ) − Δ H w ( t ) = α ( t ) z , w ( 0 ) = 0. (45) w'(t)-\Delta_{\mathrm H}w(t)=\alpha(t)z,\qquad w(0)=0. \tag*{(45)} w ′ ( t ) − Δ H w ( t ) = α ( t ) z , w ( 0 ) = 0. (45) Recall that − Δ H -\Delta_{\mathrm H} − Δ H H \mathrm H H { λ i } i ≥ 1 \{\lambda_i\}_{i\geq1} { λ i } i ≥ 1 − Δ H -\Delta_{\mathrm H} − Δ H i ≥ 1 i\geq1 i ≥ 1 E λ i E_{\lambda_i} E λ i m i m_i m i { ϕ i , k } k = 1 m i \{\phi_{i,k}\}_{k=1}^{m_i} { ϕ i , k } k = 1 m i E λ i E_{\lambda_i} E λ i
z = ∑ i ≥ 1 ∑ k = 1 m i z i , k ϕ i , k , z i , k = ( z , ϕ i , k ) H . z=\sum_{i\geq1}\sum_{k=1}^{m_i}z_{i,k}\phi_{i,k},\qquad z_{i,k}=(z,\phi_{i,k})_{\mathrm H}. z = i ≥ 1 ∑ k = 1 ∑ m i z i , k ϕ i , k , z i , k = ( z , ϕ i , k ) H . By Duhamel’s formula,
w ( t ) = ∑ i ≥ 1 ∑ k = 1 m i z i , k ( ∫ 0 t e − λ i ( t − s ) α ( s ) d s ) ϕ i , k . (46) w(t)=\sum_{i\geq1}\sum_{k=1}^{m_i}z_{i,k}\left(\int_0^t e^{-\lambda_i(t-s)}\alpha(s)\,\mathrm{d}s\right)\phi_{i,k}. \tag*{(46)} w ( t ) = i ≥ 1 ∑ k = 1 ∑ m i z i , k ( ∫ 0 t e − λ i ( t − s ) α ( s ) d s ) ϕ i , k . (46) The series converges in H \mathrm H H t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ]
Case (i): observation over a time interval.
Assume that
L w ( t ) = 0 for almost every t ∈ ( 0 , T ) . \mathcal{L}w(t)=0\qquad\text{for almost every }t\in(0,T). L w ( t ) = 0 for almost every t ∈ ( 0 , T ) . We prove that z = 0 z=0 z = 0
Since α \alpha α [ 0 , T 0 ] [0,T_0] [ 0 , T 0 ] T 0 < T T_0<T T 0 < T (45) is homogeneous for t > T 0 t>T_0 t > T 0
w ( t ) = e ( t − T 0 ) Δ H w ( T 0 ) , t > T 0 . w(t)=e^{(t-T_0)\Delta_{\mathrm H}}w(T_0),\qquad t>T_0. w ( t ) = e ( t − T 0 ) Δ H w ( T 0 ) , t > T 0 . Since − Δ H -\Delta_{\mathrm H} − Δ H Δ H \Delta_{\mathrm H} Δ H e t Δ H e^{t\Delta_{\mathrm H}} e t Δ H
t ⟼ L w ( t ) t\longmapsto\mathcal{L}w(t) t ⟼ L w ( t ) is real analytic on ( T 0 , ∞ ) (T_0,\infty) ( T 0 , ∞ )
The observation vanishes almost everywhere on ( T 0 , T ) (T_0,T) ( T 0 , T )
L w ( t ) = 0 , t > T 0 . \mathcal{L}w(t)=0,\qquad t>T_0. L w ( t ) = 0 , t > T 0 . Together with the original observation equality, this yields
L w ( t ) = 0 for almost every t > 0. \mathcal{L}w(t)=0\qquad\text{for almost every }t>0. L w ( t ) = 0 for almost every t > 0. The solution w w w [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) [ 0 , T 0 ] [0,T_0] [ 0 , T 0 ] t > T 0 t>T_0 t > T 0
∥ w ( t ) ∥ H ≤ ∥ w ( T 0 ) ∥ H , \|w(t)\|_{\mathrm H}\leq\|w(T_0)\|_{\mathrm H}, ∥ w ( t ) ∥ H ≤ ∥ w ( T 0 ) ∥ H , by contractivity of the semigroup. Hence, for Re p > 0 \operatorname{Re}p>0 Re p > 0
w ^ ( p ) = ∫ 0 ∞ e − p t w ( t ) d t \widehat{w}(p)=\int_0^\infty e^{-pt}w(t)\,\mathrm{d}t w ( p ) = ∫ 0 ∞ e − pt w ( t ) d t exists in H \mathrm H H α \alpha α [ 0 , T 0 ] [0,T_0] [ 0 , T 0 ]
α ^ ( p ) = ∫ 0 T 0 e − p t α ( t ) d t . \widehat{\alpha}(p)=\int_0^{T_0}e^{-pt}\alpha(t)\,\mathrm{d}t. α ( p ) = ∫ 0 T 0 e − pt α ( t ) d t . Taking the Laplace transform of (45) gives
( p I − Δ H ) w ^ ( p ) = α ^ ( p ) z . (pI-\Delta_{\mathrm H})\widehat{w}(p)=\widehat{\alpha}(p)z. ( p I − Δ H ) w ( p ) = α ( p ) z . Therefore,
w ^ ( p ) = α ^ ( p ) ( p I − Δ H ) − 1 z , Re p > 0. \widehat{w}(p)=\widehat{\alpha}(p)(pI-\Delta_{\mathrm{H}})^{-1}z,\qquad\operatorname{Re}p>0. w ( p ) = α ( p ) ( p I − Δ H ) − 1 z , Re p > 0. Since L \mathcal{L} L L w ( t ) = 0 \mathcal{L}w(t)=0 L w ( t ) = 0
α ^ ( p ) L ( p I − Δ H ) − 1 z = 0 , Re p > 0. \widehat{\alpha}(p)\mathcal{L}(pI-\Delta_{\mathrm{H}})^{-1}z=0,\qquad\operatorname{Re}p>0. α ( p ) L ( p I − Δ H ) − 1 z = 0 , Re p > 0. The function α ^ \widehat{\alpha} α α ≢ 0 \alpha\not\equiv0 α ≡ 0
L ( p I − Δ H ) − 1 z = 0 \mathcal{L}(pI-\Delta_{\mathrm{H}})^{-1}z=0 L ( p I − Δ H ) − 1 z = 0 on that open subset.
The function
p ↦ L ( p I − Δ H ) − 1 z p\mapsto\mathcal{L}(pI-\Delta_{\mathrm{H}})^{-1}z p ↦ L ( p I − Δ H ) − 1 z is holomorphic on
C ∖ { − λ i : i ≥ 1 } . \mathbb{C}\setminus\{-\lambda_i:i\geq1\}. C ∖ { − λ i : i ≥ 1 } . This set is connected, so the identity theorem yields
L ( p I − Δ H ) − 1 z = 0 , p ∈ C ∖ { − λ i : i ≥ 1 } . \mathcal{L}(pI-\Delta_{\mathrm{H}})^{-1}z=0,\qquad p\in\mathbb{C}\setminus\{-\lambda_i:i\geq1\}. L ( p I − Δ H ) − 1 z = 0 , p ∈ C ∖ { − λ i : i ≥ 1 } . Let P i P_i P i E λ i E_{\lambda_i} E λ i p = − λ i p=-\lambda_i p = − λ i ( p I − Δ H ) − 1 (pI-\Delta_{\mathrm{H}})^{-1} ( p I − Δ H ) − 1
( p I − Δ H ) − 1 = P i p + λ i + R i ( p ) , (pI-\Delta_{\mathrm{H}})^{-1}=\frac{P_i}{p+\lambda_i}+R_i(p), ( p I − Δ H ) − 1 = p + λ i P i + R i ( p ) , where R i R_i R i p = − λ i p=-\lambda_i p = − λ i p = − λ i p=-\lambda_i p = − λ i
L P i z = 0. \mathcal{L}P_i z=0. L P i z = 0. Equivalently,
L ( ∑ k = 1 m i z i , k ϕ i , k ) = 0. \mathcal{L}\left(\sum_{k=1}^{m_i}z_{i,k}\phi_{i,k}\right)=0. L ( k = 1 ∑ m i z i , k ϕ i , k ) = 0. Since ω o b s ⊂ Ω \omega_{\mathrm{obs}}\subset\Omega ω obs ⊂ Ω [35 ] gives
∑ k = 1 m i z i , k ϕ i , k = 0 in H . \sum_{k=1}^{m_i}z_{i,k}\phi_{i,k}=0\quad\text{in }\mathbb{H}. k = 1 ∑ m i z i , k ϕ i , k = 0 in H . By orthonormality, z i , k = 0 z_{i,k}=0 z i , k = 0 k k k i i i
Therefore,
Case (ii): terminal observation.
Suppose instead that equality of the terminal observations gives
w ( T ) = 0 in H . w(T)=0\quad\text{in }\mathbb{H}. w ( T ) = 0 in H . Evaluating (46) at t = T t=T t = T
0 = ∑ i ≥ 1 ∑ k = 1 m i γ i z i , k ϕ i , k , γ i = ∫ 0 T e − λ i ( T − s ) α ( s ) d s . 0=\sum_{i\geq1}\sum_{k=1}^{m_i}\gamma_i z_{i,k}\phi_{i,k},\qquad
\gamma_i=\int_0^T e^{-\lambda_i(T-s)}\alpha(s)\,\mathrm{d}s. 0 = i ≥ 1 ∑ k = 1 ∑ m i γ i z i , k ϕ i , k , γ i = ∫ 0 T e − λ i ( T − s ) α ( s ) d s . Taking the inner product with each basis element yields
γ i z i , k = 0 , i ≥ 1 , k = 1 , … , m i . \gamma_i z_{i,k}=0,\qquad i\geq1,\quad k=1,\ldots,m_i. γ i z i , k = 0 , i ≥ 1 , k = 1 , … , m i . Assume the nonvanishing moment condition
γ i ≠ 0 for every i ≥ 1. \gamma_i\ne0\qquad\text{for every }i\geq1. γ i = 0 for every i ≥ 1. Then z i , k = 0 z_{i,k}=0 z i , k = 0 i , k i,k i , k z = 0 z=0 z = 0
Thus the observation operator is injective in both cases.
Proof of main results
We prove the stability and uniqueness results for the time-dependent velocity field. Since the background field F \mathbf{F} F t ∈ ( 0 , T ) t \in(0,T) t ∈ ( 0 , T )
By linearity, it suffices to consider the difference of two solutions with the same initial magnetic data and the same background field F \mathbf{F} F Δ F \Delta\mathbf{F} Δ F
For i = 1 , 2 i=1,2 i = 1 , 2
v i ( t , x ) = α ( t ) u i ( x ) , z i = ∇ × ( u i × F ) . \mathbf{v}_i(t,x)=\alpha(t)\mathbf{u}_i(x), \qquad\mathbf{z}_i=\nabla\times(\mathbf{u}_i \times\mathbf{F}). v i ( t , x ) = α ( t ) u i ( x ) , z i = ∇ × ( u i × F ) . Set
u = u 1 − u 2 , v = v 1 − v 2 = α u , z = z 1 − z 2 = ∇ × ( u × F ) . \mathbf{u}=\mathbf{u}_1-\mathbf{u}_2, \qquad\mathbf{v}=\mathbf{v}_1-\mathbf{v}_2=\alpha\mathbf{u}, \qquad\mathbf{z}=\mathbf{z}_1-\mathbf{z}_2=\nabla\times(\mathbf{u}\times\mathbf{F}). u = u 1 − u 2 , v = v 1 − v 2 = α u , z = z 1 − z 2 = ∇ × ( u × F ) . Consequently, the difference source is
h ( t ) = α ( t ) z , g ( t ) = γ − v ( t ) = α ( t ) γ − u . \mathbf{h}(t)=\alpha(t)\mathbf{z}, \qquad\mathbf{g}(t)=\gamma_-\mathbf{v}(t)=\alpha(t)\gamma_-\mathbf{u}. h ( t ) = α ( t ) z , g ( t ) = γ − v ( t ) = α ( t ) γ − u . Throughout the proof, we assume α ∈ L 2 ( 0 , T ) \alpha\in L^2(0,T) α ∈ L 2 ( 0 , T ) ∥ α ∥ L 2 ( 0 , T ) > 0 \|\alpha\|_{L^2(0,T)}>0 ∥ α ∥ L 2 ( 0 , T ) > 0
Proof of the stability result
Proof of Theorem 5.1. By the regularity assumption in Theorem 5.1,
γ − u ∈ [ H 1 ( Γ i n ) ] 3 . \gamma_-\mathbf{u}\in[H^1(\Gamma_{\mathrm{in}})]^3. γ − u ∈ [ H 1 ( Γ in ) ] 3 . Hence, for almost every t ∈ ( 0 , T ) t\in(0,T) t ∈ ( 0 , T )
g ( t ) = α ( t ) γ − u ∈ [ H 1 ( Γ i n ) ] 3 . \mathbf{g}(t)=\alpha(t)\gamma_-\mathbf{u}\in[H^1(\Gamma_{\mathrm{in}})]^3. g ( t ) = α ( t ) γ − u ∈ [ H 1 ( Γ in ) ] 3 . Moreover, the compatibility condition is preserved under multiplication by α ( t ) \alpha(t) α ( t ) C \mathcal{C} C C ( γ − u ) = 0 \mathcal{C}(\gamma_-\mathbf{u})=0 C ( γ − u ) = 0
C ( g ( t ) ) = α ( t ) C ( γ − u ) = 0 for a.e. t ∈ ( 0 , T ) . \mathcal{C}(\mathbf{g}(t))=\alpha(t)\mathcal{C}(\gamma_-\mathbf{u})=0 \quad\text{for a.e. }t\in(0,T). C ( g ( t )) = α ( t ) C ( γ − u ) = 0 for a.e. t ∈ ( 0 , T ) . Therefore, Proposition 4.1 applies to v ( t ) \mathbf{v}(t) v ( t ) t ∈ ( 0 , T ) t\in(0,T) t ∈ ( 0 , T )
∥ v ( t ) ∥ X F 2 ≤ C ( ∥ h ( t ) ∥ H 2 + ∥ g ( t ) ∥ [ H 1 ( Γ i n ) ] 3 2 ) . (47) \|\mathbf{v}(t)\|_{\mathcal{X}_{\mathbf{F}}}^2\leq C\left(\|\mathbf{h}(t)\|_{\mathcal{H}}^2+\|\mathbf{g}(t)\|_{[H^1(\Gamma_{\mathrm{in}})]^3}^2\right). \tag*{(47)} ∥ v ( t ) ∥ X F 2 ≤ C ( ∥ h ( t ) ∥ H 2 + ∥ g ( t ) ∥ [ H 1 ( Γ in ) ] 3 2 ) . (47) where C > 0 C>0 C > 0 t t t
Integrating (47) over ( 0 , T ) (0,T) ( 0 , T ) v = α u \mathbf{v}=\alpha\mathbf{u} v = α u h = α z \mathbf{h}=\alpha\mathbf{z} h = α z g = α γ − u \mathbf{g}=\alpha\gamma_-\mathbf{u} g = α γ − u
∥ α ∥ L 2 ( 0 , T ) 2 ∥ u ∥ X F 2 ≤ C ∥ α ∥ L 2 ( 0 , T ) 2 ( ∥ z ∥ H 2 + ∥ γ − u ∥ [ H 1 ( Γ i n ) ] 3 2 ) . \|\alpha\|_{L^2(0,T)}^2\|\mathbf{u}\|_{\mathcal{X}_{\mathbf{F}}}^2\leq C\|\alpha\|_{L^2(0,T)}^2\left(\|\mathbf{z}\|_{\mathcal{H}}^2+\|\gamma_-\mathbf{u}\|_{[H^1(\Gamma_{\mathrm{in}})]^3}^2\right). ∥ α ∥ L 2 ( 0 , T ) 2 ∥ u ∥ X F 2 ≤ C ∥ α ∥ L 2 ( 0 , T ) 2 ( ∥ z ∥ H 2 + ∥ γ − u ∥ [ H 1 ( Γ in ) ] 3 2 ) . Since ∥ α ∥ L 2 ( 0 , T ) > 0 \|\alpha\|_{L^2(0,T)}>0 ∥ α ∥ L 2 ( 0 , T ) > 0 ∥ α ∥ L 2 ( 0 , T ) 2 \|\alpha\|_{L^2(0,T)}^2 ∥ α ∥ L 2 ( 0 , T ) 2
∥ u ∥ X F 2 ≤ C ( ∥ z ∥ H 2 + ∥ γ − u ∥ [ H 1 ( Γ i n ) ] 3 2 ) . (48) \|\mathbf{u}\|_{\mathcal{X}_{\mathbf{F}}}^2\leq C\left(\|\mathbf{z}\|_{\mathcal{H}}^2+\|\gamma_-\mathbf{u}\|_{[H^1(\Gamma_{\mathrm{in}})]^3}^2\right). \tag*{(48)} ∥ u ∥ X F 2 ≤ C ( ∥ z ∥ H 2 + ∥ γ − u ∥ [ H 1 ( Γ in ) ] 3 2 ) . (48) By assumption (i) of Theorem 5.1, the auxiliary inverse-source problem for the difference field satisfies
∥ z ∥ H 2 ≤ Θ ( ∥ L ( y 1 − y 2 ) ∥ H o b s 2 ) , (49) \|\mathbf{z}\|_{\mathcal{H}}^2\leq\Theta\left(\|\mathcal{L}(\mathbf{y}_1-\mathbf{y}_2)\|_{\mathcal{H}_{\mathrm{obs}}}^2\right), \tag*{(49)} ∥ z ∥ H 2 ≤ Θ ( ∥ L ( y 1 − y 2 ) ∥ H obs 2 ) , (49) where y i \mathbf{y}_i y i z i \mathbf{z}_i z i
Since the two magnetic perturbations have the same initial data and background field, their difference satisfies
b 1 − b 2 = y 1 − y 2 . \mathbf{b}_1-\mathbf{b}_2=\mathbf{y}_1-\mathbf{y}_2. b 1 − b 2 = y 1 − y 2 . By linearity of L \mathcal{L} L
L ( y 1 − y 2 ) = L ( b 1 ) − L ( b 2 ) . (50) \mathcal{L}(\mathbf{y}_1-\mathbf{y}_2)=\mathcal{L}(\mathbf{b}_1)-\mathcal{L}(\mathbf{b}_2). \tag*{(50)} L ( y 1 − y 2 ) = L ( b 1 ) − L ( b 2 ) . (50) Combining (48) , (49) , and (50) , we conclude that
∥ u 1 − u 2 ∥ X F 2 ≤ C [ Θ ( ∥ L ( b 1 ) − L ( b 2 ) ∥ H o b s 2 ) + ∥ γ − ( u 1 − u 2 ) ∥ [ H 1 ( Γ i n ) ] 3 2 ] . \|\mathbf{u}_1-\mathbf{u}_2\|_{\mathcal{X}_{\mathbf{F}}}^2\leq C\left[\Theta\left(\|\mathcal{L}(\mathbf{b}_1)-\mathcal{L}(\mathbf{b}_2)\|_{\mathcal{H}_{\mathrm{obs}}}^2\right)+\|\gamma_-(\mathbf{u}_1-\mathbf{u}_2)\|_{[H^1(\Gamma_{\mathrm{in}})]^3}^2\right]. ∥ u 1 − u 2 ∥ X F 2 ≤ C [ Θ ( ∥ L ( b 1 ) − L ( b 2 ) ∥ H obs 2 ) + ∥ γ − ( u 1 − u 2 ) ∥ [ H 1 ( Γ in ) ] 3 2 ] . This is the asserted estimate (38).
Proof of the uniqueness result
Proof of Theorem 5.5 . Suppose that
Λ L ( v 1 ) = Λ L ( v 2 ) . \Lambda_{\mathcal{L}}(v_1)=\Lambda_{\mathcal{L}}(v_2). Λ L ( v 1 ) = Λ L ( v 2 ) . By the definition of Λ L \Lambda_{\mathcal{L}} Λ L
L ( b 1 ) = L ( b 2 ) , γ − v 1 = γ − v 2 . \mathcal{L}(b_1)=\mathcal{L}(b_2),\qquad\gamma_-v_1=\gamma_-v_2. L ( b 1 ) = L ( b 2 ) , γ − v 1 = γ − v 2 . Set
u = u 1 − u 2 , z = ∇ × ( u × F ) = z 1 − z 2 . u=u_1-u_2,\qquad z=\nabla\times(u\times F)=z_1-z_2. u = u 1 − u 2 , z = ∇ × ( u × F ) = z 1 − z 2 . As in the stability proof, the common initial contribution and known terms cancel, and therefore
b 1 − b 2 = y 1 − y 2 . b_1-b_2=y_1-y_2. b 1 − b 2 = y 1 − y 2 . It follows that
O ( z ) = L ( y 1 − y 2 ) = L ( b 1 ) − L ( b 2 ) = 0. \mathcal{O}(z)=\mathcal{L}(y_1-y_2)=\mathcal{L}(b_1)-\mathcal{L}(b_2)=0. O ( z ) = L ( y 1 − y 2 ) = L ( b 1 ) − L ( b 2 ) = 0. By assumption (i) of Theorem 5.5 , the operator O \mathcal{O} O
z = 0 , that is, ∇ × ( u × F ) = 0 in Ω . z=0,\qquad\text{that is,}\qquad\nabla\times(u\times F)=0\quad\text{in }\Omega. z = 0 , that is, ∇ × ( u × F ) = 0 in Ω. Furthermore, since v i = α u i v_i=\alpha u_i v i = α u i
α ( t ) γ − u = 0 in L 2 ( 0 , T ; L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) ) . \alpha(t)\gamma_-u=0\quad\text{in }L^2(0,T;L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,d\sigma)). α ( t ) γ − u = 0 in L 2 ( 0 , T ; L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ )) . Thus
0 = ∥ α γ − u ∥ L 2 ( 0 , T ; L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) ) = ∥ α ∥ L 2 ( 0 , T ) ∥ γ − u ∥ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) . 0=\|\alpha\gamma_-u\|_{L^2(0,T;L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,d\sigma))}
=\|\alpha\|_{L^2(0,T)}\|\gamma_-u\|_{L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,d\sigma)}. 0 = ∥ α γ − u ∥ L 2 ( 0 , T ; L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ )) = ∥ α ∥ L 2 ( 0 , T ) ∥ γ − u ∥ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ ) . Since ∥ α ∥ L 2 ( 0 , T ) > 0 \|\alpha\|_{L^2(0,T)}>0 ∥ α ∥ L 2 ( 0 , T ) > 0
γ − u = 0 in L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) . \gamma_-u=0\quad\text{in }L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,d\sigma). γ − u = 0 in L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ ) . We have therefore obtained
∇ × ( u × F ) = 0 , γ − u = 0. \nabla\times(u\times F)=0,\qquad\gamma_-u=0. ∇ × ( u × F ) = 0 , γ − u = 0. Applying Proposition 4.1 with zero curl data and zero inflow data gives
∥ u ∥ X F = 0. \|u\|_{X_F}=0. ∥ u ∥ X F = 0. Hence u 1 = u 2 u_1=u_2 u 1 = u 2 Λ L \Lambda_{\mathcal{L}} Λ L □ \square □
Numerical Simulations
We present three-dimensional numerical experiments comparing two reconstruction strategies for the spatial velocity field u u u
v ( t , x ) = α ( t ) u ( x ) , v(t,x)=\alpha(t)u(x), v ( t , x ) = α ( t ) u ( x ) , where α ∈ L 2 ( 0 , T ) \alpha\in L^2(0,T) α ∈ L 2 ( 0 , T ) α ( t ) ≡ 1 \alpha(t)\equiv1 α ( t ) ≡ 1 v = u v=u v = u
h = Δ F + z , z = ∇ × ( u × F ) . h=\Delta F+z,\qquad z=\nabla\times(u\times F). h = Δ F + z , z = ∇ × ( u × F ) . The first strategy is the two-stage approach developed above. It first reconstructs the auxiliary source z z z y y y u u u (25) . The second strategy reconstructs u u u
∇ ⋅ u = 0 in Ω , γ − u = g o b s on Γ i n . \nabla\cdot u = 0 \quad\text{in } \Omega,\qquad\gamma_{-}u = g_{\mathrm{obs}} \quad\text{on } \Gamma_{\mathrm{in}}. ∇ ⋅ u = 0 in Ω , γ − u = g obs on Γ in . We consider the terminal observation operator
L ( b ) = b ( T ) ∈ H . \mathcal{L}(b) = b(T) \in\mathbb{H}. L ( b ) = b ( T ) ∈ H . Here b ( u ) b(u) b ( u ) (7) generated by v = α u v=\alpha u v = α u u u u
Two-stage reconstruction
In the first stage, we reconstruct the auxiliary source z z z y y y α \alpha α Δ F \Delta F Δ F Δ F + α z \Delta F+\alpha z Δ F + α z (33) , however, the unknown source is α z \alpha z α z
Accordingly, y o b s ( T ) y_{\mathrm{obs}}(T) y obs ( T ) Δ F \Delta F Δ F y y y b b b
The reconstructed source must be compatible with the prescribed inflow trace in order for the second-stage curl–div problem to be solvable.
For a tolerance parameter ε ≥ 0 \varepsilon\ge0 ε ≥ 0
G a d m = { g ∈ [ H 1 ( Γ i n ) ] 3 : ∥ C ( g ) ∥ L 2 ( Γ i n ) ≤ ε } . \mathbb{G}_{\mathrm{adm}}=\left\{g\in[H^{1}(\Gamma_{\mathrm{in}})]^{3}:\|\mathcal{C}(g)\|_{L^{2}(\Gamma_{\mathrm{in}})}\le\varepsilon\right\}. G adm = { g ∈ [ H 1 ( Γ in ) ] 3 : ∥ C ( g ) ∥ L 2 ( Γ in ) ≤ ε } . where C \mathcal{C} C (26) , with its dependence on the data understood as specified in that definition.
The exact compatibility condition is C ( g ) = 0 \mathcal{C}(g)=0 C ( g ) = 0 ε > 0 \varepsilon>0 ε > 0 G a d m \mathbb{G}_{\mathrm{adm}} G adm
For the terminal observation, we reconstruct z z z
z ∗ = argmin z ∈ H { 1 2 ∥ y ( z ) ( T ) − y o b s ( T ) ∥ H 2 + γ 2 ∥ z ∥ H 2 } (52) z^{*}=\underset{z\in\mathbb{H}}{\operatorname{argmin}}\left\{\frac{1}{2}\|y(z)(T)-y_{\mathrm{obs}}(T)\|_{\mathbb{H}}^{2}+\frac{\gamma}{2}\|z\|_{\mathbb{H}}^{2}\right\} \tag*{(52)} z ∗ = z ∈ H argmin { 2 1 ∥ y ( z ) ( T ) − y obs ( T ) ∥ H 2 + 2 γ ∥ z ∥ H 2 } (52) where γ > 0 \gamma>0 γ > 0 y ( z ) y(z) y ( z ) (33) with source α z \alpha z α z
In the second stage, we solve the curl–div problem (25) with source z ∗ z^{*} z ∗ g o b s g_{\mathrm{obs}} g obs u ∗ u^{*} u ∗
The transport equation involved in the second stage is discretized using a SUPG finite-element method; see [36 ] . The complete procedure is summarized in Algorithm 1.
Direct reconstruction
In the direct approach, the auxiliary source is eliminated using
z = ∇ × ( u × F ) , z=\nabla\times(u\times F), z = ∇ × ( u × F ) , and u u u
u ∗ = argmin u ∈ Z { 1 2 ∥ b ( u ) ( T ) − b o b s ( T ) ∥ H 2 + γ 2 ∥ u ∥ X F 2 } (53) u^{*}=\underset{u\in Z}{\operatorname{argmin}}\left\{\frac{1}{2}\|b(u)(T)-b_{\mathrm{obs}}(T)\|_{\mathbb{H}}^{2}+\frac{\gamma}{2}\|u\|_{\mathbb{X}_{F}}^{2}\right\} \tag*{(53)} u ∗ = u ∈ Z argmin { 2 1 ∥ b ( u ) ( T ) − b obs ( T ) ∥ H 2 + 2 γ ∥ u ∥ X F 2 } (53) where b ( u ) b(u) b ( u ) (7) with v = α u v=\alpha u v = α u b o b s ( T ) b_{\mathrm{obs}}(T) b obs ( T )
Z = { u ∈ X F : ∇ ⋅ u = 0 in Ω , γ − u = g o b s on Γ i n } . Z=\left\{u\in\mathbb{X}_{F}:\nabla\cdot u=0\ \text{in }\Omega,\quad\gamma_{-}u=g_{\mathrm{obs}}\ \text{on }\Gamma_{\mathrm{in}}\right\}. Z = { u ∈ X F : ∇ ⋅ u = 0 in Ω , γ − u = g obs on Γ in } . The minimization is understood over a nonempty admissible set.
The two approaches differ in the variable optimized directly. The two-stage method first reconstructs z z z u u u u u u u u u
Algorithm 1 Two-stage reconstruction of u u u
Require: Initial state y 0 y_0 y 0 y o b s ( T ) y_{\mathrm{obs}}(T) y obs ( T ) F \mathbf{F} F α \alpha α g o b s g_{\mathrm{obs}} g obs γ > 0 \gamma> 0 γ > 0 ε ≥ 0 \varepsilon\ge0 ε ≥ 0
Ensure: Reconstructed velocity field u ∗ u^* u ∗
Compute ∥ C ( g o b s ) ∥ L 2 ( Γ i n ) \lVert C(g_{\mathrm{obs}})\rVert_{L^2(\Gamma_{\mathrm{in}})} ∥ C ( g obs ) ∥ L 2 ( Γ in )
if ∥ C ( g o b s ) ∥ L 2 ( Γ i n ) > ε \lVert C(g_{\mathrm{obs}})\rVert_{L^2(\Gamma_{\mathrm{in}})} > \varepsilon ∥ C ( g obs ) ∥ L 2 ( Γ in ) > ε
Reject the boundary datum, or replace it by compatible data and use those data consistently in the subsequent steps.
end if
Reconstruct z ∗ z^* z ∗
argmin z ∈ H { 1 2 ∥ y ( z ) ( T ) − y o b s ( T ) ∥ H 2 + γ 2 ∥ z ∥ H 2 } \underset{z \in\mathbb{H}}{\operatorname{argmin}}\left\{\frac{1}{2}\lVert y(z)(T)-y_{\mathrm{obs}}(T)\rVert_{\mathbb{H}}^2+\frac{\gamma}{2}\lVert z\rVert_{\mathbb{H}}^2\right\} z ∈ H argmin { 2 1 ∥ y ( z ) ( T ) − y obs ( T ) ∥ H 2 + 2 γ ∥ z ∥ H 2 } where y ( z ) y(z) y ( z )
Check that z ∗ z^* z ∗
Solve the curl–div problem (25) with source z ∗ z^* z ∗ g o b s g_{\mathrm{obs}} g obs
return u ∗ u^* u ∗
Application to a manufactured solution
We test both reconstruction strategies on the three-dimensional unit ball
Ω = { ( x 1 , x 2 , x 3 ) ∈ R 3 : x 1 2 + x 2 2 + x 3 2 < 1 } . \Omega= \{(x_1,x_2,x_3) \in\mathbb{R}^3 : x_1^2+x_2^2+x_3^2 < 1\}. Ω = {( x 1 , x 2 , x 3 ) ∈ R 3 : x 1 2 + x 2 2 + x 3 2 < 1 } . whose outward unit normal is n = ( x 1 , x 2 , x 3 ) T \mathbf{n}=(x_1,x_2,x_3)^{T} n = ( x 1 , x 2 , x 3 ) T
F = ( 1 , 0 , 0 ) T . \mathbf{F}=(1,0,0)^{T}. F = ( 1 , 0 , 0 ) T . Thus F ∈ C 2 ( Ω ‾ ) 3 \mathbf{F} \in C^2(\overline{\Omega})^3 F ∈ C 2 ( Ω ) 3 ∇ ⋅ F = 0 \nabla\cdot\mathbf{F}=0 ∇ ⋅ F = 0 Δ F = 0 \Delta\mathbf{F}=0 Δ F = 0 α ≡ 1 \alpha\equiv1 α ≡ 1
v = u , h = z , z = ∇ × ( u × F ) . \mathbf{v}=\mathbf{u},\qquad h=z,\qquad z=\nabla\times(\mathbf{u}\times\mathbf{F}). v = u , h = z , z = ∇ × ( u × F ) . The flow generated by F \mathbf{F} F
X ( t , x 0 ) = x 0 + t ( 1 , 0 , 0 ) . \mathbf{X}(t,\mathbf{x}_0)=\mathbf{x}_0+t(1,0,0). X ( t , x 0 ) = x 0 + t ( 1 , 0 , 0 ) . Since F ⋅ n = x 1 \mathbf{F}\cdot\mathbf{n}=x_1 F ⋅ n = x 1
Γ i n = { x ∈ ∂ Ω : x 1 < 0 } . \Gamma_{\mathrm{in}}=\{\mathbf{x}\in\partial\Omega:x_1<0\}. Γ in = { x ∈ ∂ Ω : x 1 < 0 } . Every point in Ω \Omega Ω Γ i n \Gamma_{\mathrm{in}} Γ in x 1 = 0 x_1=0 x 1 = 0
u t r u e ( x 1 , x 2 , x 3 ) = ( 0 x 1 x 3 − x 1 x 2 ) (54) \mathbf{u}_{\mathrm{true}}(x_1,x_2,x_3)=
\begin{pmatrix}
0\\
x_1x_3\\
-x_1x_2
\end{pmatrix} \tag*{(54)} u true ( x 1 , x 2 , x 3 ) = 0 x 1 x 3 − x 1 x 2 (54) and prescribe the inflow datum
g = γ − u t r u e . g=\gamma_-\mathbf{u}_{\mathrm{true}}. g = γ − u true . We verify the properties of the manufactured source. First,
∇ ⋅ u t r u e = ∂ 2 ( x 1 x 3 ) + ∂ 3 ( − x 1 x 2 ) = 0. \nabla\cdot\mathbf{u}_{\mathrm{true}}=\partial_2(x_1x_3)+\partial_3(-x_1x_2)=0. ∇ ⋅ u true = ∂ 2 ( x 1 x 3 ) + ∂ 3 ( − x 1 x 2 ) = 0. Moreover, on ∂ Ω \partial\Omega ∂ Ω
u t r u e ⋅ n = x 1 x 2 x 3 − x 1 x 2 x 3 = 0. \mathbf{u}_{\mathrm{true}}\cdot\mathbf{n}=x_1x_2x_3-x_1x_2x_3=0. u true ⋅ n = x 1 x 2 x 3 − x 1 x 2 x 3 = 0. Thus the exact velocity is divergence-free and satisfies the no-penetration condition.
Since F F F
z t r u e = ∇ × ( u t r u e × F ) = ∂ 1 u t r u e = ( 0 x 3 − x 2 ) . z_{\mathrm{true}}=\nabla\times(u_{\mathrm{true}}\times F)=\partial_1u_{\mathrm{true}}=
\begin{pmatrix}0\\x_3\\-x_2\end{pmatrix}. z true = ∇ × ( u true × F ) = ∂ 1 u true = 0 x 3 − x 2 . It follows that
∇ ⋅ z t r u e = 0. \nabla\cdot z_{\mathrm{true}}=0. ∇ ⋅ z true = 0. Furthermore, on ∂ Ω \partial\Omega ∂ Ω
z t r u e ⋅ n = ( 0 , x 3 , − x 2 ) ⋅ ( x 1 , x 2 , x 3 ) = x 2 x 3 − x 2 x 3 = 0. z_{\mathrm{true}}\cdot n=(0,x_3,-x_2)\cdot(x_1,x_2,x_3)=x_2x_3-x_2x_3=0. z true ⋅ n = ( 0 , x 3 , − x 2 ) ⋅ ( x 1 , x 2 , x 3 ) = x 2 x 3 − x 2 x 3 = 0. Hence z t r u e ∈ H z_{\mathrm{true}}\in\mathbb{H} z true ∈ H H \mathbb{H} H
The compatibility condition for the manufactured boundary datum holds because g g g u t r u e u_{\mathrm{true}} u true z t r u e z_{\mathrm{true}} z true
Implementation details
The computational mesh is generated using MATLAB’s multisphere(1) geometry with mesh-size parameter 0.15, yielding a tetrahedral mesh and continuous piecewise linear (P 1 P_1 P 1
Weak formulation of the b b b Let T h \mathcal{T}_h T h
V h = { φ h ∈ [ P 1 ( T h ) ] 3 : φ h ⋅ n = 0 on ∂ Ω h } . V_h=\left\{\varphi_h\in[P_1(\mathcal{T}_h)]^3:\varphi_h\cdot n=0\text{ on }\partial\Omega_h\right\}. V h = { φ h ∈ [ P 1 ( T h ) ] 3 : φ h ⋅ n = 0 on ∂ Ω h } . where the boundary condition is imposed with respect to the normal on the polyhedral boundary ∂ Ω h \partial\Omega_h ∂ Ω h
Using backward Euler with time step Δ t \Delta t Δ t b h n ∈ V h b_h^n\in V_h b h n ∈ V h φ h ∈ V h \varphi_h\in V_h φ h ∈ V h
( b h n − b h n − 1 Δ t , φ h ) + ( ∇ × b h n , ∇ × φ h ) = ( h h n , φ h ) . (55) \left(\frac{b_h^n-b_h^{n-1}}{\Delta t},\varphi_h\right)+(\nabla\times b_h^n,\nabla\times\varphi_h)=(h_h^n,\varphi_h). \tag*{(55)} ( Δ t b h n − b h n − 1 , φ h ) + ( ∇ × b h n , ∇ × φ h ) = ( h h n , φ h ) . (55) The discrete source is
h h n = Δ F n + ∇ × ( v h n × F n ) = Δ F n + α n z h n . (56) h_h^n=\Delta F^n+\nabla\times(v_h^n\times F^n)=\Delta F^n+\alpha^n z_h^n. \tag*{(56)} h h n = Δ F n + ∇ × ( v h n × F n ) = Δ F n + α n z h n . (56) where
z h n = ∇ × ( u h n × F n ) . z_h^n=\nabla\times(u_h^n\times F^n). z h n = ∇ × ( u h n × F n ) . For the manufactured test, α ≡ 1 \alpha\equiv1 α ≡ 1 Δ F = 0 \Delta F=0 Δ F = 0 h h n = z h n h_h^n=z_h^n h h n = z h n
The curl–curl term gives the natural boundary condition n × ( ∇ × b ) = 0 n\times(\nabla\times b)=0 n × ( ∇ × b ) = 0 b h ⋅ n = 0 b_h\cdot n=0 b h ⋅ n = 0 V h V_h V h
The curl–curl formulation alone does not enforce ∇ ⋅ b h = 0 \nabla\cdot b_h=0 ∇ ⋅ b h = 0
Finite-element scheme. Let M v M_v M v A h A_h A h H h n H_h^n H h n h h n h_h^n h h n
( M v + Δ t A h ) b n = M v b n − 1 + Δ t H h n . (M_v+\Delta t\,A_h)b^n=M_vb^{n-1}+\Delta t\,H_h^n. ( M v + Δ t A h ) b n = M v b n − 1 + Δ t H h n . If the code enforces the divergence constraint by projection, the computed field is then updated according to
b h n ← P b n . b_h^n\leftarrow\mathbb{P}b^n. b h n ← P b n . where P \mathbb{P} P P \mathbb{P} P [37 ] ; the precise projection used here must be consistent with the implementation.
For the time discretization, we take
T = 1 , N t = 100 , Δ t = 0.01 , α ( t ) ≡ 1. T = 1,\qquad N_t = 100,\qquad\Delta t = 0.01,\qquad\alpha(t) \equiv1. T = 1 , N t = 100 , Δ t = 0.01 , α ( t ) ≡ 1. The regularization parameter is set to γ = 10 − 6 \gamma= 10^{-6} γ = 1 0 − 6
To assess sensitivity to measurement noise, we perturb the exact terminal observation according to
Y o b s ( T ) = Y o b s e x a c t ( T ) + η , \mathbf{Y}_{\mathrm{obs}}(T) = \mathbf{Y}_{\mathrm{obs}}^{\mathrm{exact}}(T) + \boldsymbol{\eta}, Y obs ( T ) = Y obs exact ( T ) + η , where the Gaussian noise realization is normalized so that
∥ η ∥ 2 ∥ Y o b s e x a c t ( T ) ∥ 2 = δ , δ ∈ { 0 , 0.05 , 0.10 , 0.20 } . \frac{\lVert\boldsymbol{\eta}\rVert_2}{\lVert\mathbf{Y}_{\mathrm{obs}}^{\mathrm{exact}}(T)\rVert_2} = \delta,\qquad\delta\in\{0,0.05,0.10,0.20\}. ∥ Y obs exact ( T ) ∥ 2 ∥ η ∥ 2 = δ , δ ∈ { 0 , 0.05 , 0.10 , 0.20 } . The same realization of the noisy terminal observation is supplied to both reconstruction methods at each noise level. The norm used in this normalization is the discrete Euclidean norm of the observation vector; it should not be confused with the H H H
We measure the reconstruction error using the relative X F X_F X F
E u = ∥ u r e c − u t r u e ∥ X F ∥ u t r u e ∥ X F . E_u = \frac{\lVert u_{\mathrm{rec}}-u_{\mathrm{true}}\rVert_{X_F}}{\lVert u_{\mathrm{true}}\rVert_{X_F}}. E u = ∥ u true ∥ X F ∥ u rec − u true ∥ X F . The reported values are specific to the mesh, time step, regularization parameter, noise realizations, and discrete norms used in the implementation. Although the same numerical value of γ \gamma γ
Results and Discussion
We compare the reconstructed velocity fields with the exact field u t r u e u_{\mathrm{true}} u true (54) . Table 1 reports the relative errors for the two methods at the four noise levels.
Table 1. Relative reconstruction error E u E_u E u
For the noiseless observation, both methods yield small reconstruction errors, with the two-stage method producing the smaller error in this test. As the noise level increases, the difference between the reported errors becomes more pronounced. At 20% noise, the relative error is 0.078 for the two-stage method and 0.482 for the direct method.
These results indicate that, for this manufactured example and the reported numerical settings, the two-stage method is less sensitive to the tested observation perturbations. In the two-stage approach, the auxiliary source is reconstructed through a regularized optimization problem before the curl–div system is solved. The second stage uses the transport structure of the stationary problem; in the implementation, the source is integrated along characteristics using a 12-point Gauss–Legendre quadrature.
The direct method instead reconstructs the velocity in a single optimization step, with the divergence and inflow constraints imposed simultaneously. The larger errors in the table suggest greater sensitivity to the tested noise realizations for this discretization and regularization choice. The observed errors alone, however, do not identify a unique cause for this difference. In particular, the two methods use different regularization norms, and their effective regularization strengths need not be equal even when the numerical parameter γ \gamma γ
Appendix
Proof of Proposition 4.1
Proof. We split the argument into four steps.
Figure 1. Slice at z = 0 z=0 z = 0 u t r u e \mathbf{u}_{\mathrm{true}} u true (54) .
Figure 2. Slice at z = 0 z=0 z = 0 u \mathbf{u} u two-stage method from the terminal observation under varying noise levels. (a) 0% noise (b) 5% noise (c) 10% noise (d) 20% noise
Figure 3. Slice at z = 0 z=0 z = 0 u \mathbf{u} u direct method from the same observation under varying noise levels. (a) 0% noise (b) 5% noise (c) 10% noise (d) 20% noise
Step 1: Reduction to a linear transport system. By using (19) and since [ D F u ] i = ∑ j = 1 3 ( ∂ j F i ) u j = ( u ⋅ ∇ ) F i [D F\mathbf{u}]_i=\sum_{j=1}^{3}(\partial_j F_i)u_j=(\mathbf{u}\cdot\nabla)F_i [ D F u ] i = ∑ j = 1 3 ( ∂ j F i ) u j = ( u ⋅ ∇ ) F i ( u ⋅ ∇ ) F = D F u (\mathbf{u}\cdot\nabla)F=D F\mathbf{u} ( u ⋅ ∇ ) F = D F u ( F ⋅ ∇ ) u = D u F (F\cdot\nabla)\mathbf{u}=D\mathbf{u}\,F ( F ⋅ ∇ ) u = D u F (25) is equivalent to the first-order linear transport system with zero-order coupling
{ D u F − D F u = z in Ω , u = g on Γ i n , (57) \begin{cases}
D\mathbf{u}\,F-D F\mathbf{u}=z & \text{in }\Omega,\\
\mathbf{u}=g & \text{on }\Gamma_{\mathrm{in}}, \tag*{(57)}
\end{cases} { D u F − D F u = z u = g in Ω , on Γ in , (57) together with the divergence constraint ∇ ⋅ u = 0 \nabla\cdot\mathbf{u}=0 ∇ ⋅ u = 0 Ω \Omega Ω
Step 2: Existence and energy estimate. Because F ∈ [ C 2 ( Ω ‾ ) ] 3 F\in[C^2(\overline{\Omega})]^3 F ∈ [ C 2 ( Ω ) ] 3 X X X C 1 C^1 C 1 Ω \Omega Ω x ∈ Ω x\in\Omega x ∈ Ω ( t , x 0 ) ∈ [ 0 , T f i l l ] × Γ i n (t,x_0)\in[0,T_{\mathrm{fill}}]\times\Gamma_{\mathrm{in}} ( t , x 0 ) ∈ [ 0 , T fill ] × Γ in x = X ( t , x 0 ) x=X(t,x_0) x = X ( t , x 0 ) τ ( x 0 ) ∈ ( 0 , T f i l l ] \tau(x_0)\in(0,T_{\mathrm{fill}}] τ ( x 0 ) ∈ ( 0 , T fill ] x 0 x_0 x 0
For x 0 ∈ Γ i n x_0\in\Gamma_{\mathrm{in}} x 0 ∈ Γ in R ( t , s ; x 0 ) R(t,s;x_0) R ( t , s ; x 0 ) x 0 x_0 x 0 C 1 C^1 C 1
∂ t R ( t , s ; x 0 ) = D F ( X ( t , x 0 ) ) R ( t , s ; x 0 ) , R ( s , s ; x 0 ) = I , (58) \partial_t R(t,s;x_0)=D F(X(t,x_0))R(t,s;x_0),\qquad R(s,s;x_0)=I, \tag*{(58)} ∂ t R ( t , s ; x 0 ) = D F ( X ( t , x 0 )) R ( t , s ; x 0 ) , R ( s , s ; x 0 ) = I , (58) for 0 ≤ s , t ≤ τ ( x 0 ) 0\le s,t\le\tau(x_0) 0 ≤ s , t ≤ τ ( x 0 ) F ∈ [ C 2 ( Ω ‾ ) ] 3 F\in[C^2(\overline{\Omega})]^3 F ∈ [ C 2 ( Ω ) ] 3 D F ∘ X D F\circ X D F ∘ X (58) has a unique C 1 C^1 C 1
∥ R ( t , s ; x 0 ) ∥ ≤ e ∥ D F ∥ ∞ ∣ t − s ∣ , 0 ≤ s , t ≤ τ ( x 0 ) , x 0 ∈ Γ i n . (59) \lVert R(t,s;x_0)\rVert\le e^{\lVert D F\rVert_{\infty}|t-s|},\qquad0\le s,t\le\tau(x_0),\quad x_0\in\Gamma_{\mathrm{in}}. \tag*{(59)} ∥ R ( t , s ; x 0 )∥ ≤ e ∥ D F ∥ ∞ ∣ t − s ∣ , 0 ≤ s , t ≤ τ ( x 0 ) , x 0 ∈ Γ in . (59) Since the space [ C c ∞ ( Ω ) ] 3 ∩ { ∇ ⋅ = 0 } [C^\infty_c(\Omega)]^3 \cap\{\nabla\cdot= 0\} [ C c ∞ ( Ω ) ] 3 ∩ { ∇ ⋅ = 0 } H \mathbb{H} H [38 Theorem 1.4] and [ C ∞ ( Γ i n ) ] 3 [C^\infty(\Gamma_{\mathrm{in}})]^3 [ C ∞ ( Γ in ) ] 3 L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) L^2(\Gamma_{\mathrm{in}};\tau|F \cdot n|\,\mathrm{d}\sigma) L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ ) z ∈ [ C c ∞ ( Ω ) ] 3 ∩ { ∇ ⋅ z = 0 } z \in[C^\infty_c(\Omega)]^3 \cap\{\nabla\cdot z = 0\} z ∈ [ C c ∞ ( Ω ) ] 3 ∩ { ∇ ⋅ z = 0 } g ∈ [ C ∞ ( Γ i n ) ] 3 g \in[C^\infty(\Gamma_{\mathrm{in}})]^3 g ∈ [ C ∞ ( Γ in ) ] 3 u u u
u ( X ( t , x 0 ) ) : = R ( t , 0 ; x 0 ) g ( x 0 ) + ∫ 0 t R ( t , s ; x 0 ) z ( X ( s , x 0 ) ) d s , (60) u(X(t,x_0)) := R(t,0;x_0)g(x_0) + \int_0^t R(t,s;x_0)z(X(s,x_0))\,\mathrm{d}s, \tag*{(60)} u ( X ( t , x 0 )) := R ( t , 0 ; x 0 ) g ( x 0 ) + ∫ 0 t R ( t , s ; x 0 ) z ( X ( s , x 0 )) d s , (60) for x 0 ∈ Γ i n x_0 \in\Gamma_{\mathrm{in}} x 0 ∈ Γ in 0 ≤ t ≤ τ ( x 0 ) 0 \le t \le\tau(x_0) 0 ≤ t ≤ τ ( x 0 ) (60) with respect to t t t (58) , we obtain
∂ t u ( X ( t , x 0 ) ) = D F ( X ( t , x 0 ) ) u ( X ( t , x 0 ) ) + z ( X ( t , x 0 ) ) . \partial_t u(X(t,x_0)) = DF(X(t,x_0))u(X(t,x_0)) + z(X(t,x_0)). ∂ t u ( X ( t , x 0 )) = D F ( X ( t , x 0 )) u ( X ( t , x 0 )) + z ( X ( t , x 0 )) . Since ∂ t u ( X ( t , x 0 ) ) = D u F ( X ( t , x 0 ) ) \partial_t u(X(t,x_0)) = Du F(X(t,x_0)) ∂ t u ( X ( t , x 0 )) = D u F ( X ( t , x 0 )) u u u (57) . Moreover, setting t = 0 t = 0 t = 0 (60) gives u ( x 0 ) = g ( x 0 ) u(x_0) = g(x_0) u ( x 0 ) = g ( x 0 )
To establish the stability estimate, let x 0 ∈ Γ i n x_0 \in\Gamma_{\mathrm{in}} x 0 ∈ Γ in t ∈ [ 0 , τ ( x 0 ) ] t \in[0,\tau(x_0)] t ∈ [ 0 , τ ( x 0 )] (60) and (59) ,
∣ u ( X ( t , x 0 ) ) ∣ ≤ e ∥ D F ∥ ∞ T f i l l ∣ g ( x 0 ) ∣ + ∫ 0 t e ∥ D F ∥ ∞ ( t − s ) ∣ z ( X ( s , x 0 ) ) ∣ d s . |u(X(t,x_0))| \le e^{\|DF\|_\infty T_{\mathrm{fill}}}|g(x_0)| + \int_0^t e^{\|DF\|_\infty(t-s)}|z(X(s,x_0))|\,\mathrm{d}s. ∣ u ( X ( t , x 0 )) ∣ ≤ e ∥ D F ∥ ∞ T fill ∣ g ( x 0 ) ∣ + ∫ 0 t e ∥ D F ∥ ∞ ( t − s ) ∣ z ( X ( s , x 0 )) ∣ d s . Squaring, using ( a + b ) 2 ≤ 2 a 2 + 2 b 2 (a+b)^2 \le2a^2+2b^2 ( a + b ) 2 ≤ 2 a 2 + 2 b 2
∣ u ( X ( t , x 0 ) ) ∣ 2 ≤ 2 e 2 ∥ D F ∥ ∞ T f i l l ∣ g ( x 0 ) ∣ 2 + 2 T f i l l e 2 ∥ D F ∥ ∞ T f i l l ∫ 0 t ∣ z ( X ( s , x 0 ) ) ∣ 2 d s . |u(X(t,x_0))|^2 \le2e^{2\|DF\|_\infty T_{\mathrm{fill}}}|g(x_0)|^2 + 2T_{\mathrm{fill}}e^{2\|DF\|_\infty T_{\mathrm{fill}}}\int_0^t |z(X(s,x_0))|^2\,\mathrm{d}s. ∣ u ( X ( t , x 0 )) ∣ 2 ≤ 2 e 2∥ D F ∥ ∞ T fill ∣ g ( x 0 ) ∣ 2 + 2 T fill e 2∥ D F ∥ ∞ T fill ∫ 0 t ∣ z ( X ( s , x 0 )) ∣ 2 d s . Integrating this inequality with respect to t t t 0 0 0 τ ( x 0 ) \tau(x_0) τ ( x 0 ) τ ( x 0 ) \tau(x_0) τ ( x 0 )
∫ 0 τ ( x 0 ) ∣ u ( X ( t , x 0 ) ) ∣ 2 d t ≤ 2 e 2 ∥ D F ∥ ∞ T f i l l τ ( x 0 ) ∣ g ( x 0 ) ∣ 2 + 2 T f i l l 2 e 2 ∥ D F ∥ ∞ T f i l l ∫ 0 τ ( x 0 ) ∣ z ( X ( s , x 0 ) ) ∣ 2 d s . \int_0^{\tau(x_0)} |u(X(t,x_0))|^2\,\mathrm{d}t \le2e^{2\|DF\|_\infty T_{\mathrm{fill}}}\tau(x_0)|g(x_0)|^2 + 2T_{\mathrm{fill}}^2e^{2\|DF\|_\infty T_{\mathrm{fill}}}\int_0^{\tau(x_0)} |z(X(s,x_0))|^2\,\mathrm{d}s. ∫ 0 τ ( x 0 ) ∣ u ( X ( t , x 0 )) ∣ 2 d t ≤ 2 e 2∥ D F ∥ ∞ T fill τ ( x 0 ) ∣ g ( x 0 ) ∣ 2 + 2 T fill 2 e 2∥ D F ∥ ∞ T fill ∫ 0 τ ( x 0 ) ∣ z ( X ( s , x 0 )) ∣ 2 d s . Multiplying by ∣ F ( x 0 ) ⋅ n ( x 0 ) ∣ |F(x_0)\cdot n(x_0)| ∣ F ( x 0 ) ⋅ n ( x 0 ) ∣ Γ i n \Gamma_{\mathrm{in}} Γ in (68) yields
∥ u ∥ H 2 ≤ C ( ∥ z ∥ H 2 + ∥ g ∥ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) 2 ) , (61) \|u\|_{\mathbb{H}}^2 \le C\left(\|z\|_{\mathbb{H}}^2+\|g\|_{L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,\mathrm{d}\sigma)}^2\right), \tag*{(61)} ∥ u ∥ H 2 ≤ C ( ∥ z ∥ H 2 + ∥ g ∥ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ ) 2 ) , (61) with C = C ( T f i l l , ∥ D F ∥ ∞ ) > 0 C=C(T_{\mathrm{fill}},\|DF\|_\infty)>0 C = C ( T fill , ∥ D F ∥ ∞ ) > 0 (57) we have D u F = D F u + z Du F=DF u+z D u F = D F u + z
∥ D u F ∥ H ≤ ∥ D F ∥ L ∞ ∥ u ∥ H + ∥ z ∥ H . \|Du F\|_{\mathbb{H}} \le\|DF\|_{L^\infty}\|u\|_{\mathbb{H}}+\|z\|_{\mathbb{H}}. ∥ D u F ∥ H ≤ ∥ D F ∥ L ∞ ∥ u ∥ H + ∥ z ∥ H . Squaring and combining with (61) gives the estimate
∥ u ∥ H 2 + ∥ D u F ∥ H 2 ≤ C ( ∥ z ∥ H 2 + ∥ g ∥ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) 2 ) , (62) \|u\|_{\mathbb{H}}^2+\|Du F\|_{\mathbb{H}}^2 \le C\left(\|z\|_{\mathbb{H}}^2+\|g\|_{L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,\mathrm{d}\sigma)}^2\right), \tag*{(62)} ∥ u ∥ H 2 + ∥ D u F ∥ H 2 ≤ C ( ∥ z ∥ H 2 + ∥ g ∥ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ ) 2 ) , (62) for smooth data.
Density argument: extension to general data. The estimate (62) was proved for smooth data z ∈ [ C c ∞ ( Ω ) ] 3 ∩ { ∇ ⋅ z = 0 } z \in[C^\infty_c(\Omega)]^3 \cap\{\nabla\cdot z=0\} z ∈ [ C c ∞ ( Ω ) ] 3 ∩ { ∇ ⋅ z = 0 } g ∈ [ C ∞ ( Γ i n ) ] 3 g \in[C^\infty(\Gamma_{\mathrm{in}})]^3 g ∈ [ C ∞ ( Γ in ) ] 3
z ∈ H , g ∈ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) . z \in\mathbb{H}, \qquad g \in L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,\mathrm{d}\sigma). z ∈ H , g ∈ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ ) . Choose sequences z n ∈ z ∈ [ C ∞ ( Ω ) ] 3 ∩ { ∇ ⋅ = 0 } z_n \in z \in[C^\infty(\Omega)]^3 \cap\{\nabla\cdot=0\} z n ∈ z ∈ [ C ∞ ( Ω ) ] 3 ∩ { ∇ ⋅ = 0 } g n ∈ [ C ∞ ( Γ i n ) ] 3 g_n \in[C^\infty(\Gamma_{\mathrm{in}})]^3 g n ∈ [ C ∞ ( Γ in ) ] 3 z n → z z_n \to z z n → z H \mathbb{H} H g n → g g_n \to g g n → g L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,\mathrm{d}\sigma) L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ ) n n n u n u_n u n (60) . Applying (62) to u n − u m u_n-u_m u n − u m
∥ u n − u m ∥ H 2 + ∥ D ( u n − u m ) F ∥ H 2 ≤ C ( ∥ z n − z m ∥ H 2 + ∥ g n − g m ∥ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) 2 ) . \|u_n-u_m\|_{\mathbb{H}}^2+\|D(u_n-u_m)F\|_{\mathbb{H}}^2 \le C\left(\|z_n-z_m\|_{\mathbb{H}}^2+\|g_n-g_m\|_{L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,\mathrm{d}\sigma)}^2\right). ∥ u n − u m ∥ H 2 + ∥ D ( u n − u m ) F ∥ H 2 ≤ C ( ∥ z n − z m ∥ H 2 + ∥ g n − g m ∥ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ ) 2 ) . Hence ( u n ) (u_n) ( u n ) ( D u n F ) (Du_n F) ( D u n F ) [ L 2 ( Ω ) ] 3 [L^2(\Omega)]^3 [ L 2 ( Ω ) ] 3 u u u ( u n ) (u_n) ( u n ) φ ∈ C c ∞ ( Ω ) \varphi\in C^\infty_c(\Omega) φ ∈ C c ∞ ( Ω ) D u F − D F u = z Du F-DFu=z D u F − D F u = z H \mathbb{H} H γ − \gamma_- γ − L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,\mathrm{d}\sigma) L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ ) u n u_n u n u ∈ [ L 2 ( Ω ) ] 3 u \in[L^2(\Omega)]^3 u ∈ [ L 2 ( Ω ) ] 3 D u F ∈ [ L 2 ( Ω ) ] 3 Du F \in[L^2(\Omega)]^3 D u F ∈ [ L 2 ( Ω ) ] 3
∫ Γ i n ∣ γ − u ∣ 2 τ ∣ F ⋅ n ∣ d σ ≤ 2 ∥ u ∥ H 2 + T f i l l 2 ∥ D u F ∥ H 2 . (63) \int_{\Gamma_{\mathrm{in}}}|\gamma_-u|^2\tau|F\cdot n|\,\mathrm{d}\sigma\le2\|u\|_{\mathbb{H}}^2+T_{\mathrm{fill}}^2\|Du F\|_{\mathbb{H}}^2. \tag*{(63)} ∫ Γ in ∣ γ − u ∣ 2 τ ∣ F ⋅ n ∣ d σ ≤ 2∥ u ∥ H 2 + T fill 2 ∥ D u F ∥ H 2 . (63) Applying (63) to u n − u u_n-u u n − u γ − u n = g n → g \gamma_-u_n=g_n\to g γ − u n = g n → g γ − u = g \gamma_-u=g γ − u = g (62) ,
∥ u ∥ H 2 + ∥ D u F ∥ H 2 ≤ C ( ∥ z ∥ H 2 + ∥ g ∥ L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) 2 ) . (64) \|u\|_{\mathbb{H}}^2+\|Du F\|_{\mathbb{H}}^2 \le C\left(\|z\|_{\mathbb{H}}^2+\|g\|_{L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,\mathrm{d}\sigma)}^2\right). \tag*{(64)} ∥ u ∥ H 2 + ∥ D u F ∥ H 2 ≤ C ( ∥ z ∥ H 2 + ∥ g ∥ L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ ) 2 ) . (64) Step 3: Necessity and sufficiency of the compatibility condition. Necessity. Assume u ∈ X F u\in X_F u ∈ X F ∇ ⋅ u = 0 \nabla\cdot u=0 ∇ ⋅ u = 0
Approximate ( z , g ) (z,g) ( z , g ) ( z n , g n ) (z_n,g_n) ( z n , g n ) z n ∈ [ C ∞ ( Ω ‾ ) ] 3 ∩ H z_n\in[C^\infty(\overline{\Omega})]^3\cap\mathbb{H} z n ∈ [ C ∞ ( Ω ) ] 3 ∩ H g n ∈ [ C ∞ ( Γ i n ) ] 3 g_n\in[C^\infty(\Gamma_{\mathrm{in}})]^3 g n ∈ [ C ∞ ( Γ in ) ] 3 z n → z z_n\to z z n → z H \mathbb{H} H g n → g g_n\to g g n → g [ H 1 ( Γ i n ) ] 3 [H^1(\Gamma_{\mathrm{in}})]^3 [ H 1 ( Γ in ) ] 3 u n u_n u n
∥ u n − u ∥ H + ∥ D ( u n − u ) F ∥ H → 0 , \|u_n-u\|_{\mathbb{H}}+\|D(u_n-u)F\|_{\mathbb{H}}\to0, ∥ u n − u ∥ H + ∥ D ( u n − u ) F ∥ H → 0 , and γ − u n = g n → g \gamma_-u_n=g_n\to g γ − u n = g n → g L 2 ( Γ i n ; τ ∣ F ⋅ n ∣ d σ ) L^2(\Gamma_{\mathrm{in}};\tau|F\cdot n|\,d\sigma) L 2 ( Γ in ; τ ∣ F ⋅ n ∣ d σ )
For smooth u n u_n u n F F F
F Γ = F − ( F ⋅ n ) n . F_{\Gamma}=F-(F\cdot n)n. F Γ = F − ( F ⋅ n ) n . On Γ i n \Gamma_{\mathrm{in}} Γ in D u n F = D Γ u n F Γ + ( F ⋅ n ) ∂ n u n D u_nF=D_{\Gamma}u_nF_{\Gamma}+(F\cdot n)\partial_nu_n D u n F = D Γ u n F Γ + ( F ⋅ n ) ∂ n u n n n n u n = g n u_n=g_n u n = g n Γ i n \Gamma_{\mathrm{in}} Γ in
D u n F ⋅ n = ( D Γ g n F Γ ) ⋅ n + ( F ⋅ n ) ( ∂ n u n ⋅ n ) . (65) D u_nF\cdot n=(D_{\Gamma}g_nF_{\Gamma})\cdot n+(F\cdot n)(\partial_nu_n\cdot n). \tag*{(65)} D u n F ⋅ n = ( D Γ g n F Γ ) ⋅ n + ( F ⋅ n ) ( ∂ n u n ⋅ n ) . (65) From the transport equation (57), we have D u n F − D F u n = z n D u_nF-DFu_n=z_n D u n F − D F u n = z n z n ⋅ n = 0 z_n\cdot n=0 z n ⋅ n = 0 ∂ Ω \partial\Omega ∂ Ω
D u n F ⋅ n = ( D F g n ) ⋅ n . (66) D u_nF\cdot n=(DFg_n)\cdot n. \tag*{(66)} D u n F ⋅ n = ( D F g n ) ⋅ n . (66) From ∇ ⋅ u n = 0 \nabla\cdot u_n=0 ∇ ⋅ u n = 0 [39 formula (5.19), p. 522] gives
∂ n u n ⋅ n = − ∇ Γ i n ⋅ g n . (67) \partial_nu_n\cdot n=-\nabla_{\Gamma_{\mathrm{in}}\!}\cdot g_n. \tag*{(67)} ∂ n u n ⋅ n = − ∇ Γ in ⋅ g n . (67) Substituting (67) into (65) and comparing with (66),
( F ⋅ n ) ∇ Γ i n ⋅ g n + ( D F g n − D Γ g n F Γ ) ⋅ n = 0 , (F\cdot n)\nabla_{\Gamma_{\mathrm{in}}\!}\cdot g_n+(DFg_n-D_{\Gamma}g_nF_{\Gamma})\cdot n=0, ( F ⋅ n ) ∇ Γ in ⋅ g n + ( D F g n − D Γ g n F Γ ) ⋅ n = 0 , i.e. C ( g n ) = 0 C(g_n)=0 C ( g n ) = 0 C C C g n → g g_n\to g g n → g [ H 1 ( Γ i n ) ] 3 [H^1(\Gamma_{\mathrm{in}})]^3 [ H 1 ( Γ in ) ] 3 C ( g n ) → C ( g ) C(g_n)\to C(g) C ( g n ) → C ( g ) L 2 ( Γ i n ) L^2(\Gamma_{\mathrm{in}}) L 2 ( Γ in ) C ( g ) = 0 C(g)=0 C ( g ) = 0 Γ i n \Gamma_{\mathrm{in}} Γ in
Sufficiency. We assume C ( g ) = 0 C(g)=0 C ( g ) = 0 Γ i n \Gamma_{\mathrm{in}} Γ in u u u q : = ∇ ⋅ u q:=\nabla\cdot u q := ∇ ⋅ u
For smooth data, q q q z = D u F − D F u z=D uF-DFu z = D u F − D F u ∇ ⋅ z = 0 \nabla\cdot z=0 ∇ ⋅ z = 0 ∇ ⋅ ( D u F ) − ∇ ⋅ ( D F u ) = 0 \nabla\cdot(DuF)-\nabla\cdot(DFu)=0 ∇ ⋅ ( D u F ) − ∇ ⋅ ( D F u ) = 0
∇ ⋅ ( D u F ) = ( F ⋅ ∇ ) q + ∑ i , j ( ∂ i F j ) ( ∂ j u i ) , \nabla\cdot(DuF)=(F\cdot\nabla)q+\sum_{i,j}(\partial_iF_j)(\partial_ju_i), ∇ ⋅ ( D u F ) = ( F ⋅ ∇ ) q + i , j ∑ ( ∂ i F j ) ( ∂ j u i ) , ∇ ⋅ ( D F u ) = u ⋅ ∇ ( ∇ ⋅ F ) + ∑ i , j ( ∂ j F i ) ( ∂ i u j ) , \nabla\cdot(DFu)=u\cdot\nabla(\nabla\cdot F)+\sum_{i,j}(\partial_jF_i)(\partial_iu_j), ∇ ⋅ ( D F u ) = u ⋅ ∇ ( ∇ ⋅ F ) + i , j ∑ ( ∂ j F i ) ( ∂ i u j ) , and ∇ ⋅ F = 0 \nabla\cdot F=0 ∇ ⋅ F = 0 i ↔ j i\leftrightarrow j i ↔ j ( F ⋅ ∇ ) q = 0 (F\cdot\nabla)q=0 ( F ⋅ ∇ ) q = 0 D ′ ( Ω ) D'(\Omega) D ′ ( Ω )
On Γ i n \Gamma_{\mathrm{in}} Γ in [39 formula (5.19), p. 522] applied to u u u
q = ∇ Γ i n ⋅ g + ∂ n u ⋅ n . q=\nabla_{\Gamma_{\mathrm{in}}\!}\cdot g+\partial_nu\cdot n. q = ∇ Γ in ⋅ g + ∂ n u ⋅ n . Combined with the boundary decomposition of the transport equation and z ⋅ n = 0 z\cdot n=0 z ⋅ n = 0 ∂ Ω \partial\Omega ∂ Ω
( F ⋅ n ) q ∣ Γ i n = ( F ⋅ n ) ∇ Γ i n ⋅ g + ( D F g − D Γ g F Γ ) ⋅ n = C ( g ) . (F\cdot n)q|_{\Gamma_{\mathrm{in}}}=(F\cdot n)\nabla_{\Gamma_{\mathrm{in}}\!}\cdot g+(DFg-D_{\Gamma}gF_{\Gamma})\cdot n=C(g). ( F ⋅ n ) q ∣ Γ in = ( F ⋅ n ) ∇ Γ in ⋅ g + ( D F g − D Γ g F Γ ) ⋅ n = C ( g ) . Since C ( g ) = 0 C(g)=0 C ( g ) = 0 q ∣ Γ i n = 0 q|_{\Gamma_{\mathrm{in}}}=0 q ∣ Γ in = 0 F ⋅ n ≠ 0 F\cdot n\ne0 F ⋅ n = 0 ( F ⋅ ∇ ) q = 0 (F\cdot\nabla)q=0 ( F ⋅ ∇ ) q = 0 Ω \Omega Ω q ∣ Γ i n = 0 q|_{\Gamma_{\mathrm{in}}}=0 q ∣ Γ in = 0 Ω \Omega Ω q ≡ 0 q\equiv0 q ≡ 0 ∇ ⋅ u = 0 \nabla\cdot u=0 ∇ ⋅ u = 0
For general data, the same conclusion follows by density: approximate ( z , g ) (z,g) ( z , g ) C ( g n ) → 0 C(g_n)\to0 C ( g n ) → 0 u n u_n u n q n q_n q n C ( g n ) C(g_n) C ( g n ) z n z_n z n
Combining necessity and sufficiency, C ( g ) = 0 C(g)=0 C ( g ) = 0
Step 4: Uniqueness. Let u 1 u_1 u 1 u 2 u_2 u 2 ( z , g ) (z,g) ( z , g ) w : = u 1 − u 2 w:=u_1-u_2 w := u 1 − u 2 z = 0 z=0 z = 0 g = 0 g=0 g = 0 w w w
∥ w ∥ H 2 + ∥ D w F ∥ H 2 ≤ C ( 0 + 0 ) = 0. \|w\|_{\mathbb{H}}^2+\|DwF\|_{\mathbb{H}}^2\le C(0+0)=0. ∥ w ∥ H 2 + ∥ D w F ∥ H 2 ≤ C ( 0 + 0 ) = 0. Hence w = 0 w=0 w = 0 Ω \Omega Ω
Lemma 8.1. Let Ω ⊂ R d \Omega\subset\mathbb{R}^{d} Ω ⊂ R d C 1 C^{1} C 1 F ∈ [ C 2 ( Ω ‾ ) ] d F \in[C^{2}(\overline{\Omega})]^{d} F ∈ [ C 2 ( Ω ) ] d ∇ ⋅ F = 0 \nabla\cdot F = 0 ∇ ⋅ F = 0 Ω \Omega Ω ( t , x 0 ) ↦ X ( t , x 0 ) (t,x_{0}) \mapsto X(t,x_{0}) ( t , x 0 ) ↦ X ( t , x 0 ) F F F Ω \Omega Ω Definition 3.1 . Then the map
( t , x 0 ) ⟼ X ( t , x 0 ) , ( t , x 0 ) ∈ D = { ( t , x 0 ) : x 0 ∈ Γ i n , 0 < t < τ ( x 0 ) } , (t,x_{0}) \longmapsto X(t,x_{0}), \qquad(t,x_{0}) \in D = \{(t,x_{0}): x_{0} \in\Gamma_{\mathrm{in}},\ 0 < t < \tau(x_{0})\}, ( t , x 0 ) ⟼ X ( t , x 0 ) , ( t , x 0 ) ∈ D = {( t , x 0 ) : x 0 ∈ Γ in , 0 < t < τ ( x 0 )} , is a C 1 C^{1} C 1 Ω ∖ N \Omega\setminus\mathcal{N} Ω ∖ N N ⊂ Ω \mathcal{N} \subset\Omega N ⊂ Ω ∣ N ∣ = 0 |\mathcal{N}| = 0 ∣ N ∣ = 0 f ∈ L 1 ( Ω ) f \in L^{1}(\Omega) f ∈ L 1 ( Ω )
∫ Ω f ( x ) d x = ∫ Γ i n ∫ 0 τ ( x 0 ) f ( X ( t , x 0 ) ) ∣ F ( x 0 ) ⋅ n ( x 0 ) ∣ d t d σ ( x 0 ) . (68) \int_{\Omega} f(x)\,\mathrm{d}x = \int_{\Gamma_{\mathrm{in}}} \int_{0}^{\tau(x_{0})} f(X(t,x_{0})) |F(x_{0}) \cdot n(x_{0})|\,\mathrm{d}t\,\mathrm{d}\sigma(x_{0}). \tag*{(68)} ∫ Ω f ( x ) d x = ∫ Γ in ∫ 0 τ ( x 0 ) f ( X ( t , x 0 )) ∣ F ( x 0 ) ⋅ n ( x 0 ) ∣ d t d σ ( x 0 ) . (68) Proof. Since the flow X X X Ω \Omega Ω Lemma 3.2 , the flow map X X X D D D Ω ∖ N \Omega\setminus\mathcal{N} Ω ∖ N N ⊂ Ω \mathcal{N} \subset\Omega N ⊂ Ω ∣ N ∣ = 0 |\mathcal{N}| = 0 ∣ N ∣ = 0 τ ( x 0 ) ≤ T f i l l \tau(x_{0}) \le T_{\mathrm{fill}} τ ( x 0 ) ≤ T fill σ \sigma σ x 0 ∈ Γ i n x_{0} \in\Gamma_{\mathrm{in}} x 0 ∈ Γ in
Fix a local C 1 C^{1} C 1 x 0 = γ ( s ) x_{0} = \gamma(s) x 0 = γ ( s ) s ∈ U ⊂ R d − 1 s \in U \subset\mathbb{R}^{d-1} s ∈ U ⊂ R d − 1 Γ i n \Gamma_{\mathrm{in}} Γ in
W ( t , s ) : = ∇ ( t , s ) X ( t , γ ( s ) ) = [ ∂ t X ∂ s 1 X ⋯ ∂ s d − 1 X ] . W(t,s) := \nabla_{(t,s)}X(t,\gamma(s)) = \left[\partial_{t}X\quad\partial_{s_{1}}X\quad\cdots\quad\partial_{s_{d-1}}X\right]. W ( t , s ) := ∇ ( t , s ) X ( t , γ ( s )) = [ ∂ t X ∂ s 1 X ⋯ ∂ s d − 1 X ] . where the derivatives are evaluated at ( t , γ ( s ) ) (t,\gamma(s)) ( t , γ ( s )) ∂ t X = F ( X ) \partial_{t}X = F(X) ∂ t X = F ( X ) s i s_{i} s i
∂ t ∂ s i X = D F ( X ) ∂ s i X , i = 1 , … , d − 1 , \partial_{t}\partial_{s_{i}}X = DF(X)\,\partial_{s_{i}}X,\qquad i=1,\ldots,d-1, ∂ t ∂ s i X = D F ( X ) ∂ s i X , i = 1 , … , d − 1 , and similarly
∂ t F ( X ) = D F ( X ) F ( X ) . \partial_{t}F(X) = DF(X)\,F(X). ∂ t F ( X ) = D F ( X ) F ( X ) . Thus every column of W ( t , s ) W(t,s) W ( t , s )
Y ′ ( t ) = D F ( X ( t , γ ( s ) ) ) Y ( t ) . Y'(t) = DF(X(t,\gamma(s)))Y(t). Y ′ ( t ) = D F ( X ( t , γ ( s ))) Y ( t ) . By Liouville’s formula,
∂ t det W ( t , s ) = tr ( D F ( X ( t , γ ( s ) ) ) ) det W ( t , s ) = ( ∇ ⋅ F ) ( X ( t , γ ( s ) ) ) det W ( t , s ) . \partial_{t}\det W(t,s) = \operatorname{tr}(DF(X(t,\gamma(s))))\det W(t,s) = (\nabla\cdot F)(X(t,\gamma(s)))\det W(t,s). ∂ t det W ( t , s ) = tr ( D F ( X ( t , γ ( s )))) det W ( t , s ) = ( ∇ ⋅ F ) ( X ( t , γ ( s ))) det W ( t , s ) . Since ∇ ⋅ F = 0 \nabla\cdot F = 0 ∇ ⋅ F = 0 Ω \Omega Ω [40 Proposition 7.1]
det W ( t , s ) = det W ( 0 , s ) for all t . \det W(t,s) = \det W(0,s)\qquad\text{for all }t. det W ( t , s ) = det W ( 0 , s ) for all t . At t = 0 t=0 t = 0
W ( 0 , s ) = [ F ( γ ( s ) ) ∂ s 1 γ ( s ) ⋯ ∂ s d − 1 γ ( s ) ] . W(0,s) = \left[F(\gamma(s))\quad\partial_{s_{1}}\gamma(s)\quad\cdots\quad\partial_{s_{d-1}}\gamma(s)\right]. W ( 0 , s ) = [ F ( γ ( s )) ∂ s 1 γ ( s ) ⋯ ∂ s d − 1 γ ( s ) ] . Decompose F = F τ + ( F ⋅ n ) n F = F_{\tau} + (F \cdot n)n F = F τ + ( F ⋅ n ) n F τ F_{\tau} F τ Γ i n \Gamma_{\mathrm{in}} Γ in γ ( s ) \gamma(s) γ ( s ) ∂ s i γ ( s ) \partial_{s_{i}}\gamma(s) ∂ s i γ ( s )
∣ det W ( 0 , s ) ∣ = ∣ F ( γ ( s ) ) ⋅ n ( γ ( s ) ) ∣ ∣ det [ n ( γ ( s ) ) ∂ s 1 γ ( s ) ⋯ ∂ s d − 1 γ ( s ) ] ∣ . \left|\det W(0,s)\right| = \left|F(\gamma(s)) \cdot n(\gamma(s))\right|\left|\det\left[n(\gamma(s))\quad\partial_{s_{1}}\gamma(s)\quad\cdots\quad\partial_{s_{d-1}}\gamma(s)\right]\right|. ∣ det W ( 0 , s ) ∣ = ∣ F ( γ ( s )) ⋅ n ( γ ( s )) ∣ ∣ det [ n ( γ ( s )) ∂ s 1 γ ( s ) ⋯ ∂ s d − 1 γ ( s ) ] ∣ . Let G ( s ) G(s) G ( s ) ( d − 1 ) × ( d − 1 ) (d-1)\times(d-1) ( d − 1 ) × ( d − 1 ) G i j ( s ) = ∂ s i γ ( s ) ⋅ ∂ s j γ ( s ) G_{ij}(s) = \partial_{s_{i}}\gamma(s)\cdot\partial_{s_{j}}\gamma(s) G ij ( s ) = ∂ s i γ ( s ) ⋅ ∂ s j γ ( s )
∣ det [ n ( γ ( s ) ) ∂ s 1 γ ( s ) ⋯ ∂ s d − 1 γ ( s ) ] ∣ = det G ( s ) . \left|\det\left[n(\gamma(s))\quad\partial_{s_{1}}\gamma(s)\quad\cdots\quad\partial_{s_{d-1}}\gamma(s)\right]\right| = \sqrt{\det G(s)}. ∣ det [ n ( γ ( s )) ∂ s 1 γ ( s ) ⋯ ∂ s d − 1 γ ( s ) ] ∣ = det G ( s ) . Hence
∣ det W ( t , s ) ∣ = ∣ det W ( 0 , s ) ∣ = ∣ F ( γ ( s ) ) ⋅ n ( γ ( s ) ) ∣ det G ( s ) . \left|\det W(t,s)\right| = \left|\det W(0,s)\right| = \left|F(\gamma(s)) \cdot n(\gamma(s))\right|\sqrt{\det G(s)}. ∣ det W ( t , s ) ∣ = ∣ det W ( 0 , s ) ∣ = ∣ F ( γ ( s )) ⋅ n ( γ ( s )) ∣ det G ( s ) . On Γ i n \Gamma_{\mathrm{in}} Γ in F ⋅ n < 0 F\cdot n < 0 F ⋅ n < 0 ∣ F ⋅ n ∣ > 0 |F\cdot n| > 0 ∣ F ⋅ n ∣ > 0 det G ( s ) > 0 \det G(s) > 0 det G ( s ) > 0 det W ( t , s ) ≠ 0 \det W(t,s) \ne0 det W ( t , s ) = 0 D D D
Thus the flow map is a C 1 C^{1} C 1 D D D C 1 C^{1} C 1 D D D Ω ∖ N \Omega\setminus\mathcal{N} Ω ∖ N
Finally, apply the change-of-variables theorem on each boundary chart and use d σ ( x 0 ) = det G ( s ) d s \mathrm{d}\sigma(x_{0}) = \sqrt{\det G(s)}\,\mathrm{d}s d σ ( x 0 ) = det G ( s ) d s ∣ N ∣ = 0 |\mathcal{N}| = 0 ∣ N ∣ = 0
Concluding remarks and open questions
Boundary conditions. The absolute boundary conditions are not used crucially beyond giving a meaning to the divergence-free Hodge–Laplacian space and to the inflow trace. Other Hodge-type realizations (relative, mixed, or perfect-conductor versus insulating walls) could be considered, provided the curl–div system (25) remains well posed and the inflow trace estimate of Lemma 24 holds.
Full nonlinear induction equation. The analysis is restricted to the linearized perturbation problem (system (7)) around ( b , v ) = ( 0 , 0 ) (\boldsymbol{b},\boldsymbol{v})=(0,0) ( b , v ) = ( 0 , 0 )
Stability for internal observations. The quantitative stability estimate is proved only for the final-time observation in Theorem (5.3). Proving a stability estimate for internal observations on ( 0 , T ) × ω o b s (0,T)\times\omega_{\mathrm{obs}} ( 0 , T ) × ω obs
Compatibility and admissible data. The curl–div system is overdetermined and requires C ( g ) = 0 \mathcal{C}(\boldsymbol{g})=0 C ( g ) = 0 Γ i n \Gamma_{\mathrm{in}} Γ in (26) ). In practice the inflow trace is measured and may not satisfy this condition exactly. Constructing a stable projection onto the admissible set and quantifying the error propagation into the reconstruction is an open problem, closely related to the feasibility issue in the two-stage numerical method.
Role of the inflow trace and the gauge. Magnetic observations alone are blind to the velocity component parallel to F \boldsymbol{F} F
The separable ansatz. The analysis assumes v ( t , x ) = α ( t ) u ( x ) \boldsymbol{v}(t,\boldsymbol{x})=\alpha(t)\boldsymbol{u}(\boldsymbol{x}) v ( t , x ) = α ( t ) u ( x ) α ∈ L 2 ( 0 , T ) \alpha\in L^2(0,T) α ∈ L 2 ( 0 , T )
Time-dependent background fields. The background F \boldsymbol{F} F F ( t , x ) \boldsymbol{F}(t,\boldsymbol{x}) F ( t , x )
