Unique weak solutions of the non-resistive magnetohydrodynamic equations with fractional dissipation

TL;DR

This paper investigates the uniqueness of weak solutions for non-resistive magnetohydrodynamic equations with fractional dissipation, revealing new phenomena for dissipation powers less than one and establishing optimal regularity conditions.

Contribution

It extends the understanding of weak solution uniqueness in MHD equations to fractional dissipation cases with $oldsymbol{ ext{ } extless 1}$, which previous methods could not address.

Findings

Uniqueness for $oldsymbol{ ext{ } extless 1}$ dissipation with initial data in specific Besov spaces.

Existence and uniqueness for $oldsymbol{ ext{ } extgreater= 1}$ dissipation under optimal regularity conditions.

Identification of new phenomena and regularity thresholds for fractional dissipation in MHD equations.

Abstract

This paper examines the uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with the fractional dissipation $(-\Delta)^\alpha u$ and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case $\alpha=1$. This paper discovers that there are new phenomena with the case $\alpha<1$. The approach for $\alpha=1$ can not be directly extended to $\alpha<1$. We establish that, for $\alpha<1$, any initial data $(u_0, b_0)$ in the inhomogeneous Besov space $B^\sigma_{2,\infty}(\mathbb R^d)$ with $\sigma> 1+\frac{d}{2}-\alpha$ leads to a unique local solution. For the case $\alpha\ge 1$, $u_0$ in the homogeneous Besov space $\mathring B^{1+\frac{d}{2}-2\alpha}_{2,1}(\mathbb R^d)$ and $b_0$ in $ \mathring B^{1+\frac{d}{2}-\alpha}_{2,1}(\mathbb R^d)$ guarantees the existence and uniqueness. These regularity requirements appear to be optimal.

Equations

$$\{\begin{array}{l}{\partial }_{t}u+u\cdot \nabla u+\nu (-\Delta {)}^{\alpha }u=-\nabla P+b\cdot \nabla b,x\in {\mathbb{R}}^d,t>0,\\ {\partial }_{t}b+u\cdot \nabla b=b\cdot \nabla u,x\in {\mathbb{R}}^d,t>0,\\ \nabla \cdot u=\nabla \cdot b=0,x\in {\mathbb{R}}^d,t>0,\\ u(x,0)=u_0(x),b(x,0)=b_0(x),x\in {\mathbb{R}}^d,\end{array}$$

$$\widehat{(-\Delta {)}^{\alpha }f}(\xi )=|\xi {|}^{2\alpha }\hat{f}(\xi ),$$

$$\hat{f}(\xi )=\frac{1}{(2\pi {)}^{d/2}}{\int }_{{\mathbb{R}}^d}{e}^{-ix\cdot \xi }f(x)dx.$$

$$(1)u_0\in B_{2,1}^{\frac{d}{2}+1-2\alpha }({\mathbb{R}}^d)\text{mbox}and{b}_0\in B_{2,1}^{\frac{d}{2}}({\mathbb{R}}^d);$$

$$(2)u_0\in B_{2,1}^{\frac{d}{2}+1-2\alpha }({\mathbb{R}}^d)\text{mbox}and{b}_0\in B_{2,1}^{\frac{d}{2}+1-\alpha }({\mathbb{R}}^d);$$

$$(3)u_0\in B_{2,1}^{\frac{d}{2}+1-\alpha }({\mathbb{R}}^d)\text{mbox}and{b}_0\in B_{2,1}^{\frac{d}{2}+1-\alpha }({\mathbb{R}}^d),$$

$$u_0\in B_{2,\infty }^{\sigma }({\mathbb{R}}^d),b_0\in B_{2,\infty }^{\sigma }({\mathbb{R}}^d),\sigma >\frac{d}{2}+1-\alpha .$$

$$\displaystyle\mbox{for $\alpha\geq 1$},\qquad u_{0}\in\mathring{B}_{2,1}^{\frac{d}{2}+1-2\alpha}(\mathbb{R}^{d}),\quad b_{0}\in\mathring{B}_{2,1}^{\frac{d}{2}+1-\alpha}(\mathbb{R}^{d}),$$

$$\displaystyle\mbox{for $\alpha<1$},\qquad u_{0}\in B_{2,\infty}^{\sigma}(\mathbb{R}^{d}),\quad b_{0}\in B_{2,\infty}^{\sigma}(\mathbb{R}^{d}),\quad\sigma>\frac{d}{2}+1-\alpha.$$

$$u\in C([0,T];{\mathring{B}}_{2,1}^{\frac{d}{2}+1-2\alpha }({\mathbb{R}}^d))\cap L^1(0,T;{\mathring{B}}_{2,1}^{\frac{d}{2}+1}({\mathbb{R}}^d)),b\in C([0,T];{\mathring{B}}_{2,1}^{\frac{d}{2}+1-\alpha }({\mathbb{R}}^d))$$

$$u\in C([0,T];B_{2,\infty }^{\sigma }({\mathbb{R}}^d))\cap {\tilde{L}}^2(0,T;B_{2,\infty }^{\alpha +\sigma }({\mathbb{R}}^d)),b\in C([0,T];B_{2,\infty }^{\sigma }({\mathbb{R}}^d)).$$

$$\{\begin{array}{l}{u}^{(1)}={\mathring{S}}_2{u}_0,b^{(1)}={\mathring{S}}_2{b}_0,\\ {\partial }_t{u}^{(n+1)}+\nu (-\Delta {)}^{\alpha }{u}^{(n+1)}=\mathbb{P}(-u^{(n)}\cdot \nabla u^{(n+1)}+b^{(n)}\cdot \nabla b^{(n)}),\\ {\partial }_t{b}^{(n+1)}=-u^{(n)}\cdot \nabla b^{(n+1)}+b^{(n)}\cdot \nabla u^{(n)},\\ \nabla \cdot u^{(n+1)}=\nabla \cdot b^{(n+1)}=0,\\ u^{(n+1)}(x,0)={\mathring{S}}_{n+1}{u}_0,b^{(n+1)}(x,0)={\mathring{S}}_{n+1}{b}_0,\end{array}$$

$$M=2\left(\Vert u_0{\Vert }_{{\mathring{B}}_{2,1}^{\frac{d}{2}+1-2\alpha }}+\Vert b_0{\Vert }_{{\mathring{B}}_{2,1}^{\frac{d}{2}+1-\alpha }}\right)$$

$$\displaystyle Y\equiv\Big{\{}(u,b)\big{|}\,\,\|u\|_{\widetilde{L}^{\infty}(0,T;\mathring{B}^{\frac{d}{2}+1-2\alpha}_{2,1})}\leq M,\quad\|b\|_{\widetilde{L}^{\infty}(0,T;\mathring{B}_{2,1}^{\frac{d}{2}+1-\alpha})}\leq M,\,\,\,$$

$$\displaystyle\qquad\qquad\|u\|_{L^{1}(0,T;\,\mathring{B}^{\frac{d}{2}+1}_{2,1})}\leq\delta,\quad\|u\|_{\widetilde{L}^{2}(0,T;\mathring{B}^{\frac{d}{2}+1-\alpha}_{2,1})}\leq\delta\Big{\}},$$

$$\{\begin{array}{l}{u}^{(1)}=S_2{u}_0,b^{(1)}=S_2{b}_0,\\ {\partial }_t{u}^{(n+1)}+\nu (-\Delta {)}^{\alpha }{u}^{(n+1)}=\mathbb{P}(-u^{(n)}\cdot \nabla u^{(n+1)}+b^{(n)}\cdot \nabla b^{(n+1)}),\\ {\partial }_t{b}^{(n+1)}=-u^{(n)}\cdot \nabla b^{(n+1)}+b^{(n)}\cdot \nabla u^{(n+1)},\\ \nabla \cdot u^{(n+1)}=\nabla \cdot b^{(n+1)}=0,\\ u^{(n+1)}(x,0)=S_{n+1}{u}_0,b^{(n+1)}(x,0)=S_{n+1}{b}_0\end{array}$$

$$M=2\max \left\{\Vert (u_0,b_0){\Vert }_{B_{2,\infty }^{\sigma }},\frac{1}{\sqrt{C_0}}\Vert (u_0,b_0){\Vert }_{B_{2,\infty }^{\sigma }}\right\},$$

$$\displaystyle Y\equiv\Big{\{}(u,b)\big{|}\,\,\|(u,b)\|_{\widetilde{L}^{\infty}(0,T;B^{\sigma}_{2,\infty})}\leq M,\,\,\|u\|_{\widetilde{L}^{2}(0,T;\,B^{\alpha+\sigma}_{2,\infty})}\leq M\Big{\}},$$

$${\int }_{{\mathbb{R}}^d}{\Delta }_j(b^{(n)}\cdot \nabla b^{(n+1)})\cdot {\Delta }_j{u}^{(n+1)}dx+{\int }_{{\mathbb{R}}^d}{\Delta }_j(b^{(n)}\cdot \nabla u^{(n+1)})\cdot {\Delta }_j{b}^{(n+1)}dx,$$

$${\int }_{{\mathbb{R}}^d}\left(S_j{b}^{(n)}\cdot \nabla {\Delta }_j{b}^{(n+1)}\cdot {\Delta }_j{u}^{(n+1)}+S_j{b}^{(n)}\cdot \nabla {\Delta }_j{u}^{(n+1)}\cdot {\Delta }_j{b}^{(n+1)}\right)dx=0$$

$$\Vert b^{(n)}\cdot \nabla b^{(n+1)}{\Vert }_{B_{2,1}^{\frac{d}{2}-2\alpha +1}({\mathbb{R}}^d)}$$

$$\Vert fg{\Vert }_{{\dot{B}}_{p,1}^{s_1+s_2-\frac{d}{p}}({\mathbb{R}}^d)}\le C\Vert f{\Vert }_{{\dot{B}}_{p,1}^{s_1}({\mathbb{R}}^d)}\Vert g{\Vert }_{{\dot{B}}_{p,1}^{s_2}({\mathbb{R}}^d)}.$$

$$u^{(1)}={\mathring{S}}_2{u}_0,b^{(1)}={\mathring{S}}_2{b}_0.$$

$$\Vert u^{(1)}{\Vert }_{{\tilde{L}}^{\infty }(0,T;{\mathring{B}}_{2,1}^{\frac{d}{2}+1-2\alpha })}\le M,\Vert b^{(1)}{\Vert }_{{\tilde{L}}^{\infty }(0,T;{\mathring{B}}_{2,1}^{\frac{d}{2}+1-\alpha })}\le M.$$

$$\Vert u^{(1)}{\Vert }_{L^1(0,T;{\mathring{B}}_{2,1}^{\frac{d}{2}+1})}\le T\Vert {\mathring{S}}_2{u}_0{\Vert }_{{\mathring{B}}_{2,1}^{\frac{d}{2}+1}}\le TC\Vert u_0{\Vert }_{{\mathring{B}}_{2,1}^{\frac{d}{2}+1-2\alpha }}\le \delta ,$$

$$\Vert u^{(1)}{\Vert }_{{\tilde{L}}^2(0,T;{\mathring{B}}_{2,1}^{\frac{d}{2}+1-\alpha })}\le \sqrt{T}C\Vert u_0{\Vert }_{{\mathring{B}}_{2,1}^{\frac{d}{2}+1-2\alpha }}\le \delta .$$

$$\Vert u^{(n)}{\Vert }_{{\tilde{L}}^{\infty }(0,T;{\mathring{B}}_{2,1}^{\frac{d}{2}+1-2\alpha })}\le M,\Vert b^{(n)}{\Vert }_{{\tilde{L}}^{\infty }(0,T;{\mathring{B}}_{2,1}^{\frac{d}{2}+1-\alpha })}\le M,$$

$$\Vert u^{(n)}{\Vert }_{L^1(0,T;{\mathring{B}}_{2,1}^{\frac{d}{2}+1})}\le \delta ,\Vert u^{(n)}{\Vert }_{{\tilde{L}}^2(0,T;{\mathring{B}}_{2,1}^{\frac{d}{2}+1-\alpha })}\le \delta ,$$

$$\frac{1}{2}\frac{d}{dt}\Vert {\Delta }_j{u}^{(n+1)}{\Vert }_{L^2}^2+\nu \Vert {\Lambda }^{\alpha }{\Delta }_j{u}^{(n+1)}{\Vert }_{L^2}^2=A_1+A_2,$$

$$A_1$$

$$A_2$$

$$\nu \Vert {\Lambda }^{\alpha }{\Delta }_j{u}^{(n+1)}{\Vert }_{L^2}^2\ge C_0{2}^{2\alpha j}\Vert {\Delta }_j{u}^{(n+1)}{\Vert }_{L^2}^2,$$

$$|A_1|$$

$$|A_2|$$

$$\frac{d}{dt}\Vert {\Delta }_j{u}^{(n+1)}{\Vert }_{L^2}+C_0{2}^{2\alpha j}\Vert {\Delta }_j{u}^{(n+1)}{\Vert }_{L^2}\le J_1+\cdots +J_6,$$

Full text

1. Introduction

The MHD equations govern the motion of electrically conducting fluids such as plasmas, liquid metals, and electrolytes. They consist of a coupled system of the Navier-Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. Since their initial derivation by the Nobel Laureate H. Alfvén in 1942 [1], the MHD equations have played pivotal roles in the study of many phenomena in geophysics, astrophysics, cosmology and engineering (see, e.g., [5, 13]). Besides their wide physical applicability, the MHD equations are also of great interest in mathematics. As a coupled system, the MHD equations contain much richer structures than the Navier-Stokes equations. They are not merely a combination of two parallel Navier-Stokes type equations but an interactive and integrated system. Their distinctive features make analytic studies a great challenge but offer new opportunities.

Attention here is focused on the d-dimensional non-resistive MHD equation with fractional dissipation,

[TABLE]

where $u$, $P$ and $b$ represent the velocity, the pressure and the magnetic field, respectively, and $\nu>0$ is the kinematic viscosity and $\alpha>0$ is a parameter. The fractional Laplacian operator $(-\Delta)^{\alpha}$ is defined via the Fourier transform,

[TABLE]

where

[TABLE]

When $\alpha=1$, (1.1) reduces to the standard MHD equation without magnetic diffusion, which models electrically conducting fluids that can be treated as perfect conductors such as strongly collisional plasmas. When $\alpha>0$ is fractional, (1.1) may be used to model nonlocal and long-range diffusive interactions. Mathematically (1.1) allows us to study a family of equations simultaneously and provides a broad view on how the solutions are related to the sizes of $\alpha$.

One of the most fundamental issues on the MHD equations is the well-posedness problem. Mathematically rigorous foundational work has been laid by G. Duvaut and J. L. Lions in [16] and by M. Sermange and R. Temam in [33]. The MHD equations have recently gained renewed interests and there have been substantial developments on the well-posedness problem, especially when the MHD equations involve only partial or fractional dissipation (see, e.g., [7, 8, 9, 12, 14, 15, 17, 24, 25, 39, 43, 44, 45, 46, 47, 48]). A summary on some of the recent results can be found in a review paper [40]. (1.1) with $\alpha\geq 1+\frac{d}{2}$ always has a unique global solution (see [38]). Yamazaki was able to improve this result by weakening the dissipation by a logarithm [46]. It remains an outstanding open problem whether or not (1.1) with $\alpha<1+\frac{d}{2}$ can have finite-time singular classical solutions. Even the global existence of Leray-Hopf weak solutions is not known due to the lack of suitable strong convergence in $b$. In spite of the difficulties due to the lack of magnetic diffusion, significant progress has been made on the global well-posedness of solutions near background magnetic fields and many exciting results have been obtained (see, e.g., [3, 6, 20, 21, 22, 27, 29, 30, 31, 34, 37, 41, 42, 49]).

Another direction of research on the non-resistive MHD equation has resulted in steady stream of progress. This direction has been seeking the weakest possible functional setting for which one still has the uniqueness. The results currently available are for (1.1) with $\alpha=1$. Q. Jiu and D. Niu [23] proved the local well-poseness of (1.1) with $\alpha=1$ in the Sobolev space $H^{s}$ with $s\geq 3$. Fefferman, McCormick, Robinson and Rodrigo were able to weaken the regularity assumption to $(u_{0},b_{0})\in H^{s}$ with $s>\frac{d}{2}$ in [18] and then to $u_{0}\in H^{s-1+\epsilon}$ and $b_{0}\in H^{s}$ with $s>\frac{d}{2}$ in [19]. Chemin, McCormick, Robinson and Rodrigo [11] made further improvement by assuming only $u_{0}\in B^{\frac{d}{2}-1}_{2,1}$ and $b_{0}\in B^{\frac{d}{2}}_{2,1}$. Here $B^{s}_{p,q}$ denotes an inhomogeneous Besov space. They obtained the local existence for $d=2$ and $3$, and the uniqueness for $d=3$. R. Wan [36] obtained the uniqueness for $d=2$. J. Li, W. Tan and Z. Yin [26] recently made an important progress by reducing the functional setting to homogeneous Besov space $u_{0}\in\mathring{B}^{\frac{d}{p}-1}_{p,1}$ and $b_{0}\in\mathring{B}^{\frac{d}{p}}_{p,1}$ with $p\in[1,2d]$. The definitions of the Besov spaces are provided in the appendix.

The aim of this paper is to establish the local existence and uniqueness of weak solutions with the minimal initial regularity assumption and for the largest possible range of $\alpha$’s. The case when $\alpha>1$ can be handled similarly as the case when $\alpha=1$. We can show that, for $\alpha>1$, any initial data $(u_{0},b_{0})$ with $u_{0}\in\mathring{B}^{\frac{d}{2}+1-2\alpha}_{2,1}(\mathbb{R}^{d})$ and $b_{0}\in\mathring{B}^{\frac{d}{2}+1-\alpha}_{2,1}(\mathbb{R}^{d})$ leads to a unique local solution.

However, when $\alpha<1$, the situation is different and there are new phenomena. The approach for the case $\alpha=1$ can not be directly extended to $\alpha<1$. We tested several seemingly natural classes of initial data:

[TABLE]

but it appears impossible to prove the local existence and uniqueness of weak solutions in these functional settings. Our investigation with these initial data leads to several discoveries. First, we realize that, in order to attain the uniqueness, the regularity level of the Besov space for $b_{0}$ has to have at least $\frac{d}{2}-\alpha+1$-derivative, which is more than $\frac{d}{2}$ for $\alpha<1$. Second, we discover that if the derivative of the Besov setting for $b_{0}$ exceeds $\frac{d}{2}$, then $u_{0}$ and $b_{0}$ should have the same Besov setting in order to establish the existence of solutions. Furthermore, one needs to combine the term of $b\cdot\nabla b$ in the equation of $u$ and $u\cdot\nabla b$ in the equation of $b$ in order to generate the cancellation. More technical explanations are given in Section 5. As a consequence of these findings, we choose the following Besov spaces for $\alpha<1$,

[TABLE]

These functional settings appear to be optimal when $\alpha<1$. More technical evidence is provided in Section 5. Our precise result is stated in the following theorem.

Theorem 1.1**.**

Let $d\geq 2$ and consider (1.1) with $0\leq\alpha<1+\frac{d}{4}$. Assume the initial data $(u_{0},b_{0})$ satisfies $\nabla\cdot u_{0}=\nabla\cdot b_{0}=0$, and is in the following Besov spaces

[TABLE]

Then there exist $T>0$ and a unique local solution $(u,b)$ of (1.1) satisfying, in the case of $\alpha\geq 1$,

[TABLE]

and, in the case of $\alpha<1$,

[TABLE]

Theorem 1.1 covers a full range of $\alpha\in[0,1+\frac{d}{4})$ and includes $\alpha=1$ and $\alpha=0$ as two special cases. $\alpha<1+\frac{d}{4}$ is imposed to satisfy a technical requirement in bounding the high frequency interaction terms in the paraproduct decomposition. When $\alpha$ reaches this upper bound, the functional setting for $u_{0}$ is $\mathring{B}^{-1}_{2,1}$. When $\alpha\geq 1$, the initial data $(u_{0},b_{0})$ and the corresponding solution are in the homogeneous Besov spaces. For $\alpha<1$, the functional setting are the inhomogeneous Besov spaces. We may not be able to reduce the assumption for $\alpha<1$ to the corresponding homogeneous Besov spaces.

As aforementioned, the regularity assumptions imposed on $(u_{0},b_{0})$ in Theorem 1.1 may be the minimal requirements we need for the existence and uniqueness. We present a detailed explanation in Section 5. Roughly speaking, when $\alpha\geq 1$, $u_{0}\in\mathring{B}_{2,1}^{\frac{d}{2}+1-2\alpha}(\mathbb{R}^{d})$ in (1.5) is necessary in order for the solution $u\in L^{1}(0,T;\mathring{B}^{\frac{d}{2}+1}_{2,1}(\mathbb{R}^{d}))$, which is more or less the regularity level for the uniqueness. The regularity setting for $u_{0}$ leads to the corresponding choice for $b_{0}$, namely $b_{0}\in\mathring{B}_{2,1}^{\frac{d}{2}+1-\alpha}(\mathbb{R}^{d})$. In the case when $\alpha<1$, (1.6) may be optimal due to our findings discovered in working with three other initial Besov settings described above in (1.2), (1.3) and (1.4). Another hint comes from the uniqueness requirement for the ideal MHD equations. When $\alpha$ is zero or $\alpha>0$ is small, (1.6) is the regularity class that guarantees the uniqueness of solutions to the ideal MHD equation.

The statement of Theorem 1.1 clearly indicates that the case $\alpha\geq 1$ is handled differently from the case $\alpha<1$. The existence part of Theorem 1.1 is proven through a successive approximation process. Naturally we divide the consideration into two cases: $\alpha\geq 1$ and $\alpha<1$. In the case when $\alpha\geq 1$, the successive approximation sequence $(u^{(n)},b^{(n)})$ satisfies

[TABLE]

where $\mathbb{P}$ is the standard Leray projection and $\mathring{S}_{j}$ is the standard homogeneous low frequency cutoff operator (see the Appendix for its definition). The functional setting for $(u^{(n)},b^{(n)})$ is given by

[TABLE]

where $T>0$ is chosen to be sufficiently small and $0<\delta<1$ is specified in Section 2. In the case when $\alpha<1$, $(u^{(n)},b^{(n)})$ satisfies

[TABLE]

and the corresponding functional setting is

[TABLE]

where $C_{0}>0$ is a pure constant as defined in (3.1). The main effort is devoted to showing that $(u^{(n)},b^{(n)})$ actually converges to a weak solution of (1.1). The process of obtaining a subsequence of $(u^{(n)},b^{(n)})$ that converges to a weak solution $(u,b)$ of (1.1) is divided into two main steps. The first step is to assert the uniform boundedness of $(u^{(n)},b^{(n)})$ in $Y$ while the second step is to extract a strongly convergent subsequence via the Aubin-Lions Lemma. The strong convergence then ensures that the limit is indeed a weak solution of (1.1). The uniform boundedness is shown via an iterative process. We assume $(u^{(n)},b^{(n)})\in Y$ and show $(u^{(n+1)},b^{(n+1)})\in Y$.

The technical approach to proving the uniform boundedness for the case $\alpha\geq 1$ is different from that for the case when $\alpha<1$. For $\alpha<1$, we estimate $u^{(n+1)}$ and $b^{(n+1)}$ in $\widetilde{L}^{\infty}(0,T;B^{\sigma}_{2,\infty})$, and $u^{(n+1)}$ in $\widetilde{L}^{2}(0,T;\,B^{\alpha+\sigma}_{2,\infty})$ simultaneously. The purpose is to make use of the cancellation resulting from adding the equations for $\|\Delta_{j}u^{(n+1)}\|_{L^{2}}$ and $\|\Delta_{j}b^{(n+1)}\|_{L^{2}}$. The cancellation is in the sum

[TABLE]

whose paraproduct decomposition contains

[TABLE]

due to $\nabla\cdot b^{(n)}=0$. This appears to be the only approach that allows us to show the existence of solutions in functional spaces with the order of the derivative exceeding $\frac{d}{2}$. In the case when $\alpha\geq 1$, $b_{0}\in\mathring{B}_{2,1}^{\frac{d}{2}+1-\alpha}(\mathbb{R}^{d})$ and the order of derivative is $\frac{d}{2}+1-\alpha\leq\frac{d}{2}$. The desired norms of $u^{(n+1)}$ and $b^{(n+1)}$ can be suitably estimated without the cancellation. In addition, some upper bounds on products in Besov spaces are valid only for $\alpha\geq 1$ and break down when $\alpha<1$. When $\alpha\geq 1$,

[TABLE]

based on the following lemma (see, e.g., [2, p.90] or Lemma 2.6 in [26]).

Lemma 1.2**.**

Let $1\leq p\leq\infty$, $s_{1},s_{2}\leq\frac{d}{p}$ and $s_{1}+s_{2}>d\,\max\{0,\frac{2}{p}-1\}$. Then

[TABLE]

However, when $\alpha<1$, Lemma 1.2 breaks down since $s_{1}=\frac{d}{2}-\alpha+1$ and $s_{2}=\frac{d}{2}-\alpha+1$ no longer satisfy the condition $s_{1},s_{2}\leq\frac{d}{2}$. This difficulty is overcome by performing a detailed analysis on different frequencies of this product when $\alpha<1$.

The rest of this paper is divided into four sections and an appendix. Section 2 focuses on the proof of the existence part in Theorem 1.1 for the case when $\alpha\geq 1$ while Section 3 is devoted to the case when $\alpha<1$. Section 4 presents the proof for the uniqueness part of Theorem 1.1. We again distinguish between the case when $\alpha\geq 1$ and the case when $\alpha<1$. Section 5 explains in detail why the regularity assumptions on the initial data in Theorem 1.1 may be optimal. In particular, we describe the difficulties when the regularity assumptions are reduced to those in (1.2), (1.3) and (1.4). The appendix provides the definitions of Besov spaces and other closely related tools.

2. Proof of the existence part in Theorem 1.1 for $\alpha\geq 1$

This section proves the existence part of Theorem 1.1 for the case $\alpha\geq 1$. The approach is to construct a successive approximation sequence and show that the limit of a subsequence actually solves (1.1) in the weak sense.

Proof for the existence part of Theorem 1.1 in the case when $\alpha\geq 1$.

We consider a successive approximation sequence $\{(u^{(n)},b^{(n)})\}$ satisfying (1.7). We define the functional setting $Y$ as in (1.8) . Our goal is to show that $\{(u^{(n)},b^{(n)})\}$ has a subsequence that converges to a weak solution of (1.1). This process consists of three main steps. The first step is to show that $(u^{(n)},b^{(n)})$ is uniformly bounded in $Y$. The second step is to extract a strongly convergent subsequence via the Aubin-Lions Lemma while the last step is to show that the limit is indeed a weak solution of (1.1). Our main effort is devoted to showing the uniform bound for $(u^{(n)},b^{(n)})$ in $Y$. This is proven by induction.

We show inductively that $(u^{(n)},b^{(n)})$ is bounded uniformly in $Y$. Recall that $(u_{0},b_{0})$ is in the regularity class (1.5). According to (1.7),

[TABLE]

Clearly,

[TABLE]

If $T>0$ is sufficiently small, then

[TABLE]

Assuming that $(u^{(n)},b^{(n)})$ obeys the bounds defined in $Y$, namely

[TABLE]

we prove that $(u^{(n+1)},b^{(n+1)})$ obeys the same bounds for sufficiently small $T>0$ and suitably selected $\delta>0$. For the sake of clarity, the rest of this section is divided into five subsections.

2.1. The estimate of

$u^{(n+1)}$ in $\widetilde{L}^{\infty}(0,T;\mathring{B}^{1+\frac{d}{2}-2\alpha}_{2,1}(\mathbb{R}^{d}))$

Let $j$ be an integer. Applying $\mathring{\Delta}_{j}$ (we shall just use $\Delta_{j}$ to simplify the notation) to the second equation in (1.7) and then dotting with $\Delta_{j}u^{(n+1)}$, we obtain

[TABLE]

where

[TABLE]

We remark that the projection operator $\mathbb{P}$ has been eliminated due to the divergence-free condition $\nabla\cdot u^{(n+1)}=0$. The dissipative part admits a lower bound

[TABLE]

where $C_{0}>0$ is a constant. According to Lemma A.5, $A_{1}$ can be bounded by

[TABLE]

Also by Lemma A.5, $A_{2}$ is bounded by

[TABLE]

Inserting the estimates above in (2.1) and eliminating $\|\Delta_{j}u^{(n+1)}\|_{L^{2}}$ from both sides of the inequality, we obtain

[TABLE]

where

[TABLE]

Integrating (2.2) in time yields

[TABLE]

Taking the $L^{\infty}(0,T)$ of (2.3), then multiplying by $2^{(1+\frac{d}{2}-2\alpha)j}$ and summing over $j$, we have

[TABLE]

Applying Young’s inequality to the convolution in time yields

[TABLE]

The terms on the right-hand side can be estimated as follows. Recalling the definition of $J_{1}$ and using the inductive assumption on $u^{(n)}$, we have

[TABLE]

The term involving $J_{2}$ admits the same bound. In fact,

[TABLE]

where we have used the fact that $2\alpha(m-j)\leq 0$. The term with $J_{3}$ is bounded by

[TABLE]

where we have used Young’s inequality for series convolution and the fact $(2+\frac{d}{2}-2\alpha)(j-1-k)<0$. This is the place where we need $\alpha<1+\frac{d}{4}$. We now estimate the terms involving $J_{4}$ through $J_{6}$. The term with $J_{4}$ is bounded by,

[TABLE]

where we have used the fact that $\alpha\geq 1$ and $(1-\alpha)(j-m)\leq 0$. The terms with $J_{5}$ and $J_{6}$ are estimated similarly and they obey the same bound. Inserting the bounds above in (2.4), we find

[TABLE]

2.2. The estimate of $\|b^{(n+1)}\|_{\widetilde{L}^{\infty}(0,T;\mathring{B}^{\frac{d}{2}-\alpha+1}_{2,1})}$

We use the third equation of (1.7). Applying $\Delta_{j}$ to the third equation in (1.7) and then dotting with $\Delta_{j}b^{(n+1)}$, we obtain

[TABLE]

where

[TABLE]

By Lemma A.5,

[TABLE]

and

[TABLE]

Inserting the estimates above in (2.6) and eliminating $\|\Delta_{j}b^{(n+1)}\|_{L^{2}}$ from both sides of the inequality, we obtain

[TABLE]

where

[TABLE]

Integrating (2.7) in time yields,

[TABLE]

Taking the $L^{\infty}(0,T)$ of (2.8), multiplying by $2^{(\frac{d}{2}-\alpha+1)j}$ and summing over $j$, we have

[TABLE]

The terms on the right can be bounded similarly as those in the previous subsection. In fact,

[TABLE]

Similarly,

[TABLE]

The terms with $K_{4}$, $K_{5}$ and $K_{6}$ are bounded as follows.

[TABLE]

where we have used the fact that $\alpha\geq 1$ and $(1-\alpha)(j-m)\leq 0$. Similarly,

[TABLE]

Inserting the estimates above in (2.9), we find

[TABLE]

2.3. The estimate of $\|u^{(n+1)}\|_{L^{1}\left(0,T;\,\mathring{B}^{1+\frac{d}{2}}_{2,1}\right)}$

We multiply (2.3) by $2^{(1+\frac{d}{2})j}$, sum over $j$ and integrate in time to obtain

[TABLE]

We estimate the terms on the right and start with the first term.

[TABLE]

Since $u_{0}\in\mathring{B}^{1+\frac{d}{2}-2\alpha}_{2,1}$, it follows from the Dominated Convergence Theorem that

[TABLE]

Therefore, we can choose $T$ sufficiently small such that

[TABLE]

Applying Young’s inequality for the time convolution, we have

[TABLE]

The terms with $J_{2}$ and $J_{3}$ can be estimated similarly and they share the same upper bound with the term involving $J_{1}$,

[TABLE]

We now examine the terms involving $J_{4}$ through $J_{6}$. Again by Young’s inequality,

[TABLE]

where we have used the fact that $\alpha\geq 1$ and $(1-\alpha)(j-m)\leq 0$ again. The other two terms involving $J_{5}$ and $J_{6}$ obey the same bound,

[TABLE]

Here we have used $\alpha<1+\frac{d}{4}$ in the estimate of $J_{6}$. Collecting the estimates above leads to

[TABLE]

2.4. The bound for $\|u^{(n+1)}\|_{\widetilde{L}^{2}(0,T;\mathring{B}_{2,1}^{1+\frac{d}{2}-\alpha})}$

We multiply (2.3) by $2^{(1+\frac{d}{2}-\alpha)j}$, take the $L^{2}(0,T)$-norm and sum over $j$ to obtain

[TABLE]

The first term on the right is bound by

[TABLE]

Since $u_{0}\in\mathring{B}^{1+\frac{d}{2}-2\alpha}_{2,1}$, it follows from the Dominated Convergence Theorem that

[TABLE]

Therefore we can choose $T>0$ sufficiently small such that

[TABLE]

The other six terms on the right of (2.12) are estimated as follows. Applying Young’s inequality for the time convolution, we have

[TABLE]

The terms with $J_{2}$ and $J_{3}$ share the same upper bound,

[TABLE]

The estimate of the term with $J_{4}$ is similar. Again by Young’s inequality,

[TABLE]

The other two terms involving $J_{5}$ and $J_{6}$ obey the same bound,

[TABLE]

Collecting the estimates above leads to

[TABLE]

2.5. Completion of the proof for the existence part in the case when $\alpha\geq 1$

The bounds in (2.5), (2.10), (2.11) and (2.13) allow us to conclude that, if we choose $T>0$ sufficiently small and $\delta>0$ suitably, then

[TABLE]

In fact, if $T$ and $\delta$ in (2.5) satisfy

[TABLE]

then (2.5) implies

[TABLE]

or

[TABLE]

Similarly, if $C\delta\leq\frac{1}{4}$ in (2.10), then (2.10) states

[TABLE]

According to (2.13), if we choose $C\delta\leq\frac{1}{4}$ and $C\,T\,M^{2}\leq\frac{1}{2}\delta$, then

[TABLE]

and consequently, if $C\delta\leq\frac{1}{4}$ and $C\,T\,M^{2}\leq\frac{1}{2}\delta$ in (2.11), then

[TABLE]

These uniform bounds allow us to extract a weakly convergent subsequence. That is, there is $(u,b)\in Y$ such that a subsequence of $(u^{(n)},b^{(n)})$ (still denoted by $(u^{(n)},b^{(n)})$) satisfies

[TABLE]

In order to show that $(u,b)$ is a weak solution of (1.1), we need to further extract a subsequence which converges strongly to $(u,b)$. This is done via the Aubin-Lions Lemma. We can show by making use of the equations in (1.7) that $(\partial_{t}u^{(n)},\partial_{t}b^{(n)})$ is uniformly bounded in

[TABLE]

Since we are in the case of the whole space $\mathbb{R}^{d}$, we need to combine Cantor’s diagonal process with the Aubin-Lions Lemma to show that a subsequence of the weakly convergent subsequence, still denoted by $(u^{(n)},b^{(n)})$, has the following strongly convergent property,

[TABLE]

where $\frac{d}{2}+1-2\alpha\leq\gamma<\frac{d}{2}+1-\alpha$ and $Q\subset\mathbb{R}^{d}$ is any compact subset. This strong convergence property would allow us to show that $(u,b)$ is indeed a weak solution of (1.1). This process is routine and we omit the details. This completes the proof for the existence part of Theorem 1.1 in the case when $\alpha\geq 1$. ∎

3. Proof of the existence part in Theorem 1.1 with $\alpha<1$

This section proves the existence part of Theorem 1.1 for the case when $\alpha<1$. The idea is still to construct a successive approximation sequence and show that the limit of a subsequence actually solves (1.1) in the weak sense. Some of the technical approaches here are different from those for $\alpha\geq 1$.

Proof for the existence part of Theorem 1.1 in the case when $\alpha<1$.

We consider a successive approximation sequence $\{(u^{(n)},b^{(n)})\}$ satisfying (1.9). We define the functional setting $Y$ as in (1.10). Our goal is to show that $\{(u^{(n)},b^{(n)})\}$ has a subsequence that converges to the weak solution of (1.1). This process consists of three main steps. The first step is to show that $(u^{(n)},b^{(n)})$ is uniformly bounded in $Y$. The second step is to extract a strongly convergent subsequence via the Aubin-Lions Lemma while the last step is to show that the limit is indeed a weak solution of (1.1). Our main effort is devoted to showing the uniform bound for $(u^{(n)},b^{(n)})$ in $Y$. This is proven by induction. We start with the basis step. Recall that $(u_{0},b_{0})$ is in the regularity class (1.6). According to (1.9),

[TABLE]

Clearly,

[TABLE]

If $T>0$ is sufficiently small, then

[TABLE]

Assuming that $(u^{(n)},b^{(n)})$ obeys the bounds defined in $Y$, namely

[TABLE]

we prove that $(u^{(n+1)},b^{(n+1)})$ obeys the same bound for sufficiently small $T>0$.

The proof involves inhomogeneous dyadic block operator $\Delta_{j}$ and the inhomogeneous Besov spaces. Let $j\geq 0$ be an integer. Applying $\Delta_{j}$ to the second and third equations in (1.9) and then dotting by $(\Delta_{j}u^{(n+1)},\Delta_{j}b^{(n+1)})$, we have

[TABLE]

where $C_{0}>0$ is constant and

[TABLE]

According to Lemma A.5, $E_{1}$ is bounded by

[TABLE]

$E_{2}$ is bounded by

[TABLE]

$E_{3}$ is bounded by

[TABLE]

Inserting these bounds in (3.1) and then integrating in time yield

[TABLE]

Taking $L^{\infty}(0,T)$ of (3.2), then multiplying by $2^{2\sigma j}$ and taking the sup in $j$ yield

[TABLE]

We now estimate the eleven terms on the right. By Hölder’s inequality,

[TABLE]

where we have used the fact that $\alpha+\sigma>1+\frac{d}{2}$ and we are working with inhomogeneous dyadic blocks. The terms with $L_{2}$, $L_{3}$ and $L_{4}$ can be bounded very similarly and the bounds for them are

[TABLE]

The term with $L_{5}$ is estimated slightly differently.

[TABLE]

where we have used the fact that $m-j<0$ and $1+\frac{d}{2}-\alpha-\sigma<0$. The estimates of the other terms are similar,

[TABLE]

Inserting the bounds above in (3.3) yields

[TABLE]

We choose $T>0$ to be sufficiently small such that

[TABLE]

Then (3.4) implies

[TABLE]

These uniform bounds allow us to extract a weakly convergent subsequence. That is, there is $(u,b)\in Y$ such that a subsequence of $(u^{(n)},b^{(n)})$ (still denoted by $(u^{(n)},b^{(n)})$) satisfies

[TABLE]

In order to show that $(u,b)$ is a weak solution of (1.1), we need to further extract a subsequence which converges strongly to $(u,b)$. This is done via the Aubin-Lions Lemma. We can show by making use of the equations in (1.7) that $(\partial_{t}u^{(n)},\partial_{t}b^{(n)})$ is uniformly bounded in

[TABLE]

Since the domain here is the whole space $\mathbb{R}^{d}$, we need to combine Cantor’s diagonal process with the Aubin-Lions Lemma to show that a subsequence of the weakly convergent subsequence, still denoted by $(u^{(n)},b^{(n)})$, has the following strongly convergent property,

[TABLE]

where $Q\subset\mathbb{R}^{d}$ is any compact subset. This strong convergence property would allow us to show that $(u,b)$ is indeed a weak solution of (1.1). This process is routine and we omit the details. This completes the proof for the existence part of Theorem 1.1 in the case when $\alpha<1$. ∎

4. Proof for the uniqueness part of Theorem 1.1

This section proves the uniqueness part of Theorem 1.1.

Proof.

Assume that $(u^{(1)},b^{(1)})$ and $(u^{(2)},b^{(2)})$ are two solutions. Their difference $(\widetilde{u},\widetilde{b})$ with

[TABLE]

satisfies

[TABLE]

We estimate the difference $(\widetilde{u},\widetilde{b})$ in $L^{2}(\mathbb{R}^{d})$. Dotting (4.1) by ${(\widetilde{u},\widetilde{b})}$ and applying the divergence-free condition, we find

[TABLE]

where

[TABLE]

Due to $\nabla\cdot b^{(2)}=0$, we find $Q_{2}=0$ after integration by parts. We remark that $Q_{3}+Q_{4}$ is not necessarily zero. The rest of the proof distinguish between the two cases: $\alpha\geq 1$ and $\alpha<1$. For the sake of clarity, we divide the rest of this section into two subsections.

4.1. The case $\alpha\geq 1$

The uniqueness for the case when $\alpha=1$ has been obtained in [11, 26, 36]. Our attention will be focused on $\alpha>1$. In this subsection $\Delta_{j}$ denotes the homogeneous dyadic block operators for the simplicity of notation. By Hölder’s inequality,

[TABLE]

By integration by parts,

[TABLE]

For $p$ and $q$ defined by

[TABLE]

we have, by Hölder’s inequality,

[TABLE]

$Q_{4}$ obeys the same bound. Inserting these bounds in (4.2), we find

[TABLE]

By Bernstein’s inequality,

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

Due to the time integrability

[TABLE]

Gronwall’s inequality then implies that

[TABLE]

4.2. The case $\alpha<1$

The operator $\Delta_{j}$ in this subsection denotes the inhomogeneous dyadic block operators and the Besov spaces are inhomogeneous. By Hölder’s inequality,

[TABLE]

To bound $Q_{3}$, we set

[TABLE]

and apply Hölder’s inequality and the Hardy-Littlewood-Sobolev inequality to obtain

[TABLE]

$Q_{4}$ obeys exactly the same bound. Inserting these bounds in (4.2), we find

[TABLE]

By Bernstein’s inequality,

[TABLE]

where we have used the fact that $\sigma>1+\frac{d}{2}-\alpha$. In addition,

[TABLE]

where again we have used the fact that $\sigma>1+\frac{d}{2}-\alpha$. Therefore,

[TABLE]

Applying Gronwall’s inequality to (4.3) and invoking (4.4) and (4.5), we obtain

[TABLE]

which leads to the desired uniqueness. This completes the proof of the uniqueness part of Theorem 1.1. ∎

5. Conclusion and discussions

We have established that, for $\alpha\geq 1$, any initial data $(u_{0},b_{0})$ with

[TABLE]

and, for $\alpha<1$, any initial data $(u_{0},b_{0})$ with

[TABLE]

leads to a unique local weak solution of (1.1). This purpose of this section is to explain in some detail on why these regularity assumptions may be optimal. The optimality for the case $\alpha\geq 1$ can be easily explained. The index $\frac{d}{2}+1-2\alpha$ is minimal for the velocity in order to achieve the uniqueness. As we know, the velocity $u$ should obey $\int_{0}^{T}\|\nabla u\|_{L^{\infty}}\,dt<\infty$ or a slightly weaker version in order to guarantee the uniqueness. In the Besov setting here, we need

[TABLE]

which, in turn, requires that

[TABLE]

This is how the index $\frac{d}{2}+1-2\alpha$ arises. Once the Besov space for $u_{0}$ is set, the functional setting for $b_{0}$ is determined correspondingly.

We now explain why the initial setup for the case $\alpha<1$ may be optimal. We have attempted to replace (5.1) by several weaker assumptions, but we failed to establish the desired existence and uniqueness. We now describe the difficulties associated with those weaker initial data.

5.1. Can we replace (5.1) by $u_{0}\in B_{2,1}^{\frac{d}{2}+1-2\alpha}(\mathbb{R}^{d})$ and $b_{0}\in B_{2,1}^{\frac{d}{2}}(\mathbb{R}^{d})$?

We would have difficulty proving the uniform boundedness of the successive approximation sequence in the existence proof part. If we assume that

[TABLE]

then the corresponding functional space for $(u,b)$ would be

[TABLE]

Suppose we construct the successive approximation sequence by (1.7). We can obtain suitable bounds for

[TABLE]

We would have difficulty controlling $\|u^{(n+1)}\|_{L^{1}(0,T;\,B^{\frac{d}{2}+1}_{2,1})}$ due to the term $b^{(n)}\cdot\nabla b^{(n)}$ in (1.7). A quick way to see the difficultly is to count the derivatives needed and the derivatives allowed,

[TABLE]

We explain the meaning of this inequality. The left-hand side $2\left(\frac{d}{2}+1-\alpha\right)$ represents the derivative imposed and the right-hand side $2\cdot\frac{d}{2}$ denotes the derivatives allowed on the two $b^{(n)}$’s. The first $\frac{d}{2}+1$ comes from $B^{\frac{d}{2}+1}_{2,1}$, the second $\frac{d}{2}+1$ represents the derivative when we estimates $\|b^{(n)}\cdot\nabla b^{(n)}\|_{L^{2}}$ and $-2\alpha$ is due to the dissipation. When $\alpha<1$, the derivatives imposed are more than the derivatives allowed and we can not close the estimates in $Y$.

5.2. Can we replace (5.1) by $u_{0}\in B_{2,1}^{\frac{d}{2}+1-2\alpha}(\mathbb{R}^{d})$ and $b_{0}\in B_{2,1}^{\frac{d}{2}+1-\alpha}(\mathbb{R}^{d})$?

Even though we increased the regularity of $b_{0}$ to the level that would allow us to overcome one difficulty mentioned in the previous subsection, we would still have trouble proving the uniform boundedness of the successive approximation sequence in the existence proof part. If we assume that

[TABLE]

then the corresponding functional space for $(u,b)$ would be

[TABLE]

Suppose we construct the successive approximation sequence by (1.7). We can obtain suitable bounds for

[TABLE]

But this new setup would make it impossible to control

[TABLE]

The difficulty comes from bounding the term $b^{(n)}\cdot\nabla u^{(n)}$ in the equation of $b^{(n+1)}$ in (1.7). In order to bound $\|\Delta_{j}(b^{(n)}\cdot\nabla u^{(n)})\|_{L^{2}}$, one naturally decomposes it by paraproducts as in (2.7),

[TABLE]

The trouble arises in the first term on the right-hand side. For $\alpha<1$, we can no longer bound

[TABLE]

by

[TABLE]

and, as a consequence, we are not able to control $b^{(n)}\cdot\nabla u^{(n)}$ by the desired bound $\|u^{(n)}\|_{L^{1}(0,T;\,B^{\frac{d}{2}+1}_{2,1})}\,\|b^{(n)}\|_{\widetilde{L}^{\infty}(0,T;B_{2,1}^{\frac{d}{2}+1-\alpha})}$. This problem arises when $u$ and $b$ are in different functional settings. We can no longer estimate $u$ and $b$ simultaneously and the good structure of combining the terms $b\cdot\nabla b$ and $b\cdot\nabla u$ can no longer be taken advantage of.

5.3. Can we replace (5.1) by $u_{0}\in B_{2,1}^{\frac{d}{2}+1-\alpha}(\mathbb{R}^{d})$ and $b_{0}\in B_{2,1}^{\frac{d}{2}+1-\alpha}(\mathbb{R}^{d})$?

Even $u_{0}$ and $b_{0}$ are now in the same functional setting, but we are still not able to prove the uniform boundedness of the successive approximation sequence in the existence proof part. We now explain the difficulty. Naturally the corresponding functional setting for $(u,b)$ is

[TABLE]

Suppose we construct the successive approximation sequence by (1.9). In order to make use of the cancellation in the combination of $b\cdot\nabla b$ and $b\cdot\nabla u$, we have to add the estimates at the $L^{2}$-level as in (3.1). However, if we add them at the $L^{2}$-level, it is then impossible to control the norm of $(u,b)$ in $B^{\frac{d}{2}+1-\alpha}_{2,1}$. This is exactly why we have selected the functional setting $B^{\sigma}_{2,\infty}$ with $\sigma>\frac{d}{2}+1-\alpha$ when $\alpha<1$, as in the proof in Section 3.

In conclusion, the regularity assumptions on the initial data in Theorem 1.1 may be optimal.

Appendix A Besov spaces and related tools

This appendix provides the definition of the Besov spaces and related facts that have been used in the previous sections. Some of the materials are taken from [2]. More details can be found in several books and many papers (see, e.g., [2, 4, 28, 32, 35]). In addition, we also prove several bounds on triple products involving Fourier localized functions. These bounds have been used in the previous sections.

We start with the partition of unit. Let $B(0,r)$ and ${\mathcal{C}}(0,r_{1},r_{2})$ denote the standard ball and the annulus, respectively,

[TABLE]

There are two compactly supported smooth radial functions $\phi$ and $\psi$ satisfying

[TABLE]

We use $\widetilde{h}$ and $h$ to denote the inverse Fourier transforms of $\phi$ and $\psi$ respectively,

[TABLE]

In addition, for notational convenience, we write $\psi_{j}(\xi)=\psi(2^{-j}\xi)$. By a simple property of the Fourier transform,

[TABLE]

The inhomogeneous dyadic block operator $\Delta_{j}$ are defined as follows

[TABLE]

The corresponding inhomogeneous low frequency cut-off operator $S_{j}$ is defined by

[TABLE]

For any function $f$ in the usual Schwarz class $\mathcal{S}$, (A.1) implies

[TABLE]

or, in terms of the inhomogeneous dyadic block operators,

[TABLE]

where Id denotes the identity operator. More generally, for any $F$ in the space of tempered distributions, denoted ${\mathcal{S}}^{\prime}$, (A.2) still holds but in the distributional sense. That is, for $F\in{\mathcal{S}}^{\prime}$,

[TABLE]

In fact, one can verify that

[TABLE]

(A.3) is referred to as the Littlewood-Paley decomposition for tempered distributions.

In terms of the inhomogeneous dyadic block operators, we can write the standard product in terms of the paraproducts, namely the Bony decomposition,

[TABLE]

where $\widetilde{\Delta}_{k}=\Delta_{k-1}+\Delta_{k}+\Delta_{k+1}$.

The inhomogeneous Besov space can be defined in terms of $\Delta_{j}$ specified above.

Definition A.1**.**

The inhomogeneous Besov space $B^{s}_{p,q}$ with $1\leq p,q\leq\infty$ and $s\in{\mathbb{R}}$ consists of $f\in{\mathcal{S}}^{\prime}$ satisfying

[TABLE]

The concepts defined above have their homogeneous version. The homogeneous dyadic block and the homogeneous low frequency cutoff operators are defined by, for any $j\in\mathbb{Z}$,

[TABLE]

For any function $f$ in the usual Schwarz class $\mathcal{S}$, (A.1) implies

[TABLE]

when $f$ satisfies, for any $x^{\beta}$ in the set of all polynomials $\mathcal{P}$,

[TABLE]

In order to write the Littlewood-Paley decomposition for $F\in\mathcal{S}^{\prime}$, we need to restrict to the subspace $\mathcal{S}_{h}^{\prime}$ consisting of $f\in\mathcal{S}^{\prime}$ satisfying

[TABLE]

Any $f\in\mathcal{S}^{\prime}$ that has a locally integrable Fourier transform is in $\mathcal{S}_{h}^{\prime}$.

The homogeneous Besov space can be defined in terms of $\mathring{\Delta}_{j}$ specified above.

Definition A.2**.**

The homogeneous Besov space $\mathring{B}^{s}_{p,q}$ with $1\leq p,q\leq\infty$ and $s\in{\mathbb{R}}$ consists of $f\in{\mathcal{S}}_{h}^{\prime}$ satisfying

[TABLE]

In terms of the homogeneous dyadic blocks, we can also write the standard products in terms of the paraproducts.

The following space-time spaces introduced in [10] have been used in the previous sections.

Definition A.3**.**

Let $s\in\mathbb{R}$ and $1\leq p,q,r\leq\infty$. Let $T\in(0,\infty]$. The space-time space $\widetilde{L}^{r}(0,T;B^{s}_{p,q})$ consists of tempered distributions satisfying

[TABLE]

$\widetilde{L}^{r}(0,T;\mathring{B}^{s}_{p,q})$* is similarly defined.*

By Minkowski’s inequality, the standard space-time space $L^{r}(0,T;B^{s}_{p,q})$ is related to $\widetilde{L}^{r}(0,T;B^{s}_{p,q})$ as follows

[TABLE]

Bernstein’s inequality is a useful tool on Fourier localized functions and these inequalities trade derivatives for integrability. The following proposition provides Bernstein type inequalities for fractional derivatives.

Lemma A.4**.**

Let $\alpha\geq 0$. Let $1\leq p\leq q\leq\infty$.

If $f$ satisfies

[TABLE]

for some integer $j$ and a constant $K>0$, then

[TABLE] 2. 2)

If $f$ satisfies

[TABLE]

for some integer $j$ and constants $0<K_{1}\leq K_{2}$, then

[TABLE]

where $C_{1}$ and $C_{2}$ are constants depending on $\alpha,p$ and $q$ only.

We now state and prove bounds for the triple products involving Fourier localized functions. These bounds have been used in the previous sections in the proof of Theorem 1.1.

Lemma A.5**.**

Let $j\in\mathbb{Z}$ be an integer. Let $\Delta_{j}$ be a dyadic block operator (either inhomogeneous or homogeneous).

1. (1)

Let $F$ be a divergence-free vector field. Then there exists a constant $C$ independent of $j$ such that

[TABLE] 2. (2)

Let $F$ be a divergence-free vector field. Then there exists a constant $C$ independent of $j$ such that

[TABLE] 3. (3)

Let $F$ be a divergence-free vector field. Then there exists a constant $C$ independent of $j$ such that

[TABLE]

Proof.

The proof of these inequalities essentially follow from the paraproduct decomposition. By the paraproduct decomposition,

[TABLE]

By Hölder’s inequality and Bernstein’s inequality in Lemma A.4,

[TABLE]

where we have used $\nabla\cdot F=0$ in the last part. (A.4) then follows if we invoke the inequalities of the form

[TABLE]

To prove (A.5), we further write the first term as the sum of a commutator and two correction terms,

[TABLE]

Due to $\nabla\cdot F=0$,

[TABLE]

By Hölder’s inequality, Bernstein’s inequality and a commutator estimate,

[TABLE]

(A.5) then follows when we invoke similar inequalities as (A.7). The proof of (A.6) is very similar. This completes the proof of Lemma A.5. ∎

Acknowledgments

Q. Jiu was partially supported by the National Natural Science Foundation of China (NNSFC) (No. 11671273) and Beijing Natural Science Foundation (BNSF) (No. 1192001). J. Wu was partially supported by the National Science Foundation of the United States under grant number DMS-1614246 and by the AT&T Foundation at the Oklahoma State University.