Liouville Theorems for the Stationary Fractional Magnetohydrodynamic Equations

Introduction and main results

The Liouville problem for the stationary Navier–Stokes equations in $\mathbb{R}^{3}$ asks whether a smooth solution must vanish identically under suitable assumptions on its behavior at infinity. A classical version of this problem concerns solutions with finite Dirichlet integral,

and remains open in its full generality. An important contribution was made by Galdi, who proved that the additional assumption $u \in L^{9/2}(\mathbb{R}^{3})$ is sufficient to guarantee triviality. Since then, a number of Liouville-type criteria have been established under different integrability and regularity assumptions; see, for instance, [1, 8, 11, 16, 17, 20, 21] and the references therein.

A natural extension of this problem is obtained by replacing the Laplacian with a fractional power $(-\Delta)^{\alpha/2}$. This type of dissipation, which goes back to the work of J.-L. Lions [13], allows one to study stationary flows with different orders of dissipation. For the stationary fractional Navier–Stokes equations, Liouville-type results have been obtained in several settings; see, among others, [24, 23, 10, 15, 9, 12, 14]. The relation between the dissipation exponent and the integrability of the velocity field is particularly relevant here. Indeed, the homogeneous Sobolev embedding

shows that the exponent $9/2$ in Galdi’s criterion corresponds precisely to $\alpha= 5/3$. This observation provides a natural reference point for understanding the range of fractional dissipation considered in this paper.

Our main interest is the stationary magnetohydrodynamic (MHD) system, which describes the interaction between the velocity field of an electrically conducting fluid and its magnetic field. In this system, the velocity $u$ and the magnetic field $b$ are coupled through the Lorentz force and the induction equation, while $\Pi$ denotes the total pressure. For the classical stationary MHD equations, several Liouville-type theorem are available in the literature. See, for instance, [4, 5, 6, 18, 3]. The fractional setting introduces an additional difficulty in that the velocity and magnetic fields may have different dissipation orders, denoted by $\alpha$ and $\beta$, respectively. As a consequence, the two fields need not have the same regularity, and the interaction between their dissipative effects becomes an essential part of the analysis.

To the best of our knowledge, relatively few works have addressed Liouville-type theorems for the stationary fractional MHD system. Among the relevant contributions, we mention the results of Zeng [26] and Lee–Lee [12]. These works provide important criteria for the triviality of solutions, but leave room to investigate parameter ranges in which the two dissipation orders interact more directly. Thus, our purpose is to establish such criteria under the natural energy assumptions

without imposing additional integrability assumptions on either field in the parameter ranges covered by our first theorem.

More precisely, we consider the stationary fractional MHD system in $\mathbb{R}^{3}$:

with the uniform decay conditions

Here, the fractional Laplacians are defined through the Fourier multipliers with symbols $|\xi|^{\alpha}$ and $|\xi|^{\beta}$. The system includes the stationary fractional Navier–Stokes equations when $b \equiv0$, while the choice $\alpha= \beta= 2$ formally recovers the classical stationary MHD system.

Our first result identifies a region in the $(\alpha,\beta)$-plane where the energy assumptions alone imply that both fields vanish. To continue, we introduce the set

The significance of this region is that the admissible dissipation orders are not restricted to zones of the plane where $\beta\ge1$. Instead, the mixed interaction between the velocity and magnetic fields leads to a more flexible condition which exploits the combination $\alpha+ 2\beta$.

An immediate consequence of Theorem 1.1 is a Liouville theorem for the stationary fractional Navier–Stokes equations. In fact, by setting $b\equiv0$, the MHD system reduces to the fractional Navier–Stokes system, and the magnetic dissipation exponent $\beta$ can be chosen freely, provided that $(\alpha,\beta)\in\mathcal{A}$. This observation allows us to obtain the following result throughout the range $1\leq\alpha\leq5/3$, without any additional integrability assumption on the velocity field. To the best of our knowledge, Zeng was the first to establish this conclusion in the same range; see [26 Corollary 1.5]. Our proof, however, follows a different approach.

We next consider the parameter ranges outside $\mathcal{A}$. In this setting, we show that triviality can still be obtained by imposing suitable integrability assumptions on the radial components of the velocity and magnetic fields,

This approach is motivated by recent Liouville-type theorems for the classical and fractional stationary Navier–Stokes equations [9, 22]. In contrast to conditions imposed on individual Cartesian components, assumptions on radial components are invariant under rotations about the origin. Moreover, they do not require additional integrability of the tangential components of either field.

Our second result extends this perspective to the stationary fractional MHD system. It covers the parameter region outside $\mathcal{A}$, except for a region where the requirements arising from the velocity and magnetic fluxes are incompatible within our method. In particular, our theorem distinguishes the cases in which the radial integrability assumptions are needed for the velocity field, for the magnetic field, or for both. We present the result below.

Some remarks are in order.

Remark 1.2. We stress that Theorem 1.2 requires additional integrability assumptions only on the radial components of the velocity and magnetic fields, namely $u_{\rho}$ and $b_{\rho}$, without imposing any further integrability conditions on their tangential components of $u$ and of $b$. In particular, the criterion depends only on these two components, while the remaining components are constrained only by the Sobolev regularity assumed in the statement of the theorem.

Remark 1.3. For comparison, Zeng’s Liouville theorem for the stationary fractional MHD system in [26] is based on the assumption that

where $\vec{p}_j=(p_{j,1},p_{j,2},p_{j,3})$ and $\vec{q}_j=(q_{j,1},q_{j,2},q_{j,3})$, with $p_{j,l},q_{j,l}\in[3,\infty)$ satisfying

Thus, Zeng’s result imposes anisotropic integrability conditions on each Cartesian component of both fields, combining different integrability requirements along the coordinate directions to ensure the triviality of the solution. By contrast, Theorem 1.2 assumes only

for exponents $r_1,r_2 \in[1,\infty]$ satisfying (R1)–(R3). These restrictions depend on the fractional orders $\alpha$ and $\beta$ and allow us to formulate the Liouville criterion entirely in terms of the radial components, rather than imposing additional integrability conditions on each Cartesian component of the two fields.

region	condition on $u_{\rho}$	condition on $b_{\rho}$
$\mathcal{A}$	$1 \le\alpha\le\frac{5}{3},\ 3 \le\alpha+2\beta\le5$	none	none
I	$\alpha< 1,\ \alpha+2\beta< 3$	$r_1 \ge\frac{3}{\min\{\alpha,\beta\}}$	$r_2 \ge\frac{6}{\alpha+\beta}$
II	$\alpha< 1,\ \alpha+2\beta\ge3$	$r_1 \ge\frac{3}{\alpha}$	none
III	$1 \le\alpha\le\frac{5}{3},\ \alpha+2\beta< 3$	$r_1 \ge\frac{3}{\beta}$	$r_2 \ge\frac{6}{\alpha+\beta}$
IV	$1 \le\alpha\le\frac{5}{3},\ \alpha+2\beta> 5$	$r_1 \le\frac{3}{\beta-1}$	$r_2 \le\frac{6}{\alpha+\beta-2}$
V	$\alpha> \frac{5}{3},\ 3 \le\alpha+2\beta\le5$	$r_1 \le\frac{3}{\alpha-1}$	none
VI	$\alpha> \frac{5}{3},\ \alpha+2\beta> 5$	$r_1 \le\frac{3}{\max\{\alpha,\beta\}-1}$	$r_2 \le\frac{6}{\alpha+\beta-2}$
$\mathcal{G}$	$\alpha> \frac{5}{3},\ \alpha+2\beta< 3$	incompatible requirements	—

In particular, on $\mathcal{A}$, Theorem 1.2 contains no radial hypothesis and reduces to Theorem 1.1, whereas on each of the regions I–VI it provides a result with a one-sided condition.

The paper is organized as follows. In Section 2, we recall the Caffarelli–Silvestre extension and its main properties. Sections 3 and 4 contain the proofs of Theorems 1.1 and 1.2, respectively.

Given $R > 1$, in what follows, and throughout this paper, we use the notation

We let $\psi\in C^\infty(\mathbb{R}^3)$ be a smooth radial non-increasing cut-off function such that $0\leq\psi\leq1$, $\psi(x)=1$ for $|x|\leq1$, and $\psi(x)=0$ for $|x|\geq2$. Given $R>1$, set $\psi_R(x)=\psi\left(\frac{x}{R}\right)$.

We also consider a non-increasing function $\chi\in C^\infty([0,\infty))$ such that $\chi(y)=1$ for $y\leq1$ and $\chi(y)=0$ for $y\geq2$. Given $R>1$, we define $\chi_R(y)=\chi\left(\frac{y}{R}\right)$. Note that $|\nabla\psi_R|\leq CR^{-1}$, $|\Delta\psi_R|\leq CR^{-2}$, $|\chi_R'|\leq CR^{-1}$ and $|\chi_R''|\leq CR^{-2}$, for some absolute constant $C>0$.

Preliminaries

Since the operators $(-\Delta)^{\alpha/2}$ and $(-\Delta)^{\beta/2}$ are nonlocal, some of the classical arguments used to prove Liouville type theorems for the stationary Navier–Stokes and MHD equations cannot be carried over directly. Instead, we make use of the Caffarelli–Silvestre extension, which allows one to represent a fractional Laplacian on $\mathbb{R}^3$ as the boundary trace of a local problem involving the operator $\bar{\Delta}$ in the half-space $\mathbb{R}^4_+$. In the following, we shall write $\bar{\nabla}$, $\overline{\operatorname{div}}$ and $\bar{\Delta}$ for the differential operators associated with the extended variables in $\mathbb{R}^4_+$. Since the two equations in (1) involve different fractional Laplacians, we will need two extensions, one for $u$ and one for $b$, with different weights. For this reason we state the result for a generic exponent $\sigma$.

Then $v$ admits a unique extension $v^* \in\dot{H}^1(\mathbb{R}_+^4,y^\mu)$ satisfying

and

In addition, there exists $C_\sigma>0$, depending only on $\sigma$, such that:

(1) the operator $(-\Delta)^{\frac{\sigma}{2}}v$ is given by

(2) and, the energy equality

Moreover, $v^*(\cdot,y)=P_y^{(\sigma)}*v$, where $P_y^{(\sigma)}(x)=c_{3,\sigma}y^\sigma(|x|^2+y^2)^{-\frac{3+\sigma}{2}}$ is the fractional Poisson kernel and $c_{3,\sigma}>0$ is chosen so that $\|P_y^{(\sigma)}\|_{L^1(\mathbb{R}^3)}=1$.

In what follows we write

so that, by the Sobolev embedding, $u\in L^p(\mathbb{R}^3)$ and $b\in L^q(\mathbb{R}^3)$. We denote by $u^*$ and $b^*$ the Caffarelli–Silvestre extensions of $u$ and $b$, associated with the weights $y^\mu$ and $y^\nu$ respectively, and by $C_\alpha$, $C_\beta$ the corresponding constants given by Theorem 2.1. Note that $\mu,\nu>-1$ because $\alpha,\beta<2$.

Proof of Theorem 1.1

We begin with two lemmas that will be used in the proof of Theorem 1.1. The first is stated for a generic exponent, so that it can be applied both to $u$ and to $b$. Its proof is analogous to that of [9]; we include it here for completeness. The proof of the second is postponed to the end of this section.

Proof of Theorem 1.1. To begin, note that, since $u\in\dot{H}^{\alpha/2}(\mathbb{R}^3)$ and $b\in\dot{H}^{\beta/2}(\mathbb{R}^3)$, the Sobolev embedding gives $u\in L^p(\mathbb{R}^3)$ and $b\in L^q(\mathbb{R}^3)$. Since $\alpha,\beta>0$ we have $p,q>2$, and therefore $u_i u_j\in L^{p/2}(\mathbb{R}^3)$ and $b_i b_j\in L^{q/2}(\mathbb{R}^3)$ with $1<\frac p2,\frac q2<+\infty$. Then, taking the divergence of the momentum equation and using $\nabla\cdot u = \nabla\cdot b = 0$, we obtain the elliptic equation

Now, since (2) determines $\Pi$ only up to an additive harmonic function, we fix the pressure through the Calderón–Zygmund representation

where $R^i$ denotes the $i$-th Riesz transform on $\mathbb{R}^3$. By the Calderón–Zygmund theorem,

We claim that this is the only solution of (2) in $L^{\frac{p}{2}}(\mathbb{R}^3)+L^{\frac{q}{2}}(\mathbb{R}^3)$. Indeed, if $\widetilde{\Pi}$ is another solution in this space, then $h:=\Pi-\widetilde{\Pi}=h_1+h_2$ is harmonic on $\mathbb{R}^3$, with $h_1\in L^{\frac{p}{2}}(\mathbb{R}^3)$ and $h_2\in L^{\frac{q}{2}}(\mathbb{R}^3)$. Thus, considering the mean value property and Hölder’s inequality, for every $x\in\mathbb{R}^3$ and $\rho>0$,

thus, we get $h\equiv0$. Throughout the paper, $\Pi$ always denotes this normalized pressure. On the other hand, since $u$ and $b$ are smooth, by using elliptic regularity theory on (2), we conclude that the pressure term $\Pi$ is also smooth.

Then, testing the first equation of (1) with $\psi_Ru$ and the second one with $\psi_Rb$, using $\nabla\cdot u=\nabla\cdot b=0$ and integrating by parts, we get

We next treat the fractional dissipation terms on the left-hand side by means of the Caffarelli–Silvestre extension. For this purpose, we consider $\Phi(x,y):=\psi_R(x)\chi_R(y)$ and the identity (see [23 Eq. (3.2)]), applied to each component of $u$ with $\sigma=\alpha$ and to each component of $b$ with $\sigma=\beta$,

Then plugging (6) and (7) into (5), we write

Now, considering the supports of $\chi_{R}$, $\psi_{R}$ and the facts that $\overline{\nabla}\Phi(x,y)=(\chi_{R}\nabla\psi_{R},\psi_{R}\chi_{R}')$ and, for a generic weight $y^{\kappa}$, we can write

we notice that the first term is supported in $C_{R}\times[0,2R]$, while the second one is supported in $B_{2R}\times[R,2R]$, where $y^{\kappa-1}\leq R^{-1}y^{\kappa}$. We deduce the estimate

where $C$ depends on $\kappa$ but not on $R$. Take $(v,\kappa)=(u,\mu)$ and $(v,\kappa)=(b,\nu)$. Then, applying Lemma 3.1 (with $\sigma=\alpha$, respectively $\sigma=\beta$) with $E_{R}=C_{R}$, we conclude that the first term on the right-hand side tends to zero as $R\to\infty$. On the other hand, since $y^{\kappa}|v^{*}|^{2}\geq0$, we can write

therefore, considering Lemma 3.1 with $E_{R}=B_{2R}$ we obtain that the second term tends to zero also. Considering the same approach for $b^{*}$, we conclude

Now, note that, since $\psi_{R}$ is radial and $\nabla\psi_{R}$ is supported in $C_{R}$, we can write

where we used $|(u\cdot b)(b\cdot\nabla\psi_{R})|\leq|u||b|^{2}|\nabla\psi_{R}|$. By Lemma 3.2, the right-hand side tends to zero as $R\to\infty$.

Thus, provided with these results, we take the limit as $R\to\infty$ in (8). Since $\psi$ is radial and non-increasing, and $\chi$ is non-increasing, $\Phi=\Phi_{R}$ increases pointwise to $1$ as $R\to\infty$, and the monotone convergence theorem applied to the left-hand side of (8) yields

Thus, since $C_{\alpha},C_{\beta}>0$, the both integrals above vanish. Hence, we conclude that $u^{*}$ and $b^{*}$ are constant in $\mathbb{R}^{4}_{+}$, and so are their traces $u(x)=u^{*}(x,0)$ and $b(x)=b^{*}(x,0)$. Then, considering that $u\in L^{p}(\mathbb{R}^{3})$ and $b\in L^{q}(\mathbb{R}^{3})$, and the fact that the only constant function belonging to $L^{r}(\mathbb{R}^{3})$, $r<\infty$, is the zero function, we conclude that $u\equiv0$ and $b\equiv0$. This finishes the proof. □

Proof of Lemma 3.1. In the following we denote by $q_\sigma$ the Sobolev exponent, i.e. $q_\sigma:= \frac{6}{3-\sigma} > 2$, and we write $\kappa:= 1-\sigma\in(-1,1)$.

By Theorem 2.1, the Caffarelli–Silvestre extension can be written as

By Young’s inequality,

Now, considering Hölder inequality with $\frac{1}{q_\sigma/2}+\frac{1}{q_\sigma/(q_\sigma-2)}=1$, we get

Then, since $|E_R|^{1-2/q_\sigma}\leq|B_{2R}|^{1-2/q_\sigma}=CR^{3(1-2/q_\sigma)}=CR^\sigma$, we can write for all $y>0$

With this information at hand, we can write

Then, making the change of variables $y=Rt$, $dy=R\,dt$ and using that $\kappa=1-\sigma$, we get

Let $\varepsilon>0$. Now considering that $C^\infty_c(\mathbb{R}^3)$ is dense in $L^{q_\sigma}(\mathbb{R}^3)$, we can choose $v_\varepsilon\in C^\infty_c(\mathbb{R}^3)\subset L^1\cap L^{q_\sigma}(\mathbb{R}^3)$ such that $\|v-v_\varepsilon\|_{L^{q_\sigma}}<\varepsilon/2$. Then, considering Young inequality, we can write

Now, note that, by rescaling we have $P_y^{(\sigma)}(x)=y^{-3}P_1^{(\sigma)}(x/y)$, and then we obtain

Hence, for fixed $\varepsilon$, we obtain

With this in hand, we choose $y_0$ such that $K_\sigma y_0^{-(3+\sigma)/2}\|v_\varepsilon\|_{L^1}<\varepsilon/2$. Then, for each $y>y_0$, we use Young’s inequality to write

This implies that

On the other hand, considering (9) and the fact that $\kappa=1-\sigma>-1$ (since $\sigma<2$) we have

Moreover, using (11), for all $t\in(0,2)$ we have $\lVert v^*(\cdot,Rt)\rVert_{L^{q_\sigma}}^2\to0$ as $R\to\infty$. Then, considering the Dominated Convergence Theorem and (10), we conclude the claim of the lemma. □

Proof of Lemma 3.2. To begin, recall that $p=\frac{6}{3-\alpha}$ and $q=\frac{6}{3-\beta}$. Then, since $u\in\dot{H}^{\frac{\alpha}{2}}(\mathbb{R}^3)$ and $b\in\dot{H}^{\frac{\beta}{2}}(\mathbb{R}^3)$, by the Sobolev embedding we have $u\in L^p(\mathbb{R}^3)$ and $b\in L^q(\mathbb{R}^3)$, and $3\leq p\leq\frac{9}{2}$ because $1\leq\alpha\leq\frac{5}{3}$.

We first record an elementary scaling estimate. Let $\rho_1,\rho_2,\rho_3\in[1,\infty]$ be such that

Then, by Hölder’s inequality with exponents $\rho_1,\rho_2,\rho_3,\frac{1}{\theta}$, and using that $\lVert\nabla\psi_R\rVert_{L^\infty}\leq CR^{-1}$ and $|C_R|\leq CR^3$, we have, for all $R>1$ that

where for $\theta=0$ the last norm is understood as an $L^\infty$ norm. In particular, if $3\theta-1\leq0$ then $R^{3\theta-1}\leq1$ for $R>1$.

Now, to continue, we consider the notation

where $I_1(R)$ contains $|u|^3$, $I_2(R)$ contains $|b|^2|u|$, and $I_3(R)$ contains the pressure. With this in hand, we aim to prove

The term $I_1$. We apply (12) with $(\rho_1,\rho_2,\rho_3)=(p,p,p)$, for which $\theta_1=1-\frac{3}{p}=\frac{\alpha-1}{2}\in[0,\frac{1}{3}]$. Then $3\theta_1-1=\frac{3\alpha-5}{2}\leq0$, since $\alpha\leq\frac{5}{3}$, and we obtain

Thus, considering that $\lim_{R\to\infty}\lVert u\rVert_{L^p(C_R)}=0$, we conclude $I_1(R)\to0$ as $R\to+\infty$.

The term $I_2$. We apply (12) with $(\rho_1,\rho_2,\rho_3)=(p,q,q)$, for which

and $3\theta_2-1=\frac{\alpha+2\beta-5}{2}\leq0$, by the assumption $3\leq\alpha+2\beta\leq5$. Hence

since $\|b\|_{L^q(C_R)}\leq\|b\|_{L^q(\mathbb{R}^3)}<+\infty$ and $\|u\|_{L^p(C_R)}\to0$.

The term $I_3$. We split $\Pi=\Pi_1+\Pi_2$ as in (3). For $\Pi_1$ we apply (12) with $(\rho_1,\rho_2,\rho_3)=\left(\frac{p}{2},p,\infty\right)$ and $f_3\equiv1$, which gives the same $\theta_1=1-\frac{2}{p}-\frac{1}{p}=\frac{\alpha-1}{2}$ as before. Note that $\|f_3\|_{L^\infty(C_R)}=1$, so no volume factor appears, the only power of $R$ is $R^{3\theta_1-1}\leq1$, coming from $\|\nabla\psi_R\|_{L^{1/\theta_1}(C_R)}$. Hence, by (4), we can write

Now, for $\Pi_2$ we apply (12) with $(\rho_1,\rho_2,\rho_3)=\left(\frac{q}{2},p,\infty\right)$ and $f_3\equiv1$, so that $\theta=1-\frac{2}{q}-\frac{1}{p}=\theta_2$ and $R^{3\theta_2-1}\leq1$ as before. Thus, by (4), we get

Hence, we conclude that $I_3(R)\to0$ as $R\to+\infty$. Then, by gathering together the limits for $I_1(R)$, $I_2(R)$ and $I_3(R)$, we deduce the desired limit. □

Proof of Theorem 1.2

We begin with the lemma that will be used in the proof of Theorem 1.2, whose proof is presented in Subsection 4.1. The other tool we need, namely Lemma 3.1, was already stated and proved in § 3 for a generic exponent.

Proof of Theorem 1.2. The proof follows the lines of that of Theorem 1.1, and we only indicate the changes. We use the pressure $\Pi=\Pi_1+\Pi_2$ of (3) and the cut-offs $\psi_R$, $\chi_R$ and $\Phi_R$ applied there. In this section we also write $\Pi_u:=\Pi_1$ and $\Pi_b:=-\Pi_2$, so that $\Pi=\Pi_u-\Pi_b$ and, by (4),

Since $\psi_R$ is radial, we have $\nabla\psi_R(x)=\partial_\rho\psi_R(x)\frac{x}{|x|}$, with $|\partial_\rho\psi_R|=|\nabla\psi_R|\leq CR^{-1}$. Hence, we can write

Testing the equations with $\psi_Ru$ and $\psi_Rb$, using the Caffarelli–Silvestre extension and arguing exactly as in the proof of Theorem 1.1, we arrive at (8), and the two terms on the right-hand side of (8) involving $u^*$ and $b^*$ tend to zero as $R\to\infty$, by Lemma 3.1. The only difference is the treatment of the flux term, which now we write using $u\cdot\nabla\psi_R=u_\rho\partial_\rho\psi_R$, $b \cdot\nabla\psi_R = b_\rho\partial_\rho\psi_R$, $\Pi= \Pi_u - \Pi_b$ and $|u \cdot b| \le|u||b|$:

By Lemma Lemma 4.1, the right-hand side tends to zero as $R\to\infty$. Thus, taking the limit as $R\to\infty$ in (8), and using the monotone convergence theorem as in the proof of Theorem Theorem 1.1, we get

Hence $u$ and $b$ are constant, and then, since $u\in L^{p_\alpha}(\mathbb{R}^3)$ and $b\in L^{p_\beta}(\mathbb{R}^3)$ by the Sobolev embedding, and the only constant function belonging to $L^p(\mathbb{R}^3)$, $p<\infty$, is the zero function, we conclude that $u\equiv0$ and $b\equiv0$. This finishes the proof. □

Proof of Lemma 4.1

To begin, we stress the fact that each term in the proof below is estimated by Hölder’s inequality. This requires an exponent $\sigma$ for $u_\rho$ or $b_\rho$ such that

for a suitable $\tau$. The following elementary lemma gives a criterion for the existence of such an exponent.

Proof. By interpolation, $g\in L^\sigma(\mathbb{R}^3)$ whenever $1/\sigma$ lies between $1/p$ and $1/r$. We choose $1/\sigma$ by projecting $1/p$ onto the interval $[\tau-\frac{1}{3},\tau]$, more precisely

In the following we distinguish three cases. If

we take $\sigma=p$, and the conclusion follows. If $1/p>\tau$, then $1/\sigma=\tau$. The first condition in (13) gives $1/r\le\tau<1/p$, thus $1/\sigma$ lies between $1/r$ and $1/p$, and then $g\in L^\sigma(\mathbb{R}^3)$. Finally, if $1/p < \tau- \frac{1}{3}$, then $1/\sigma= \tau- \frac{1}{3}$. The second condition in (13) gives

and thus, considering again interpolation, we get $g \in L^\sigma(\mathbb{R}^3)$. In particular, note that in all three cases we have

Thus, we conclude that $1<\sigma<\infty$, and $0\leq\tau-\frac{1}{\sigma}\leq\frac{1}{3}$. This finishes the proof. $\square$

In the next, we apply this lemma with $|u_\rho|\leq|u|$ and $|b_\rho|\leq|b|$. At this point is convenient to recall that

We use the lemma with

For clarity of the exposition, the resulting exponents can be written explicitly as

and

At this point is convenient to recall that $\frac{1}{p_\alpha}=\frac{1}{2}-\frac{\alpha}{6}$, and $\frac{1}{p_\beta}=\frac{1}{2}-\frac{\beta}{6}$. With this, note that

As a consequence of this, the conditions (13) in the three applications above are exactly the assumptions (R1), (R2), and (R3), respectively.

Hence, under the hypotheses of Theorem 1.2, we have $u_\rho\in L^{\sigma_1}(\mathbb{R}^3)\cap L^{\sigma_2}(\mathbb{R}^3)$, and $b_\rho\in L^{\sigma_3}(\mathbb{R}^3)$, where

all belong to $\left[0,\frac{1}{3}\right]$.

Proof of Lemma 4.1. To begin, recall that $p_\alpha= \frac{6}{3-\alpha}$ and $p_\beta= \frac{6}{3-\beta}$. Then, since $u \in\dot{H}^{\frac{\alpha}{2}}(\mathbb{R}^3)$ and $b \in\dot{H}^{\frac{\beta}{2}}(\mathbb{R}^3)$, by the Sobolev embedding we have $u \in L^{p_\alpha}(\mathbb{R}^3)$ and $b \in L^{p_\beta}(\mathbb{R}^3)$, and by the Calderón–Zygmund estimate (4) we have $\Pi_u \in L^{\frac{p_\alpha}{2}}(\mathbb{R}^3)$ and $\Pi_b \in L^{\frac{p_\beta}{2}}(\mathbb{R}^3)$. Moreover, let $\sigma_1,\sigma_2,\sigma_3 \in[1,\infty]$ be the exponents constructed in the previous subsection, so that

and the numbers $\theta_1,\theta_2,\theta_3$ defined in (15) belong to $[0,\frac{1}{3}]$.

We first record an elementary estimate. Let $\varrho_1,\varrho_2,\varrho_3 \in[1,\infty]$ be such that

Then, by Hölder’s inequality with exponents $\varrho_1,\varrho_2,\varrho_3,\frac{1}{\theta}$, and using that $\|\partial_\rho\psi_R\|_{L^\infty} \leq CR^{-1}$ and $|C_R| \leq CR^3$, we have, for all $R>1$,

where for $\theta=0$ the last factor in the first line is understood as an $L^\infty$ norm, and the last inequality holds because $3\theta-1\leq0$ and $R>1$.

Now, to continue, we consider

where $I_1$ contains $|u|^2|u_\rho|$, $I_2$ contains $|b|^2|u_\rho|$, $I_3$ and $I_4$ contain $|\Pi_u||u_\rho|$ and $|\Pi_b||u_\rho|$ respectively, and $I_5$ contains $|u||b||b_\rho|$. With this in hand, we aim to prove that $\lim_{R\to+\infty}I_k(R)=0$ for $k=1,\ldots,5$.

The term $I_1$. Since $|u_\rho|\leq|u|$ and $\frac{1}{p_\alpha}=\frac{1}{2}-\frac{\alpha}{6}$, we apply (16) with $(\varrho_1,\varrho_2,\varrho_3)=(p_\alpha,p_\alpha,\sigma_1)$, for which

Hence

The term $I_2$. We apply (16) with $(\varrho_1,\varrho_2,\varrho_3)=(p_\beta,p_\beta,\sigma_2)$, for which $\theta=1-\frac{2}{p_\beta}-\frac{1}{\sigma_2}=\frac{\beta}{3}-\frac{1}{\sigma_2}=\theta_2\in[0,\frac{1}{3}]$. Hence

The terms $I_3$ and $I_4$. We apply (16) with $f_3\equiv1$ and $\varrho_3=\infty$, so that no volume factor appears and the only power of $R$ is the one in (16). For $I_3$ we take $(\varrho_1,\varrho_2,\varrho_3)=(\frac{p_\alpha}{2},\sigma_1,\infty)$, which gives $\theta=1-\frac{2}{p_\alpha}-\frac{1}{\sigma_1}=\theta_1$, and for $I_4$ we take $(\varrho_1,\varrho_2,\varrho_3)=(\frac{p_\beta}{2},\sigma_2,\infty)$, which gives $\theta= \theta_2$. Then, by (4),

The term $I_5$. We apply (16) with $(\varrho_1,\varrho_2,\varrho_3)=(p_\alpha,p_\beta,\sigma_3)$, for which $\theta=1-\frac{1}{p_\alpha}-\frac{1}{p_\beta}-\frac{1}{\sigma_3}=\frac{\alpha+\beta}{6}-\frac{1}{\sigma_3}=\theta_3\in[0,\frac{1}{3}]$. Hence

Finally, since $u_\rho\in L^{\sigma_1}\cap L^{\sigma_2}$ and $b_\rho\in L^{\sigma_3}$ with $\sigma_1,\sigma_2,\sigma_3<\infty$, we have $\lVert u_\rho\rVert_{L^{\sigma_i}(C_R)}\to0$ for $i=1,2$ and $\lVert b_\rho\rVert_{L^{\sigma_3}(C_R)}\to0$ as $R\to+\infty$, being tails of convergent integrals. Since $\lVert u\rVert_{L^{p_\alpha}}$ and $\lVert b\rVert_{L^{p_\beta}}$ are finite, we get $\lim_{R\to+\infty}I_k(R)=0$ for $k=1,\ldots,5$. Then, by gathering together these limits, we deduce the desired limit. □

Datasets. Data sharing does not apply to this article as no datasets were generated or analyzed during the current study.

Conflict of interest. In addition, the authors declare no conflict of interest and confirm that they contributed equally to this paper.

References

Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road Haidian District, Beijing 100871, People’s Republic of China

Email address: [email protected]

School of Mathematics, Harbin Institute of Technology, Xidazhi Street, Harbin 150001, Heilongjiang, People’s Republic of China

Email address: [email protected] (corresponding author)