# Global existence and uniqueness of weak solutions of a Stokes-Magneto system
with fractional diffusions

Hyunseok Kim  Department of Mathematics, Sogang University, Seoul, 04107,
Republic of Korea<EMAIL_ADDRESS>and  Hyunwoo Kwon  Department of
Mathematics, Sogang University, Seoul, 04107, Republic of Korea Division of
Applied Mathematics, Brown University, 182 George Street, Providence, RI
02912, USA<EMAIL_ADDRESS>
###### Abstract.

We consider a Stokes-Magneto system in $\mathbb{R}^{d}$ ($d\geq 2$) with
fractional diffusions $\Lambda^{2\alpha}\bm{u}$ and $\Lambda^{2\beta}\bm{b}$
for the velocity $\bm{u}$ and the magnetic field $\bm{b}$, respectively. Here
$\alpha,\beta$ are positive constants and $\Lambda^{s}=(-\Delta)^{s/2}$ is the
fractional Laplacian of order $s$. We establish global existence of weak
solutions of the Stokes-Magneto system for any initial data in $L_{2}$ when
$\alpha$, $\beta$ satisfy $1/2<\alpha<(d+1)/2$, $\beta>0$, and
$\min\\{\alpha+\beta,2\alpha+\beta-1\\}>d/2$. It is also shown that weak
solutions are unique if $\beta\geq 1$ and
$\min\\{\alpha+\beta,2\alpha+\beta-1\\}\geq d/2+1$, in addition.

###### Key words and phrases:

Global existence; Uniqueness; Weak solutions; Fractional diffusions

###### 1991 Mathematics Subject Classification:

35Q35;76W05

The authors were supported by Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education (No. NRF-2016R1D1A1B02015245).

##  1\. Introduction

For several decades, many authors have studied linear and nonlinear equations
with fractional diffusions from a mathematical point of view and applied them
to diverse fields such as biology [5], financial mathematics [2], and
probability [30]. Particularly important models from fluid mechanics are
generalized Navier-Stokes equations (see e.g [24, 25, 12, 33]) and surface
quasi-geographic equations (see e.g. [11, 9]).

In this paper, we consider the following initial value problem for a Stokes-
Magneto system with fractional diffusions in $\mathbb{R}^{d}$, $d\geq 2$:

(1.1) $\left\\{\begin{aligned}
\nu\Lambda^{2\alpha}\bm{u}+\nabla{p_{*}}&=(\bm{b}\cdot\nabla)\bm{b}&&\quad\text{in
}\mathbb{R}^{d}\times(0,\infty),\\\
\partial_{t}\bm{b}+\eta\Lambda^{2\beta}\bm{b}+(\bm{u}\cdot\nabla)\bm{b}&=(\bm{b}\cdot\nabla)\bm{u}&&\quad\text{in
}\mathbb{R}^{d}\times(0,\infty),\\\
\operatorname{div}\bm{u}=\operatorname{div}\bm{b}&=0&&\quad\text{in
}\mathbb{R}^{d}\times(0,\infty),\\\ \bm{b}(\cdot,0)&=\bm{b}_{0}&&\quad\text{on
}\mathbb{R}^{d}.\end{aligned}\right.$

Here $\bm{u}:\mathbb{R}^{d}\times[0,\infty)\rightarrow\mathbb{R}^{d}$ is the
velocity field,
$\bm{b}:\mathbb{R}^{d}\times[0,\infty)\rightarrow\mathbb{R}^{d}$ is the
magnetic field, and
${p_{*}}:\mathbb{R}^{d}\times[0,\infty)\rightarrow\mathbb{R}$ denotes the
total pressure: ${p_{*}}=p+\frac{1}{2}|\bm{b}|^{2}$ with $p$ being the
pressure on the fluid. The positive constants $\nu$ and $\eta$ stand for the
viscosity constant and the magnetic diffusivity, respectively. For
$s\in\mathbb{R}$, $\Lambda^{s}=(-\Delta)^{s/2}$ denotes the fractional
Laplacian of order $s$ (see Section 2 for more details).

Our model (1.1) was motivated by Moffatt [27, 28] who proposed the magnetic
relaxation method to construct magnetohydrodynamics (MHD) equilibria from a
given magnetic field $\bm{b}_{0}$. More precisely, Moffatt [27] first
suggested the method of magnetic relaxation to construct a magnetic
equilibrium from a solution of ideal MHD equations by letting
$t\rightarrow\infty$. Later, in 2009, Moffatt suggested another possible way
to construct an MHD equilibrium by introducing the system (1.1) with
$\alpha=1$ and $\eta=0$. See Beekie-Friedlander-Vicol [4], Constantin-
Pasqualotto [13], and references therein for further motivation of magnetic
relaxation.

In comparison to (1.1), we recall the generalized MHD system:

(1.2) $\left\\{\begin{aligned}
\partial_{t}\bm{u}+\nu\Lambda^{2\alpha}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}+\nabla{p_{*}}&=(\bm{b}\cdot\nabla)\bm{b}&&\quad\text{in
}\mathbb{R}^{d}\times(0,\infty),\\\
\partial_{t}\bm{b}+\eta\Lambda^{2\beta}\bm{b}+(\bm{u}\cdot\nabla)\bm{b}&=(\bm{b}\cdot\nabla)\bm{u}&&\quad\text{in
}\mathbb{R}^{d}\times(0,\infty),\\\
\operatorname{div}\bm{u}=\operatorname{div}\bm{b}&=0&&\quad\text{in
}\mathbb{R}^{d}\times(0,\infty),\\\
\bm{u}(\cdot,0)=\bm{u}_{0},\quad\bm{b}(\cdot,0)&=\bm{b}_{0}&&\quad\text{on
}\mathbb{R}^{d}.\end{aligned}\right.$

When $\nu,\eta>0$ and $\alpha=\beta=1$, Duvaut-Lions [15] proved global
existence of weak solutions of (1.2) with $L_{2}$-initial data. Later, Wu [37]
extended this result to arbitrary $\alpha,\beta>0$ and also proved existence
of global classical solutions for smooth initial data when $\alpha,\beta$
satisfy $\alpha,\beta\geq d/4+1/2$. The existence results in [37] were later
improved by the same author [38] when $\alpha\geq d/4+1/2$, $\beta>0$, and
$\alpha+\beta\geq d/2+1$. In addition to these results, many authors proved
global regularity of classical solutions of (1.2) in two-dimensional spaces
under various hypotheses on $\alpha$ and $\beta$; see Yamazaki [39] and
references therein. On the other hand, when $\nu=0$ and $\eta>0$, Kozono [23]
proved local existence and uniqueness of classical solutions of (1.2) when
$\beta=1$. This result was extended to any $\beta>0$ by Wu [38] for
sufficiently smooth initial data. Recently, Fefferman et.al [16, 17]
established local existence of strong solutions of (1.2) when $\nu>0$,
$\eta=0$, and $\alpha=1$, which was extended by Kim-Zhou [22] to general
$\alpha$.

In contrast to the generalized MHD system, there are few results for the
Stokes-Magneto system with or without fractional diffusions. When
$\nu,\eta>0$, $\alpha=\beta=1$, and $d=2,3$, McCormick-Robinson-Rodrigo [26]
proved global existence of weak solutions of (1.1). For the two-dimensional
case, they also proved uniqueness and regularity of weak solutions. Recently,
Ji-Tan [21] proved global existence of strong solutions of (1.1) when $d=3$,
$\alpha=1$, and $\beta\geq 3/2$. On the other hand, when $\nu>0$, $\eta=0$,
and $\alpha=0$, Brenier [7] proved global existence of dissipative weak
solutions on the two-dimensional torus $\mathbb{T}^{2}$. When $\nu>0$,
$\eta=0$, and $\alpha=1$, Fefferman et.al. [16] proved local existence and
uniqueness of strong solutions of (1.1) on $\mathbb{R}^{d}$, $d=2,3$.
Recently, when $\nu>0$, $\eta=0$, and $\alpha>d/2+1$, Beekie-Friedlander-Vicol
[4] established global existence of strong solutions of (1.1) on the torus
$\mathbb{T}^{d}$, $d=2,3$. Moreover, they also investigated 2D stability and
3D instability as well as the long-time behavior of solutions.

In this paper, we establish global existence and uniqueness of weak solutions
of the Stokes-Magneto system (1.1) with fractional diffusions. To elaborate a
motivation of our notion of weak solutions, we multiply the first equation in
(1.1) by $\bm{u}$ and the second equation by $\bm{b}$, and then integrate the
resulting equations on $\mathbb{R}^{d}$. Using the divergence-free condition
on $\bm{u}$ and $\bm{b}$, we have

$\nu\int_{\mathbb{R}^{d}}|\Lambda^{\alpha}\bm{u}|^{2}\,d{x}=-\int_{\mathbb{R}^{d}}(\bm{b}\otimes\bm{b}):\nabla\bm{u}\,d{x}$

and

$\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{d}}|\bm{b}|^{2}\,d{x}+\eta\int_{\mathbb{R}^{d}}|\Lambda^{\beta}\bm{b}|^{2}\,d{x}=\int_{\mathbb{R}^{d}}(\bm{b}\cdot\nabla)\bm{u}\cdot\bm{b}\,d{x}.$

Adding these two identities and integrating it in time, we derive the energy
identity

(1.3)
$\int_{\mathbb{R}^{d}}|\bm{b}(t)|^{2}\,d{x}+2\int_{0}^{t}\int_{\mathbb{R}^{d}}\left(\nu|\Lambda^{\alpha}\bm{u}(s)|^{2}+\eta|{\Lambda^{\beta}\bm{b}(s)}|^{2}\right)\,d{x}\,d{s}=\int_{\mathbb{R}^{d}}|\bm{b}_{0}|^{2}\,d{x}$

for all $t\geq 0$. Furthermore, it will be shown in Subsection 3.2 that if
$1/2<\alpha<(d+1)/2$, then $\bm{u}$ is given by

(1.4) $\bm{u}=\mathbf{U}_{\alpha}*(\bm{b}\otimes\bm{b})$

where $\mathbf{U}_{\alpha}$ is a matrix-valued function satisfying

$|\mathbf{U}_{\alpha}(x)|\leq\frac{C}{|x|^{d+1-2\alpha}}$

for some constant $C$ depending only on $d$ and $\alpha$. By Young’s
convolution inequality in weak spaces (Lemma 2.5), we deduce that

(1.5) $\|{\bm{u}(t)}\|_{L_{d/(d+1-2\alpha),\infty}}\leq
C\|{\bm{b}(t)\otimes\bm{b}(t)}\|_{L_{1}}\leq C\|{\bm{b}(t)}\|_{L_{2}}^{2}$

for some constant $C=C(d,\alpha)$. Here $L_{p}$ and $L_{p,q}$ denote the
Lebesgue spaces and Lorentz spaces on $\mathbb{R}^{d}$, respectively, where
$1\leq p<\infty$ and $1\leq q\leq\infty$. For $s\in\mathbb{R}$, let
$\dot{H}^{s}$ be the homogeneous Sobolev spaces on $\mathbb{R}^{d}$ (see
Section 2 for details). Then motivated by (1.3) and (1.5), we define weak
solutions of (1.1) as follows.

######  Definition 1.1.

Let $1/2<\alpha<(d+1)/2$ and $0<\beta<\infty$. Suppose that $\bm{b}_{0}\in
L_{2}$, ${\rm div}\,\bm{b}_{0}=0$ in $\mathbb{R}^{d}$, and $0<T\leq\infty$.
Then a pair $(\bm{u},\bm{b})$ of vector fields satisfying

$\displaystyle\bm{u}$ $\displaystyle\in
L_{\infty}(0,T;L_{{d}/{(d+1-2\alpha)},\infty})\cap
L_{2}(0,T;\dot{H}^{\alpha}),$ $\displaystyle\bm{b}$ $\displaystyle\in
L_{\infty}(0,T;L_{2})\cap L_{2}(0,T;\dot{H}^{\beta})$

is called a _weak solution_ of (1.1) in $[0,T)$ if the following are
satisfied:

  1. (1)

For all $\Phi\in C_{c}^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$ with
$\operatorname{div}\Phi=0$,

(1.6)
$\int_{\mathbb{R}^{d}}\nu\Lambda^{\alpha}\bm{u}(t)\cdot\Lambda^{\alpha}\Phi\,d{x}=-\int_{\mathbb{R}^{d}}\bm{b}(t)\otimes\bm{b}(t):\nabla\Phi\,d{x}$

for almost all $t\in(0,T)$.

  2. (2)

For all $\Phi\in C_{c}^{\infty}(\mathbb{R}^{d}\times[0,T);\mathbb{R}^{d})$,

(1.7)
$\displaystyle\mathrel{\phantom{=}}\int_{0}^{T}\int_{\mathbb{R}^{d}}\bm{b}\cdot(-\partial_{t}\Phi+\eta\Lambda^{2\beta}\Phi)\,d{x}dt$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{R}^{d}}(\bm{u}\otimes\bm{b}-\bm{b}\otimes\bm{u}):\nabla\Phi\,d{x}dt$
$\displaystyle=\int_{\mathbb{R}^{d}}\bm{b}_{0}\cdot\Phi(0)\,dx.$

  3. (3)

For all $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$,

$\int_{\mathbb{R}^{d}}\bm{u}(t)\cdot\nabla\psi\,d{x}=\int_{\mathbb{R}^{d}}\bm{b}(t)\cdot\nabla\psi\,d{x}=0$

for almost all $t\in(0,T)$.

A weak solution of (1.1) in $[0,\infty)$ will be called a _global weak
solution_.

Now we are ready to present the main results of the paper. We first establish
existence of global weak solutions of (1.1) under a suitable assumption on
$\alpha$ and $\beta$.

######  Theorem 1.2.

Let $\alpha$ and $\beta$ satisfy

$\frac{1}{2}<\alpha<\frac{d+1}{2},\quad\beta>0,\quad\mbox{and}\quad\min\\{\alpha+\beta,2\alpha+\beta-1\\}>\frac{d}{2}.$

Then for any $\bm{b}_{0}\in L_{2}$ with ${\rm div}\,\bm{b}_{0}=0$, there
exists at least one global weak solution of (1.1).

Next, we prove the time-continuity, energy identity, and uniqueness of weak
solutions of (1.1) for larger values of $\alpha$ and $\beta$.

######  Theorem 1.3.

Let $\alpha$ and $\beta$ satisfy

$\frac{1}{2}<\alpha<\frac{d+1}{2},\quad\beta\geq
1,\quad\text{and}\quad\min\\{\alpha+\beta,2\alpha+\beta-1\\}\geq\frac{d}{2}+1.$

Then for any $\bm{b}_{0}\in L_{2}$ with ${\rm div}\,\bm{b}_{0}=0$, there
exists a unique global weak solution $(\bm{u},\bm{b})$ of (1.1). Moreover, the
solution $(\bm{u},\bm{b})$ is continuous in
$L_{{d}/{(d+1-2\alpha)},\infty}\times L_{2}$ and satisfies the energy identity
(1.3) for any $t\geq 0$.

######  Remark.

(a) Assume that $\alpha=1$ and $\beta>d/2-1$. Then it follows from Theorem 1.2
that for any $\bm{b}_{0}\in L_{2}$ with ${\rm div}\,\bm{b}_{0}=0$, there
exists a global weak solution $(\bm{u},\bm{b})$ of (1.1). This result extends
the existence result of McCormick-Robinson-Rodrigo [26] who considered the
case when $d=2,3$ and $\beta=1$. Moreover, by Theorem 1.3, weak solutions of
(1.1) are unique if $\beta\geq d/2$. Hence it follows from the existence
theorem for strong solutions due to Ji-Tan [21] that if $d=3$ and
$\bm{b}_{0}\in H^{1}$, then $(\bm{u},\bm{b})$ satisfies the following strong
regularity properties:

$\bm{u}\in C([0,T];H^{3/2})\cap L_{2}(0,T;H^{2}),\quad\bm{b}\in
C([0,T];H^{1})\cap L_{2}(0,T;H^{1+\beta})$

for any $T<\infty$.

(b) Assume that $\beta=1$ and $\max\\{1/2,d/2-1,d/4\\}<\alpha<(d+1)/2$. Then
it follows from Theorem 1.2 that for any $\bm{b}_{0}\in L_{2}$ with ${\rm
div}\,\bm{b}_{0}=0$, the problem (1.1) has global weak solutions. Moreover, by
Theorem 1.3, weak solutions of (1.1) are unique if $\alpha\geq d/2$ in
addition.

(c) Assume that $2\leq d\leq 4$ and $(d+2)/6<\alpha=\beta<(d+1)/2$. Then it
follows from Theorems 1.2 and 1.3 that for any $\bm{b}_{0}\in L_{2}$ with
${\rm div}\,\bm{b}_{0}=0$, the problem (1.1) has global weak solutions, which
are unique if $(d+2)/4\leq\alpha=\beta<(d+1)/2$.

One of our main tools to prove Theorems 1.2 and 1.3 is the following product
estimate (see Lemma 2.8): if $1/2<\alpha<(d+1)/2$, $\beta>0$, and
$\alpha+\beta>d/2$, then there exist constants $0<\theta_{1},\theta_{2}<1$
such that

(1.8)
$\|{fg}\|_{{2}}\apprle\|{f}\|_{{d/(d+1-2\alpha),\infty}}^{1-\theta_{1}}\|{f}\|_{\dot{H}^{\alpha}}^{\theta_{1}}\|{g}\|_{{2}}^{1-\theta_{2}}\|{g}\|_{\dot{H}^{\beta}}^{\theta_{2}}$

for all $f\in L_{d/(d+1-2\alpha),\infty}\cap\dot{H}^{\alpha}$ and
$g\in{H}^{\beta}$. Moreover, if $\alpha+(1-\mu)\beta\geq d/2+1$ for some
$0\leq\mu\leq 1$, we can choose $\theta_{1}$ and $\theta_{2}$ so that
$(1+\mu)\theta_{1}+\theta_{2}+1/\beta\leq 2$. This additional condition
enables us to establish uniqueness of weak solutions of (1.1). The estimate
(1.8) will be derived from generalized Gagliardo-Nirenberg interpolation
inequalities that were obtained recently by Byeon-Kim-Oh [8] and Wang-Wei-Ye
[36].

To prove existence of weak solutions (Theorem 1.2) of (1.1), we first
construct a family of solutions $\\{(\bm{u}_{R},\bm{b}_{R})\\}$ of the
$R$-truncated problems (5.4). Moreover, it is shown that these solutions
satisfy uniform bounds. Then by a standard Aubin-Lions compactness argument
combined with the estimate (1.8), we construct a sequence of functions
$\\{\bm{b}_{R_{k}}\\}$ that has strong convergence in $L_{2}(0,T;L_{2}(K))$
for every finite $T>0$ and compact $K\subset\mathbb{R}^{d}$. To show strong
convergence of $\bm{u}_{R_{k}}$, we utilize the solution formula (1.4) for the
fractional Stokes system that will be derived in Section 3 (Theorem 3.9). Very
weak solutions of the fractional Stokes system are introduced to apply the
solution formula (1.4) for $L^{1}$-data. Finally, making crucial use of the
assumption $2\alpha+\beta>d/2+1$, we show that the nonlinear terms
$\bm{u}_{R_{k}}\otimes\bm{b}_{R_{k}}$ converge strongly in
$L_{1,\mathrm{loc}}(\mathbb{R}^{d}\times(0,\infty))$. The strong convergence
properties of $\bm{u}_{R_{k}}$ and $\bm{b}_{R_{k}}$ enable us to complete the
proof of existence of weak solutions of (1.1) by passing to the limit
$k\to\infty$. To prove uniqueness of weak solutions (Theorem 1.3), we first
study weak solutions of perturbed fractional heat equations. Then applying a
standard Gronwall-type argument combined with product estimates (Lemmas 2.8
and 2.9), we show that weak solutions are unique.

This paper proceeds in six sections and three appendix sections. In Section 2,
we introduce our notations, the fractional Laplacian, homogeneous Sobolev
spaces, and Lorentz spaces. We also provide proofs of product estimates that
play crucial roles in proving existence and uniqueness of weak solutions. In
Section 3, we prove the solution formula (1.4) for the fractional Stokes
system. The notion of very weak solutions is also introduced. In Section 4, we
prove existence and regularity properties of weak solutions of perturbed
fractional heat equations that will be used in the proof of uniqueness of weak
solutions of (1.1). The main theorems are proved in Sections 5 and 6. Finally
we give somewhat detailed proofs of some technical results in Appendices.

##  2\. Preliminaries

###  2.1. Notations

Let $\mathbb{N}_{0}=\\{0,1,2,\ldots\\}$ be the set of nonnegative integers.
For $\gamma=(\gamma_{1},\dots,\gamma_{d})\in\mathbb{N}^{d}_{0}$ and a function
$u:\mathbb{R}^{d}\rightarrow\mathbb{R}$, we write

$D^{\gamma}u=D_{1}^{\gamma_{1}}\cdots
D_{d}^{\gamma_{d}}u\quad\mbox{and}\quad|\gamma|=\gamma_{1}+\cdots+\gamma_{d}.$

We write $\nabla u=(D_{1}u,\dots,D_{d}u)$ for the gradient of $u$.

We use bolditalic to denote vector fields, e.g.,
$\bm{u}:\mathbb{R}^{d}\times(0,\infty)\rightarrow\mathbb{R}^{d}$. Similarly we
use boldroman to denote 2-tensors, e.g.,
$\mathbf{F}:\mathbb{R}^{d}\times(0,\infty)\rightarrow\mathbb{R}^{d\times d}$.
For two vector fields $\bm{u}=(u^{1},\dots,u^{d})$ and
$\bm{v}=(v^{1},\dots,v^{d})$, let $\bm{u}\otimes\bm{v}$ be the $2$-tensor
whose $(i,j)$ component is $u^{i}v^{j}$ for $1\leq i,j\leq d$. Also,
$(\bm{u}\cdot\nabla)\bm{v}$ is the vector field defined by

$[(\bm{u}\cdot\nabla)\bm{v}]^{i}=\sum_{j=1}^{d}u^{j}\frac{\partial
v^{i}}{\partial x_{j}}\quad(1\leq i\leq d).$

For two 2-tensors $\mathbf{F}=[F^{ij}]_{i,j=1}^{d}$ and
$\mathbf{G}=[G^{ij}]_{i,j=1}^{d}$, their inner product is defined by

$\mathbf{F}:\mathbf{G}=\sum_{i,j=1}^{d}F^{ij}G^{ij}$

For $\Omega\subset\mathbb{R}^{d}$ and $1\leq p\leq\infty$, $L_{p}(\Omega)$
denotes the $L_{p}$-space over $\Omega$ with the norm
$\|{\cdot}\|_{{p};\Omega}$. For $1\leq p<\infty$ and $1\leq q\leq\infty$, we
denote by $L_{p,q}(\Omega)$ the Lorentz space over $\Omega$ with the quasi-
norm $\|{\cdot}\|_{{p,q};\Omega}$. Let $C^{\infty}_{c}(\Omega)$ denote the
space of all smooth functions (or vector fields) that have compact supports in
$\Omega$. We denote by $C_{c,\sigma}^{\infty}(\Omega)$ the set of all vector
fields in $C_{c}^{\infty}(\Omega)$ that are divergence-free. For the sake of
convenience, we drop $\Omega$ in function spaces and their (quasi-)norms if
$\Omega=\mathbb{R}^{d}$, e.g., $L_{p}=L_{p}(\mathbb{R}^{d})$ and
$\|\cdot\|_{p}=\|\cdot\|_{p;\mathbb{R}^{d}}$. For $x\in\mathbb{R}^{d}$ and
$r>0$, let $B_{r}(x)$ denote the open ball of radius $r$ centered at $x$. We
write $B_{r}$ for open balls centered at the origin.

For two nonnegative quantities $A$ and $B$, we write
$A\apprle_{\alpha,\beta,\ldots}B$ if $A\leq CB$ for some positive constant $C$
depending on the parameters $\alpha$, $\beta$, …. If the dependence is
evident, we usually omit the subscripts and simply write $A\apprle B$. We
write $A\approx B$ if $A\apprle B$ and $B\apprle A$. For two topological
vector spaces $X$ and $Y$ with $X\subset Y$, we write $X\hookrightarrow Y$ if
$X$ is continuously embedded into $Y$, that is, the embedding $x\mapsto x$ is
continuous. For a topological vector space $X$, we denote by $X^{*}$ the
(topological) dual space of $X$. The dual pairing between $X^{*}$ and $X$ is
denoted by $\left<\cdot,\cdot\right>$.

Let $\mathscr{S}$ and $\mathscr{S}^{\prime}$ be the Schwartz class and the
space of tempered distributions on $\mathbb{R}^{d}$, respectively. For an
integrable function $f$ on $\mathbb{R}^{d}$, we denote by $\hat{f}$ and
$f^{\vee}$ the Fourier transform and the inverse Fourier transform,
respectively, of $f$:

$\hat{f}(\xi)=\int_{\mathbb{R}^{d}}e^{-2\pi
ix\cdot\xi}f(x)\,d{x}\quad\text{and}\quad
f^{\vee}(x)=\int_{\mathbb{R}^{d}}e^{2\pi ix\cdot\xi}f(\xi)\,d{\xi}.$

If $f$ is a tempered distribution, then $\hat{f}$ and $f^{\vee}$ are defined
via duality by

$\langle{\hat{f},\phi}\rangle=\langle{f,\hat{\phi}}\rangle\quad\mbox{and}\quad\left<f^{\vee},\phi\right>=\left<f,\phi^{\vee}\right>\quad\mbox{for
all}\,\,\phi\in\mathscr{S}.$

###  2.2. Sobolev spaces

For $s\in\mathbb{R}$ and $1<p<\infty$, the _inhomogeneous Sobolev space_
${H}^{s}_{p}$ is defined by

${H}^{s}_{p}=\\{f\in\mathscr{S}^{\prime}:J^{s}f\in L_{p}\\},$

where
$J^{s}=(I-\Delta)^{s/2}:\mathscr{S}^{\prime}\rightarrow\mathscr{S}^{\prime}$
is defined via the Fourier transform by

$\widehat{J^{s}f}=\left(1+4\pi^{2}|\cdot|^{2}\right)^{s/2}\hat{f}\quad\text{for
}f\in\mathscr{S}^{\prime}.$

For $p=2$, we write ${H}^{s}={H}^{s}_{2}$. Then the Sobolev space ${H}^{s}$ is
indeed a Hilbert space equipped with the inner product

$(u,v)_{{H}^{s}}=\left(J^{s}u,J^{s}v\right)$

with $(\cdot,\cdot)$ denoting the inner product on $L_{2}$. By the Plancherel
theorem, we have

$(u,v)_{{H}^{s}}=\int_{\mathbb{R}^{d}}\left(1+4\pi^{2}|\xi|^{2}\right)^{s}\hat{u}(\xi)\overline{\hat{v}(\xi)}\,\,d{\xi}.$

To define homogeneous Sobolev spaces, we need to introduce the following
subspace of $\mathscr{S}$:

$\mathscr{S}_{0}=\\{\phi\in\mathscr{S}:D^{\gamma}\hat{\phi}(0)=0\quad\text{for
all }\gamma\in\mathbb{N}^{d}_{0}\\}.$

Let $\mathscr{S}_{0}^{\prime}$ be the dual space of $\mathscr{S}_{0}$. Then
since $\mathscr{S}_{0}\hookrightarrow\mathscr{S}\hookrightarrow L_{1}\cap
L_{\infty}$, we have

$L_{1}+L_{\infty}\hookrightarrow\mathscr{S}^{\prime}\hookrightarrow\mathscr{S}_{0}^{\prime}.$

Moreover, it follows from the Hahn-Banach theorem that every
$f\in\mathscr{S}_{0}^{\prime}$ can be extended to a tempered distribution $F$.
Such an $F$ is unique up to additive polynomials, thanks to the following
simple lemma (see e.g. [19, Proposition 1.1.3]).

######  Lemma 2.1.

Let $f$ be a tempered distribution. Then $f$ is a polynomial if and only if
$\left<f,\psi\right>=0$ for all $\psi\in\mathscr{S}_{0}$.

The fractional Laplacian
$\Lambda^{s}:\mathscr{S}_{0}\rightarrow\mathscr{S}_{0}$ of order
$s\in\mathbb{R}$ is defined by

$\Lambda^{s}\phi=\left((2\pi|\cdot|)^{s}\hat{\phi}\right)^{\vee}\quad\text{for
all }\phi\in\mathscr{S}_{0}.$

Then since $\Lambda^{s+t}=\Lambda^{s}\Lambda^{t}$ for any $s,t\in\mathbb{R}$,
it follows that $\Lambda^{s}:\mathscr{S}_{0}\rightarrow\mathscr{S}_{0}$ is an
isomorphism with the inverse $\Lambda^{-s}$. Hence for each
$u\in\mathscr{S}_{0}^{\prime}$, we define
$\Lambda^{s}u\in\mathscr{S}_{0}^{\prime}$ via duality by

$\left<\Lambda^{s}u,\phi\right>=\left<u,\Lambda^{s}\phi\right>\quad\text{for
all }\phi\in\mathscr{S}_{0}.$

Note also that
$\Lambda^{s}:\mathscr{S}_{0}^{\prime}\rightarrow\mathscr{S}_{0}^{\prime}$ is
an isomorphism.

For $s\in\mathbb{R}$ and $1<p<\infty$, the _homogeneous Sobolev space_
$\dot{H}^{s}_{p}$ is defined by

$\dot{H}^{s}_{p}=\\{f\in\mathscr{S}_{0}^{\prime}:\Lambda^{s}f\in L_{p}\\}.$

For $p=2$, we write $\dot{H}^{s}=\dot{H}^{s}_{p}$. The space $\dot{H}^{s}$ is
a Hilbert space equipped with the inner product

$(u,v)_{\dot{H}^{s}}=\left(\Lambda^{s}u,\Lambda^{s}v\right).$

It should be remarked that $\Lambda^{t}:\dot{H}^{s+t}\rightarrow\dot{H}^{s}$
is an isometric isomorphism for any $s,t\in\mathbb{R}$. It is standard that
${H}^{s}=L_{2}\cap\dot{H}^{s}$ for any $s\geq 0$ (see e.g. [6, Theorem
6.3.2]).

The following density results will be used several times in the paper.

######  Lemma 2.2.

  1. (1)

$C_{c}^{\infty}$ is dense in $\mathscr{S}$.

  2. (2)

For each $\phi\in\mathscr{S}$, there exists a sequence $\\{\phi_{k}\\}$ in
$\mathscr{S}_{0}$ such that $\phi_{k}\rightarrow\phi$ in ${H}^{s}$ for any
$s\geq 0$

  3. (3)

$\mathscr{S}_{0}$ is dense in both ${H}^{s}$ and $\dot{H}^{s}$ for
$s\in\mathbb{R}$.

  4. (4)

$\mathscr{S}_{0}$ is dense in $L_{p}\cap\dot{H}^{s}$ for $1<p<\infty$ and
$s\in\mathbb{R}$.

###### Proof.

To prove Parts (1) and (2), we choose $\psi\in C^{\infty}$ such that
$0\leq\psi\leq 1$, $\psi=0$ in $B_{1}$, and $\psi=1$ outside $B_{2}$. Given
$\phi\in\mathscr{S}$, we define $\phi_{k}(x)=\left[1-\psi(x/k)\right]\phi(x)$
for $k\in\mathbb{N}$. Then $\phi_{k}\in C_{c}^{\infty}$ and
$\phi_{k}\rightarrow\phi$ in $\mathscr{S}$. Moreover, if
$\psi_{k}=\left(\psi(k\cdot)\hat{\phi}\right)^{\vee}$ for $k\in\mathbb{N}$,
then $\psi_{k}\in\mathscr{S}_{0}$ and

$\|{\psi_{k}-\phi}\|_{{H}^{s}}^{2}=\int_{\mathbb{R}^{d}}\left(1+4\pi^{2}|\xi|^{2}\right)^{s}\left|\psi(k\xi)-1\right|^{2}|\hat{\phi}(\xi)|^{2}\,\,d{\xi}\rightarrow
0$

for any $s\geq 0$, by the dominated convergence theorem.

Suppose that $f\in\dot{H}^{s}$ and $u=\Lambda^{s}f$. Then since $\mathscr{S}$
is dense in $L_{2}$, it follows from Part (2) that there exists a sequence
$\\{u_{k}\\}$ in $\mathscr{S}_{0}$ such that $u_{k}\to u$ in $L_{2}$. If
$f_{k}=\Lambda^{-s}u_{k}$, then $f_{k}\in\mathscr{S}_{0}$ and
$\|{f_{k}-f}\|_{\dot{H}^{s}}=\|{u_{k}-u}\|_{2}\to 0$. This proves that
$\mathscr{S}_{0}$ is dense in $\dot{H}^{s}$. Similarly, using $J^{s}$ instead
of $\Lambda^{s}$, one easily shows that $\mathscr{S}_{0}$ is dense in
${H}^{s}$. This proves Part (3). A proof of (4) will be given later in
Appendix A. ∎

An immediate consequence of Lemma 2.2 (3) is that every $f\in\dot{H}^{-s}$ can
be extended to a bounded linear functional on $\dot{H}^{s}$ in such a way that
the inequality

(2.1) $\left<f,g\right>\leq\|f\|_{\dot{H}^{-s}}\|g\|_{\dot{H}^{s}}$

holds for all $g\in\dot{H}^{s}$. Moreover, since $L^{2}$ is self-dual and
$\Lambda^{t}:\dot{H}^{t}\to L^{2}$ is an isomorphism for any $t\in\mathbb{R}$,
it is easily shown that $\dot{H}^{-s}$ is the dual space of $\dot{H}^{s}$.

Assume now that $s>0$. Then it can be shown (see e.g. [20, Lemma 1]) that
$\Lambda^{s}$ is well-defined on $\mathscr{S}$ and maps $\mathscr{S}$ to
$\mathscr{S}_{s}$, where

$\mathscr{S}_{s}=\left\\{f\in
C^{\infty}:\left(1+|\cdot|^{d+s}\right)D^{\gamma}\psi\in L_{\infty}\,\,\text{
for all }\gamma\in\mathbb{N}_{0}^{d}\right\\}.$

In particular, $\Lambda^{s}$ is bounded from $\mathscr{S}$ to $L_{1}\cap
L_{\infty}$. Hence for each $u\in L_{1}+L_{\infty}$, we can extend
$\Lambda^{s}u$ to a tempered distribution by defining

$\left<\Lambda^{s}u,\phi\right>=\int_{\mathbb{R}^{d}}u\Lambda^{s}\phi\,d{x}\quad\mbox{for
all}\,\,\phi\in\mathscr{S}.$

###  2.3. Lorentz spaces

Let $\Omega$ be any domain in $\mathbb{R}^{d}$. For a measurable function
$f:\Omega\rightarrow\mathbb{C}$, the _decreasing rearrangement_ $f^{*}$ of $f$
is defined by

$f^{*}(t)=\inf\\{s>0:d_{f}(s)\leq t\\}\quad(t>0),$

where $d_{f}(s)=|\\{x\in\Omega:|f(x)|>s\\}|$. For $1\leq p<\infty$ and $1\leq
q\leq\infty$, we define

$\|{f}\|_{{p,q};\Omega}=\begin{dcases*}\left(\int_{0}^{\infty}[t^{1/p}f^{*}(t)]^{q}\frac{dt}{t}\right)^{1/q}&if
$q<\infty$,\\\ \sup_{t>0}t^{1/p}f^{*}(t)&if $q=\infty$.\end{dcases*}$

The set of all $f$ satisfying $\|{f}\|_{{p,q};\Omega}<\infty$ is denoted by
$L_{p,q}(\Omega)$ and called the _Lorentz space_ over $\Omega$ with indices
$p$ and $q$. It is well-known that $L_{p,q}(\Omega)$ is a complete quasi-
normed space and is normable if $1<p<\infty$ (see e.g. [32, Theorem 3.22]).
Moreover, for $1\leq p<\infty$, we have

$L_{p,1}(\Omega)\hookrightarrow L_{p,p}(\Omega)=L_{p}(\Omega)\hookrightarrow
L_{p,\infty}(\Omega).$

Using Lorentz spaces, we can refine several well-known inequalities in
analysis. The first one is Hölder’s inequality in Lorentz spaces which was
proved by O’Neil [29, Theorems 3.4, 3.5].

######  Lemma 2.3.

Let $1\leq p,p_{1},p_{2}<\infty$ and $1\leq q,q_{1},q_{2}\leq\infty$ satisfy

$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\quad\text{and}\quad\frac{1}{q}\leq\frac{1}{q_{1}}+\frac{1}{q_{2}}.$

Then for all $f\in L_{p_{1},q_{1}}(\Omega)$ and $g\in
L_{p_{2},q_{2}}(\Omega)$,

$\|{fg}\|_{{p,q};\Omega}\apprle\|{f}\|_{{p_{1},q_{1}};\Omega}\|{g}\|_{{p_{2},q_{2}};\Omega}.$

The second one is the following refinement of Sobolev’s inequality (see e.g.
[1, Theorem 7.34] and [6, Theorem 6.5.1]).

######  Lemma 2.4.

Let $0<s<d/2$. Then for all $u\in\dot{H}^{s}$, we have

$\|{u}\|_{{{2d}/{(d-2s)},2}}\apprle\|{u}\|_{\dot{H}^{s}}.$

The third one is the following generalization of Young’s convolution
inequality due to O’Neil [29, Theorem 2.6] (see also [18, Theorems 1.2.13 and
1.4.24]).

######  Lemma 2.5.

Let $1<p,p_{1}<\infty$ and $1\leq p_{2}<\infty$ satisfy

$\frac{1}{p}+1=\frac{1}{p_{1}}+\frac{1}{p_{2}}.$

Then for all $f\in L^{p_{1},\infty}$ and $g\in L^{p_{2}}$, we have

$\displaystyle\|f*g\|_{p}$
$\displaystyle\apprle\|f\|_{p_{1},\infty}\|g\|_{p_{2}}\quad\mbox{if}\,\,1<p_{2}<\infty;$
$\displaystyle\|f*g\|_{p,\infty}$
$\displaystyle\apprle\|f\|_{p_{1},\infty}\|g\|_{1}\quad\mbox{if}\,\,p_{2}=1.$

Recall the following classical Gagliardo-Nirenberg interpolation inequality
(see e.g. [8, Theorem 5.10]).

######  Proposition 2.6.

Let $0\leq s_{0}<s$, $1<p,p_{1}<\infty$, and $0<\theta<1$ satisfy

$\frac{1}{p}-\frac{s_{0}}{d}=\frac{1-\theta}{p_{1}}+\theta\left(\frac{1}{2}-\frac{s}{d}\right)\quad\text{and}\quad\theta\geq\frac{s_{0}}{s}.$

Then for all $f\in L_{p_{1}}\cap\dot{H}^{s}$, we have

$\|{\Lambda^{s_{0}}f}\|_{{p}}\apprle\|{f}\|_{{p_{1}}}^{1-\theta}\|{\Lambda^{s}f}\|_{{2}}^{\theta}.$

Particularly, if $s_{0}=0$ and $p_{1}=2$ in Proposition 2.6, we deduce the
well-known Sobolev embedding $H^{s}\hookrightarrow L^{p}$ for $s>0$ and $2\leq
p<\infty$ satisfying $1/p\geq 1/2-s/d$.

The Gagliardo-Nirenberg interpolation inequality can be refined by using
Lorentz spaces. The following is a special case of the Gagliardo-Nirenberg
interpolation inequalities in Sobolev-Lorentz spaces (see [36, Theorem 1.1]
and [8, Theorem 5.17]).

######  Proposition 2.7.

Let $s>0$, $1<p,p_{1}<\infty$, and $0<\theta<1$ satisfy

$\frac{1}{p}=\frac{1-\theta}{p_{1}}+\theta\left(\frac{1}{2}-\frac{s}{d}\right)\quad\text{and}\quad\frac{1}{p_{1}}\neq\frac{1}{2}-\frac{s}{d}.$

Then for all $f\in L_{p_{1},\infty}\cap\dot{H}^{s}$, we have

$\|{f}\|_{{p,1}}\apprle\|{f}\|_{{p_{1},\infty}}^{1-\theta}\|{\Lambda^{s}f}\|_{{2}}^{\theta}.$

###  2.4. Product estimates

In this subsection, we prove product estimates that will be used in the paper.
The first product estimate plays a crucial role in the proof of existence and
uniqueness of weak solutions of (1.1).

######  Lemma 2.8.

Let $\alpha,\beta>0$ satisfy $1/2<\alpha<(d+1)/2$ and $\alpha+\beta>d/2$. Then
there exist $0<\theta_{1},\theta_{2}<1$ such that

$\|{fg}\|_{{2}}\apprle\|{f}\|_{{d/(d+1-2\alpha),\infty}}^{1-\theta_{1}}\|{f}\|_{\dot{H}^{\alpha}}^{\theta_{1}}\|{g}\|_{{2}}^{1-\theta_{2}}\|{g}\|_{\dot{H}^{\beta}}^{\theta_{2}}$

for all $f\in L_{d/(d+1-2\alpha),\infty}\cap\dot{H}^{\alpha}$ and
$g\in{H}^{\beta}$. Moreover, if $\alpha+(1-\mu)\beta\geq d/2+1$ for some
$0\leq\mu\leq 1$, then $\theta_{1}$ and $\theta_{2}$ can be chosen so that
$(1+\mu)\theta_{1}+\theta_{2}+1/\beta\leq 2$.

###### Proof.

We will show that there exists $2<p<\infty$ satisfying

(2.2)
$\frac{d}{2}-\frac{d}{p}<\alpha<\frac{d+1}{2}-\frac{d}{2p}\quad\text{and}\quad\beta>\frac{d}{p}.$

For such $p$, we define

$\theta_{1}=\frac{d+1-2\alpha-d/p}{{d}/{2}+1-\alpha}\quad\text{and
}\quad\theta_{2}=\frac{d}{p\beta}.$

Then since

(2.3) $\displaystyle\frac{1}{p}$
$\displaystyle=(1-\theta_{1})\frac{d+1-2\alpha}{d}+\theta_{1}\left(\frac{1}{2}-\frac{\alpha}{d}\right),$
$\displaystyle\frac{p-2}{2p}$
$\displaystyle=\frac{1-\theta_{2}}{2}+\theta_{2}\left(\frac{1}{2}-\frac{\beta}{d}\right),\quad\mbox{and}\quad
0<\theta_{1},\theta_{2}<1,$

it follows from Hölder’s inequality and Proposition 2.7 that

$\displaystyle\|{fg}\|_{{2}}\leq\|{f}\|_{{p}}\|{g}\|_{{2p/(p-2)}}\apprle\|{f}\|_{{d/(d+1-2\alpha),\infty}}^{1-\theta_{1}}\|{f}\|_{\dot{H}^{\alpha}}^{\theta_{1}}\|{g}\|_{{2}}^{1-\theta_{2}}\|{g}\|_{\dot{H}^{\beta}}^{\theta_{2}}.$

Moreover, since

${\beta\theta_{2}}=\frac{d}{p}=(1-\theta_{1})\left({d+1-2\alpha}\right)+\theta_{1}\left(\frac{d}{2}-{\alpha}\right),$

we deduce that if $\alpha+(1-\mu)\beta\geq d/2+1$ for some $0\leq\mu\leq 1$,
then

$\displaystyle\beta\left[(1+\mu)\theta_{1}+\theta_{2}-2\right]+1$
$\displaystyle=\left[\alpha+(1+\mu)\beta-\frac{d}{2}-1\right]\theta_{1}-2\alpha-2\beta+d+2$
$\displaystyle\leq-\alpha-(1-\mu)\beta+\frac{d}{2}+1\leq 0.$

Therefore it remains to prove existence of $2<p<\infty$ satisfying (2.2).

First, if ${d}/{2}\leq\alpha<{(d+1)}/{2}$ and $\beta>0$, then choosing a
sufficiently large $2<p<\infty$, we obviously have

$\frac{d}{2}-\frac{d}{p}<\frac{d}{2}\leq\alpha<\frac{d+1}{2}-\frac{d}{2p}\quad\text{and}\quad\beta>\frac{d}{p}.$

Suppose next that ${1}/{2}<\alpha<{d}/{2}$ and $\beta>d/2-\alpha$. Then

$\max\left\\{2,\frac{d}{\beta},\frac{d}{d+1-2\alpha}\right\\}<\frac{d}{{d}/{2}-\alpha}<\infty.$

Choosing any $p$ such that

$\max\left\\{2,\frac{d}{\beta},\frac{d}{d+1-2\alpha}\right\\}<p<\frac{d}{{d}/{2}-\alpha},$

we have

$2<p<\infty,\quad\frac{d}{2}-\frac{d}{p}<\alpha<\frac{d+1}{2}-\frac{d}{2p},\quad\text{and}\quad\beta>\frac{d}{p}.$

This completes the proof of Lemma 2.8. ∎

To prove uniqueness of weak solutions of (1.1) particularly when
$1/2<\alpha<1$, we also need the following product estimate.

######  Lemma 2.9.

Let $\gamma>d/2$ and $0<s\leq\gamma$. Then for all $f\in{H}^{s}$ and
$g\in{H}^{\gamma}$,

$\|{\Lambda^{s}(fg)}\|_{{2}}\leq C\|{f}\|_{{H}^{s}}\|{g}\|_{{H}^{\gamma}},$

where $C=C(d,\gamma,s)>0$.

###### Proof.

A proof is given in [22, Lemma 3.2]. However, we provide a detailed proof of
the lemma for the sake of convenience.

Suppose that $0<s<\gamma$. Then since $\gamma>d/2$, there exists $2<p<\infty$
such that

$\max\left\\{\frac{1}{2}-\frac{s}{d},0\right\\}<\frac{1}{p}<\min\left\\{\frac{1}{2},\frac{\gamma-s}{d}\right\\}.$

By the Sobolev embedding theorem, we have

$\|{f}\|_{{p}}\apprle\|{f}\|_{{H}^{s}}\quad\text{and}\quad\|{J^{s}g}\|_{{{2p}/{(p-2)}}}\apprle\|{J^{s}g}\|_{{H}^{\gamma-s}}.$

Therefore, it follows from the fractional Leibniz rule (see e.g. [20, Theorem
1]) and Sobolev embedding theorem that

$\displaystyle\|{\Lambda^{s}(fg)}\|_{{2}}$
$\displaystyle\apprle\|{f}\|_{{p}}\|{\Lambda^{s}g}\|_{{{2p}/{(p-2)}}}+\|{\Lambda^{s}f}\|_{{2}}\|{g}\|_{{\infty}}$
$\displaystyle\apprle\|{f}\|_{{H}^{s}}\|{J^{s}g}\|_{{H}^{\gamma-s}}+\|{\Lambda^{s}f}\|_{{2}}\|{g}\|_{{H}^{\gamma}}$
$\displaystyle\apprle\|{f}\|_{{H}^{s}}\|{g}\|_{{H}^{\gamma}}.$

Similarly, if $s=\gamma$, then

$\displaystyle\|{\Lambda^{s}(fg)}\|_{{2}}$
$\displaystyle\apprle\|{f}\|_{{\infty}}\|{\Lambda^{s}g}\|_{{2}}+\|{\Lambda^{s}f}\|_{{2}}\|{g}\|_{{\infty}}$
$\displaystyle\apprle\|{f}\|_{{H}^{s}}\|{g}\|_{{H}^{\gamma}}.$

This completes the proof of Lemma 2.9. ∎

######  Lemma 2.10.

Let $\alpha$ and $\beta$ satisfy either one of the following conditions:

  1. (1)

$\alpha<1$, $\beta>d/2$, and $\alpha+\beta\geq 1$.

  2. (2)

$1\leq\alpha<d/2+1$ and $\alpha+2\beta\geq d/2+1$.

Then for all $f,g\in{H}^{\beta}$,

$\|{fg}\|_{\dot{H}^{1-\alpha}}\leq
C\|{f}\|_{{H}^{\beta}}\|{g}\|_{{H}^{\beta}},$

where $C=C(d,\alpha,\beta)>0$.

###### Proof.

Suppose that $\alpha<1$, $\beta>d/2$, and $\alpha+\beta\geq 1$. Then since
$0<1-\alpha\leq\beta$, it immediately follows from Lemma 2.9 that

$\|{fg}\|_{\dot{H}^{1-\alpha}}\leq
C\|{f}\|_{{H}^{1-\alpha}}\|{g}\|_{{H}^{\beta}}\leq
C\|{f}\|_{{H}^{\beta}}\|{g}\|_{{H}^{\beta}}.$

Suppose next that $1\leq\alpha<d/2+1$ and $\alpha+2\beta\geq d/2+1$. Recall
then that

$\dot{H}^{1-\alpha}=\left(\dot{H}^{\alpha-1}\right)^{*}.$

Let $2\leq r<\infty$ be defined by

$\frac{1}{r}=\frac{1}{2}-\frac{\alpha-1}{d}.$

Note that

$\beta>0\quad\mbox{and}\quad\frac{1}{2r^{\prime}}=\frac{1}{2}-\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha-1}{d}\right)\geq\frac{1}{2}-\frac{\beta}{d}.$

Hence by the Sobolev embedding theorem (see Lemma 2.4 and Proposition 2.6 with
$p_{1}=2$ e.g.), we have

$\int_{\mathbb{R}^{d}}(fg)\phi\,d{x}\leq\|{f}\|_{{2r^{\prime}}}\|{g}\|_{{2r^{\prime}}}\|{\phi}\|_{{r}}\apprle\|{f}\|_{{H}^{\beta}}\|{g}\|_{{H}^{\beta}}\|{\phi}\|_{\dot{H}^{\alpha-1}}$

for any $\phi\in\dot{H}^{\alpha-1}$. This completes the proof of Lemma 2.10. ∎

##  3\. Fractional Stokes equations

In this section, we consider the fractional Stokes equations in
$\mathbb{R}^{d},d\geq 2$:

(3.1)
$\Lambda^{2\alpha}\bm{u}+\nabla{p_{*}}=\operatorname{div}\mathbf{F},\quad\operatorname{div}\bm{u}=0\quad\mbox{in}\,\,\mathbb{R}^{d},$

where $\alpha$ is a real number. Here $\mathbf{F}=[F^{jk}]_{1\leq j,k\leq d}$
is a given matrix-valued function and $\operatorname{div}\mathbf{F}$ is the
vector field whose $j$-th component is given by

$(\operatorname{div}\mathbf{F})^{j}=\sum_{k=1}^{d}D_{k}F^{jk};$

hence

$\operatorname{div}\operatorname{div}\mathbf{F}=\sum_{j,k=1}^{d}D_{jk}F^{jk}$

and

$\widehat{\operatorname{div}\operatorname{div}\mathbf{F}}(\xi)=(2\pi
i\xi)\otimes(2\pi
i\xi):\widehat{\mathbf{F}}(\xi)=-4\pi^{2}\sum_{j,k=1}^{d}\xi_{j}\xi_{k}\widehat{F}^{jk}(\xi).$

This section consists of three subsections. In Subsection 3.1, we prove
solvability results in Sobolev spaces for (3.1). In Subsection 3.2, we derive
a solution formula for the problem (3.1) that will be used in several places
in the rest of the paper. Finally, very weak solutions of (3.1) are studied in
Subsection 3.3.

###  3.1. Solutions in Sobolev spaces

The problem (3.1) is first solved in the class $\mathscr{S}_{0}$ by decoupling
the fractional Stokes system into two elliptic equations.

######  Lemma 3.1.

Let $\alpha$ be any real number. Then for each $\mathbf{F}\in\mathscr{S}_{0}$,
there exists a unique solution
$(\bm{u},{p_{*}})\in\mathscr{S}_{0}\times\mathscr{S}_{0}$ of (3.1).

###### Proof.

Suppose that $\mathbf{F}\in\mathscr{S}_{0}$. Then since
$\operatorname{div}\mathbf{F}\in\mathscr{S}_{0}$ and $\Delta=-\Lambda^{2}$ is
an isomorphism from $\mathscr{S}_{0}$ onto itself, there exists a unique
${p_{*}}\in\mathscr{S}_{0}$ such that

(3.2)
$\Delta{p_{*}}=\operatorname{div}\operatorname{div}\mathbf{F}\quad\mbox{in}\,\,\mathbb{R}^{d}.$

Let $\mathbf{I}$ denote the identity matrix. Then since
$\Lambda^{2\alpha}:\mathscr{S}_{0}\rightarrow\mathscr{S}_{0}$ is an
isomorphism, there exists a unique $\bm{u}\in\mathscr{S}_{0}$ such that

(3.3)
$\Lambda^{2\alpha}\bm{u}=\operatorname{div}(\mathbf{F}-{p_{*}}\mathbf{I})\quad\mbox{in}\,\,\mathbb{R}^{d}.$

Since $\operatorname{div}({p_{*}}\mathbf{I})=\nabla{p_{*}}$, we have

$\Lambda^{2\alpha}\bm{u}+\nabla{p_{*}}=\operatorname{div}\mathbf{F}\quad\mbox{in}\,\,\mathbb{R}^{d}.$

Moreover, taking the divergence in (3.3), we have

$\Lambda^{2\alpha}(\operatorname{div}\bm{u})=\operatorname{div}\operatorname{div}\mathbf{F}-\Delta{p_{*}}=0\quad\mbox{in}\,\,\mathbb{R}^{d}.$

Since $\operatorname{div}\bm{u}\in\mathscr{S}_{0}$ and
$\Lambda^{2\alpha}:\mathscr{S}_{0}\rightarrow\mathscr{S}_{0}$ is an
isomorphism, it follows that $\operatorname{div}\bm{u}=0$ in $\mathbb{R}^{d}$.
This proves existence of a solution in $\mathscr{S}_{0}\times\mathscr{S}_{0}$
of (3.1).

To prove the uniqueness assertion, let
$(\bm{u},{p_{*}})\in\mathscr{S}_{0}\times\mathscr{S}_{0}$ be a solution of
(3.1) with the trivial data $\mathbf{F}={\mathbf{0}}$. Then since
$\operatorname{div}\bm{u}=0$ in $\mathbb{R}^{d}$ and
$\Lambda^{2\alpha}\operatorname{div}=\operatorname{div}\Lambda^{2\alpha}$ in
$\mathscr{S}_{0}$, we take the divergence in (3.1) to obtain

$\Delta{p_{*}}=0\quad\mbox{in}\,\,\mathbb{R}^{d}.$

Since $\Delta:\mathscr{S}_{0}\rightarrow\mathscr{S}_{0}$ is an isomorphism, it
follows that ${p_{*}}=0$ in $\mathbb{R}^{d}$. Then since
$\Lambda^{2\alpha}\bm{u}=\mathbf{0}$ in $\mathbb{R}^{d}$, we conclude that
$\bm{u}=\mathbf{0}$ in $\mathbb{R}^{d}$. This completes the proof of Lemma
3.1. ∎

Using the decomposition of (3.1) into (3.2) and (3.3), we prove unique
solvability of (3.1) in homogeneous Sobolev spaces of arbitrary orders.

######  Theorem 3.2.

Let $\alpha,s\in\mathbb{R}$. Then for each $\mathbf{F}\in\dot{H}^{s}$, there
exists a unique solution $(\bm{u},{p_{*}})$ in
$\dot{H}^{2\alpha+s-1}\times\dot{H}^{s}$ of (3.1). Moreover, we have

$\|{\bm{u}}\|_{\dot{H}^{2\alpha+s-1}}+\|{{p_{*}}}\|_{\dot{H}^{s}}\apprle\|{\mathbf{F}}\|_{\dot{H}^{s}}.$

###### Proof.

Suppose that $\mathbf{F}\in\dot{H}^{s}$. Then since
$\operatorname{div}\operatorname{div}\mathbf{F}\in\dot{H}^{s-2}$ and $\Delta$
is an isometric isomorphism from $\dot{H}^{t+2}$ to $\dot{H}^{t}$ for any
$t\in\mathbb{R}$, there exists a unique solution ${p_{*}}\in\dot{H}^{s}$ of
(3.2), which satisfies

$\|{{p_{*}}}\|_{\dot{H}^{s}}\apprle\|{\mathbf{F}}\|_{\dot{H}^{s}}.$

Moreover, since $\Lambda^{2\alpha}$ is an isomorphism from
$\dot{H}^{2\alpha+s-1}$ to $\dot{H}^{s-1}$, there exists a unique
$\bm{u}\in\dot{H}^{2\alpha+s-1}$ satisfying (3.3). Note also that

$\|{\bm{u}}\|_{\dot{H}^{2\alpha+s-1}}\apprle\|{\mathbf{F}}\|_{\dot{H}^{s}}+\|{{p_{*}}}\|_{\dot{H}^{s}}\apprle\|{\mathbf{F}}\|_{\dot{H}^{s}}.$

Finally, since $\operatorname{div}\bm{u}\in\dot{H}^{2\alpha+s-2}$ and
$\Lambda^{2\alpha}(\operatorname{div}\bm{u})=0$ in $\dot{H}^{s-2}$, it follows
that $\operatorname{div}\bm{u}=0$ in $\mathbb{R}^{d}$. This proves the
existence assertion of the theorem.

To show the uniqueness, suppose that
$(\bm{u},{p_{*}})\in\dot{H}^{2\alpha+s-1}\times\dot{H}^{s}$ satisfies (3.1)
with $\mathbf{F}=\mathbf{0}$. Since $\operatorname{div}\bm{u}=0$ in
$\mathbb{R}^{d}$ and
$\Lambda^{2\alpha}\operatorname{div}=\operatorname{div}\Lambda^{2\alpha}$ in
$\mathscr{S}_{0}^{\prime}$, it follows from (3.1) that $\Delta{p_{*}}=0$ in
$\mathbb{R}^{d}$. Since $\Delta$ is an isomorphism, we then deduce that
${p_{*}}=0$ in $\mathbb{R}^{d}$. From (3.1) again, we have
$\Lambda^{2\alpha}\bm{u}=\mathbf{0}$ in $\mathbb{R}^{d}$. Since
$\Lambda^{2\alpha}$ is an isomorphism, we then conclude that
$\bm{u}=\mathbf{0}$ in $\mathbb{R}^{d}$. This completes the proof of Theorem
3.2. ∎

######  Theorem 3.3.

Let $\alpha$ be any real number. Then for each
$\mathbf{F}\in\dot{H}^{1-\alpha}$, there exists a unique solution
$(\bm{u},{p_{*}})$ in $\dot{H}^{\alpha}\times\dot{H}^{1-\alpha}$ of (3.1).
Moreover, we have

$\|{\bm{u}}\|_{\dot{H}^{\alpha}}+\|{{p_{*}}}\|_{\dot{H}^{1-\alpha}}\apprle\|{\mathbf{F}}\|_{\dot{H}^{1-\alpha}}.$

In addition, $\bm{u}$ satisfies the energy identity

(3.4)
$\int_{\mathbb{R}^{d}}|\Lambda^{\alpha}\bm{u}|^{2}\,d{x}=\left<\operatorname{div}\mathbf{F},\bm{u}\right>.$

###### Proof.

By Theorem 3.2, it remains to show that the solution $(\bm{u},{p_{*}})$ in
$\dot{H}^{\alpha}\times\dot{H}^{1-\alpha}$ of (3.1) satisfies the energy
identity (3.4). Note first that

$\int_{\mathbb{R}^{d}}\Lambda^{\alpha}\bm{u}\cdot\Lambda^{\alpha}\Phi\,d{x}-\left<{p_{*}},\operatorname{div}\Phi\right>=\left<\Lambda^{2\alpha}\bm{u},\Phi\right>+\left<\nabla{p_{*}},\Phi\right>=\left<\operatorname{div}\mathbf{F},\Phi\right>$

for all $\Phi\in\mathscr{S}_{0}$. By Lemma 2.2, there exists a sequence
$\\{\bm{u}_{k}\\}$ in $\mathscr{S}_{0}$ such that
$\|{\bm{u}_{k}-\bm{u}}\|_{\dot{H}^{\alpha}}\rightarrow 0$. For each $k$, we
have

$\int_{\mathbb{R}^{d}}\Lambda^{\alpha}\bm{u}\cdot\Lambda^{\alpha}\bm{u}_{k}\,d{x}-\left<{p_{*}},\operatorname{div}\bm{u}_{k}\right>=\left<\operatorname{div}\mathbf{F},\bm{u}_{k}\right>.$

Hence by the duality inequality (2.1), $\bm{u}$ satisfies

$\int_{\mathbb{R}^{d}}|\Lambda^{\alpha}\bm{u}|^{2}\,d{x}-\left<{p_{*}},\operatorname{div}\bm{u}\right>=\left<\operatorname{div}\mathbf{F},\bm{u}\right>.$

Since $\operatorname{div}\bm{u}=0$, we get the energy identity (3.4). This
completes the proof of Theorem 3.3. ∎

###  3.2. A solution formula

Given $\mathbf{F}=[F^{jk}]_{1\leq j,k\leq d}\in\mathscr{S}_{0}$, let
$(\bm{u},{p_{*}})\in\mathscr{S}_{0}\times\mathscr{S}_{0}$ be the solution of
(3.1). First, taking the Fourier transform in (3.2), we have

$-(2\pi|\xi|)^{2}\widehat{{p_{*}}}(\xi)=(2\pi i\xi)\otimes(2\pi
i\xi):\widehat{\mathbf{F}}(\xi)\quad\mbox{for all }\xi\in{\mathbb{R}}^{d}$

or equivalently

$\widehat{{p_{*}}}(\xi)=\frac{\xi\otimes\xi}{|\xi|^{2}}:\widehat{\mathbf{F}}(\xi)\quad\mbox{for
all }\xi\neq 0.$

Next, taking the Fourier transform in (3.3), we have

$\displaystyle(2\pi|\xi|)^{2\alpha}\widehat{u}^{j}(\xi)$
$\displaystyle=\sum_{l=1}^{d}(2\pi i\xi_{l})\widehat{F}^{jl}(\xi)-(2\pi
i\xi_{j})\widehat{{p_{*}}}(\xi)$ $\displaystyle=\sum_{k,l=1}^{d}(2\pi
i\xi_{l})\left(\delta_{jk}-\frac{\xi_{j}\xi_{k}}{|\xi|^{2}}\right)\widehat{F}^{kl}(\xi)$

and so

$\widehat{u}^{j}(\xi)=\sum_{k,l=1}^{d}\frac{(2\pi
i\xi_{l})}{(2\pi|\xi|)^{2\alpha}}\left(\delta_{jk}-\frac{\xi_{j}\xi_{k}}{|\xi|^{2}}\right)\widehat{F}^{kl}(\xi)$

for $j=1,\ldots,d$.

Assuming that $2\alpha<d+1$, let $U_{\alpha,j}^{kl}\in\mathscr{S}^{\prime}$ be
the inverse Fourier transform of the locally integrable function

$\frac{(2\pi
i\xi_{l})}{(2\pi|\xi|)^{2\alpha}}\left(\delta_{jk}-\frac{\xi_{j}\xi_{k}}{|\xi|^{2}}\right)$

in $\mathscr{S}^{\prime}$. The following lemma will be proved later in
Appendix B.

######  Lemma 3.4.

Suppose that $j,k,l\in\\{1,...,d\\}$, $\lambda\in\mathbb{C}$, and
$\psi\in\mathscr{S}$. Let $\Gamma$ be the Gamma function.

  1. (1)

If $1<\operatorname{Re}\lambda<d+1$, then

$\int_{\mathbb{R}^{d}}\frac{-2\pi
i\xi_{j}}{|\xi|^{\lambda}}{\psi}^{\vee}(\xi)\,d\xi=\frac{2\Gamma((d+2-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\frac{x_{j}}{|x|^{d+2-\lambda}}\psi(x)\,dx.$

  2. (2)

If $2<\operatorname{Re}\lambda<d+2$, then

$\displaystyle\int_{\mathbb{R}^{d}}\frac{(-2\pi i\xi_{j})(-2\pi
i\xi_{k})}{|\xi|^{\lambda}}{\psi}^{\vee}(\xi)\,d\xi$
$\displaystyle\quad=\frac{2\Gamma((d+2-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\frac{(d+2-\lambda)x_{j}x_{k}-\delta_{jk}|x|^{2}}{|x|^{d+4-\lambda}}{\psi}(x)\,dx.$

  3. (3)

If $3<\operatorname{Re}\lambda<d+3$, then

$\displaystyle\int_{\mathbb{R}^{d}}\frac{(-2\pi i\xi_{j})(-2\pi
i\xi_{k})(-2\pi i\xi_{l})}{|\xi|^{\lambda}}{\psi}^{\vee}(\xi)\,d\xi$
$\displaystyle\quad=\frac{4\Gamma((d+4-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\left[-D_{l}\left(\frac{x_{j}x_{k}}{|x|^{d+4-\lambda}}\right)-\frac{\delta_{jk}x_{l}}{|x|^{d+4-\lambda}}\right]{\psi}(x)\,d{x}.$

Assume that $1<2\alpha<d+1$. Then using Lemma 3.4, we can find
$U_{\alpha,j}^{kl}$ explicitly. First of all, it follows from Lemma 3.4 (1)
that the inverse Fourier transform of $(2\pi i\xi_{l})(2\pi|\xi|)^{-2\alpha}$
is

$-\frac{2\Gamma(1+d/2-\alpha)}{2^{2\alpha}\pi^{d/2}\Gamma(\alpha)}\frac{x_{l}}{|x|^{d+2-2\alpha}}.$

Next, by Lemma 3.4 (3) with $\lambda=2\alpha+2$, we have

$\displaystyle\int_{\mathbb{R}^{d}}\frac{2\pi
i\xi_{j}\xi_{k}\xi_{l}}{|\xi|^{2\alpha+2}}{\psi}^{\vee}(\xi)\,d\xi$
$\displaystyle\quad=\frac{\Gamma(1+d/2-\alpha)}{\Gamma(1+\alpha)\pi^{d/2-2\alpha}}\int_{\mathbb{R}^{d}}\left[-D_{l}\left(\frac{x_{j}x_{k}}{|x|^{d+2-2\alpha}}\right)-\frac{\delta_{jk}x_{l}}{|x|^{d+2-2\alpha}}\right]{\psi}(x)\,d{x}$

for all $\psi\in\mathscr{S}$. Hence the inverse Fourier transform of $(2\pi
i\xi_{l}\xi_{j}\xi_{k})(2\pi|\xi|)^{-2\alpha}|\xi|^{-2}$ is

$\frac{\Gamma(1+d/2-\alpha)}{2^{2\alpha}\pi^{d/2}\Gamma(1+\alpha)}\left[-D_{l}\left(\frac{x_{j}x_{k}}{|x|^{d+2-2\alpha}}\right)-\frac{\delta_{jk}x_{l}}{|x|^{d+2-2\alpha}}\right].$

Therefore,

(3.5)
$U_{\alpha,j}^{kl}(x)=\frac{\Gamma(1+d/2-\alpha)}{2^{2\alpha}\pi^{d/2}\Gamma(1+\alpha)}\left[D_{l}\left(\frac{x_{j}x_{k}}{|x|^{d+2-2\alpha}}\right)-\frac{\delta_{jk}(2\alpha-1)x_{l}}{|x|^{d+2-2\alpha}}\right].$

Note that

$U_{\alpha,j}^{kl}\in L_{d/(d+1-2\alpha),\infty}.$

Now, setting $\mathbf{U}_{\alpha}=[U_{\alpha,j}^{kl}]_{1\leq j,k,l\leq d}$, we
have

$\bm{u}=\mathbf{U}_{\alpha}*\mathbf{F}$

or componentwise

$u^{j}=\sum_{k,l=1}^{d}U_{\alpha,j}^{kl}*F^{kl}\quad(j=1,\ldots,d).$

###  3.3. Very weak solutions

The notion of very weak solutions is introduced to solve the fractional Stokes
system (3.1) for $\mathbf{F}\in L_{1}$. Recall that if $s\geq 0$, then
$\Lambda^{s}$ is bounded from $\mathscr{S}$ to $L_{1}\cap L_{\infty}$.

######  Definition 3.5.

Let $\alpha\geq 0$. Suppose that $\mathbf{F}$ is a distribution. Then we say
that $\bm{u}\in L_{1}+L_{\infty}$ is a _very weak solution_ of (3.1) if

(3.6)
$\int_{\mathbb{R}^{d}}\bm{u}\cdot\left(\Lambda^{2\alpha}\Phi+\nabla\psi\right)dx=-\left<\mathbf{F},\nabla\Phi\right>$

for all $\Phi\in C_{c,\sigma}^{\infty}$ and $\psi\in C_{c}^{\infty}$.

Indeed, taking $\Phi=0$ in (3.6), we deduce that if $\bm{u}$ is a very weak
solution of (3.1), then it is divergence-free:

$\int_{\mathbb{R}^{d}}\bm{u}\cdot\nabla\psi\,d{x}=0\quad\mbox{for
all}\,\,\psi\in C_{c}^{\infty}.$

Existence of a pressure associated with a very weak solution of (3.1) is an
easy consequence of the following more or less standard result, which will be
proved later in Appendix C.

######  Lemma 3.6.

If $\bm{u}\in\mathscr{S}^{\prime}$ satisfies $\left<\bm{u},\Phi\right>=0$ for
all $\Phi\in C_{c,\sigma}^{\infty}$, then there exists
$p\in\mathscr{S}^{\prime}$ such that $\bm{u}=\nabla p$.

######  Lemma 3.7.

Let $\alpha\geq 0$. If $\bm{u}$ is a very weak solution of (3.1) with
$\mathbf{F}\in\mathscr{S}^{\prime}$, then there exists
${p_{*}}\in\mathscr{S}^{\prime}$ such that

$\int_{\mathbb{R}^{d}}\bm{u}\cdot\left(\Lambda^{2\alpha}\Phi+\nabla\psi\right)dx-\left<{p_{*}},\operatorname{div}\Phi\right>=-\left<\mathbf{F},\nabla\Phi\right>$

for all $(\Phi,\psi)\in\mathscr{S}$.

###### Proof.

Since $C_{c}^{\infty}$ is dense in $\mathscr{S}$, we easily have

$\int_{\mathbb{R}^{d}}\bm{u}\cdot\nabla\psi\,d{x}=0\quad\mbox{for
all}\,\,\psi\in\mathscr{S}.$

Note that
$\Lambda^{2\alpha}\bm{u}-\operatorname{div}\mathbf{F}\in\mathscr{S}^{\prime}$.
Moreover, for all $\Phi\in C_{c,\sigma}^{\infty}$, we have

$\left<\Lambda^{2\alpha}\bm{u}-\operatorname{div}\mathbf{F},\Phi\right>=\int_{\mathbb{R}^{d}}\bm{u}\cdot\Lambda^{2\alpha}\Phi\,d{x}+\left<\mathbf{F},\nabla\Phi\right>=0.$

Hence by Lemma 3.6, there exists ${p_{*}}\in\mathscr{S}^{\prime}$ such that

$\int_{\mathbb{R}^{d}}\bm{u}\cdot\Lambda^{2\alpha}\Phi\,d{x}+\left<\mathbf{F},\nabla\Phi\right>=\left<\Lambda^{2\alpha}\bm{u}-\operatorname{div}\mathbf{F},\Phi\right>=\left<{p_{*}},\operatorname{div}\Phi\right>$

for all $\Phi\in\mathscr{S}$. This completes the proof of Lemma 3.7. ∎

We are now able to prove uniqueness of very weak solutions of (3.6).

######  Lemma 3.8.

Let $\alpha\geq 0$. If $\bm{u}\in L_{1}+L_{\infty}$ is a very weak solution of
(3.1) with $\mathbf{F}=\mathbf{0}$, then $\bm{u}$ is constant. In addition, if
$\bm{u}\in L_{q,\infty}$ for some $1<q<\infty$, then $\bm{u}$ is identically
zero.

###### Proof.

By Lemma 3.7, there exists ${p_{*}}\in\mathscr{S}^{\prime}$ such that

$\int_{\mathbb{R}^{d}}\bm{u}\cdot\Lambda^{2\alpha}\Phi\,d{x}-\left<{p_{*}},\operatorname{div}\Phi\right>=0\quad\mbox{and}\quad\int_{\mathbb{R}^{d}}\bm{u}\cdot\nabla\psi\,d{x}=0$

for all $(\Phi,\psi)\in\mathscr{S}$.

Let $\psi\in\mathscr{S}_{0}$ be given. Then since
$\nabla\psi\in\mathscr{S}_{0}$, $\Lambda^{2\alpha}\psi\in\mathscr{S}_{0}$, and
$\Lambda^{2\alpha}\nabla\psi=\nabla\Lambda^{2\alpha}\psi$, we have

$\left<{p_{*}},\Delta\psi\right>=\int_{\mathbb{R}^{d}}\bm{u}\cdot\Lambda^{2\alpha}\nabla\psi\,d{x}=\int_{\mathbb{R}^{d}}\bm{u}\cdot\nabla(\Lambda^{2\alpha}\psi)\,d{x}=0.$

Since $\Delta=-\Lambda^{2}:\mathscr{S}_{0}\rightarrow\mathscr{S}_{0}$ is an
isomorphism, it follows from Lemma 2.1 that ${p_{*}}$ is a polynomial. Hence,
for all $\Phi\in\mathscr{S}_{0}$, we have

$\int_{\mathbb{R}^{d}}\bm{u}\cdot\Lambda^{2\alpha}\Phi\,d{x}=\left<{p_{*}},\operatorname{div}\Phi\right>=0.$

Therefore, by Lemma 2.1 again, $\bm{u}$ is a polynomial. It is easy to check
that polynomials in $L_{1}+L_{\infty}$ are constant and the only constant
function belonging to $L_{q,\infty}$ is the zero function. This completes the
proof of Lemma 3.8. ∎

The following theorem shows existence and uniqueness of very weak solutions in
$L_{q}$ of (3.1) when $\mathbf{F}\in L_{p}$, where $1<p<d/(2\alpha-1)$ and
$1/q=1/p-(2\alpha-1)/d$.

######  Theorem 3.9.

Let $1/2<\alpha<(d+1)/2$, $1<p<d/(2\alpha-1)$, and $1/q=1/p-(2\alpha-1)/d$.
Then for each $\mathbf{F}\in L_{p}$, there exists a unique very weak solution
$\bm{u}$ in $L_{q}$ of (3.1). Moreover, the solution $\bm{u}$ is given by
$\bm{u}=\mathbf{U}_{\alpha}*\mathbf{F}$ and satisfies

$\|{\bm{u}}\|_{{q}}\apprle\|{\mathbf{F}}\|_{{p}}.$

In addition, if $\mathbf{F}\in\dot{H}^{1-\alpha}$, then $\bm{u}$ belongs to
$\dot{H}^{\alpha}$ and satisfies the energy identity (3.4).

###### Proof.

Define $\bm{u}=\mathbf{U}_{\alpha}*\mathbf{F}$. Then since
$\mathbf{U}_{\alpha}\in L_{d/(d+1-2\alpha),\infty}$, it follows from Lemma 2.5
that $\bm{u}\in L_{q}$ and $\|{\bm{u}}\|_{{q}}\apprle\|{\mathbf{F}}\|_{{p}}$.

We now show that $\bm{u}$ is a very weak solution of (3.1). To show this,
choose a sequence $\\{\mathbf{F}_{l}\\}$ in $\mathscr{S}_{0}$ such that
$\mathbf{F}_{l}\rightarrow\mathbf{F}$ in $L_{p}$. By Lemma 3.1, there exists a
unique solution $(\bm{u}_{l},p_{*,l})\in\mathscr{S}_{0}\times\mathscr{S}_{0}$
of (3.1) with $\mathbf{F}=\mathbf{F}_{l}$. Moreover, it was shown in
Subsection 3.2 that

$\bm{u}_{l}=\mathbf{U}_{\alpha}*\mathbf{F}_{l}.$

For all $(\Phi,\psi)\in\mathscr{S}_{0}$, we have

$\int_{\mathbb{R}^{d}}\bm{u}_{l}\cdot\Lambda^{2\alpha}\Phi\,d{x}=\int_{\mathbb{R}^{d}}(\Lambda^{2\alpha}\bm{u}_{l})\cdot\Phi\,d{x}=-\int_{\mathbb{R}^{d}}(\mathbf{F}_{l}:\nabla\Phi+\nabla
p_{*,l}\cdot\Phi)\,d{x}$

and

$\int_{\mathbb{R}^{d}}\bm{u}_{l}\cdot\nabla\psi\,d{x}=0.$

Since $\bm{u}_{l},{p_{*}}_{l},\mathbf{F}_{l}\in\mathscr{S}_{0}$, it follows
from Lemma 2.2 that

$\displaystyle\int_{\mathbb{R}^{d}}\bm{u}_{l}\cdot\left(\Lambda^{2\alpha}\Phi+\nabla\psi\right)dx$
$\displaystyle=-\int_{\mathbb{R}^{d}}(\mathbf{F}_{l}:\nabla\Phi+\nabla
p_{*,l}\cdot\Phi)\,d{x}$ for any $(\Phi,\psi)\in\mathscr{S}$. Hence for all
$\Phi\in C_{c,\sigma}^{\infty}$ and $\psi\in C_{c}^{\infty}$, we get
$\displaystyle\int_{\mathbb{R}^{d}}\bm{u}_{l}\cdot\left(\Lambda^{2\alpha}\Phi+\nabla\psi\right)dx$
$\displaystyle=-\int_{\mathbb{R}^{d}}\mathbf{F}_{l}:\nabla\Phi\,d{x}.$

Since $\mathbf{F}_{l}\rightarrow\mathbf{F}$ in $L_{p}$, it follows from Lemma
2.5 that $\bm{u}_{l}\rightarrow\bm{u}$ in $L_{q}$. Therefore, for all $\Phi\in
C_{c,\sigma}^{\infty}$ and $\psi\in C_{c}^{\infty}$, we have

$\int_{\mathbb{R}^{d}}\bm{u}\cdot\left(\Lambda^{2\alpha}\Phi+\nabla\psi\right)dx=-\int_{\mathbb{R}^{d}}\mathbf{F}:\nabla\Phi\,d{x}.$

Hence $\bm{u}$ is a very weak solution of (3.1), which is unique due to Lemma
3.8.

To complete the proof, suppose in addition that
$\mathbf{F}\in\dot{H}^{1-\alpha}$. Then by Lemma 2.2, there is a sequence
$\\{\mathbf{G}_{l}\\}$ in $\mathscr{S}_{0}$ such that
$\mathbf{G}_{l}\rightarrow\mathbf{F}$ in both $L_{p}$ and
$\dot{H}^{1-\alpha}$. By Lemma 3.1, there exists a unique solution
$(\bm{v}_{l},\tilde{p}_{*,l})\in\mathscr{S}_{0}\times\mathscr{S}_{0}$ of (3.1)
with $\mathbf{F}=\mathbf{G}_{l}$. Moreover, by Theorem 3.3, $\\{\bm{v}_{l}\\}$
converges in $\dot{H}^{\alpha}$ to some $\bm{v}\in\dot{H}^{\alpha}$ satisfying

$\int_{\mathbb{R}^{d}}|\Lambda^{\alpha}\bm{v}|^{2}\,d{x}=\left<\operatorname{div}\mathbf{F},\bm{v}\right>.$

On the other hand, it was already shown that
$\bm{v}_{l}=\mathbf{U}_{\alpha}*\mathbf{G}_{l}$ and $\bm{v}_{l}\to\bm{u}$ in
$L_{q}$. Hence, for all $\Phi\in\mathscr{S}$, we have

$\displaystyle\int_{\mathbb{R}^{d}}\bm{u}\cdot\Lambda^{\alpha}\Phi\,dx$
$\displaystyle=\lim_{l\to\infty}\int_{\mathbb{R}^{d}}\bm{v}_{l}\cdot\Lambda^{\alpha}\Phi\,dx$
$\displaystyle=\lim_{l\to\infty}\int_{\mathbb{R}^{d}}\Lambda^{\alpha}\bm{v}_{l}\cdot\Phi\,dx=\int_{\mathbb{R}^{d}}\Lambda^{\alpha}\bm{v}\cdot\Phi\,dx,$

which implies that $\Lambda^{\alpha}\bm{u}=\Lambda^{\alpha}\bm{v}$ in
$\mathscr{S}^{\prime}$. Since
$\Lambda^{\alpha}:\mathscr{S}_{0}\rightarrow\mathscr{S}_{0}$ is an
isomorphism, we also have

$\int_{\mathbb{R}^{d}}\bm{u}\cdot\Phi\,dx=\int_{\mathbb{R}^{d}}\bm{v}\cdot\Phi\,dx\quad\mbox{for
all}\,\,\Phi\in\mathscr{S}_{0}.$

Therefore,

$\left<{\rm
div}\,\mathbf{F},\bm{u}\right>=\lim_{l\to\infty}\int_{\mathbb{R}^{d}}\bm{u}\cdot{\rm
div}\,\mathbf{G}_{l}\,dx=\lim_{l\to\infty}\int_{\mathbb{R}^{d}}\bm{v}\cdot{\rm
div}\,\mathbf{G}_{l}\,dx=\left<{\rm div}\,\mathbf{F},\bm{v}\right>.$

This shows that $\bm{u}$ satisfies the energy identity (3.4). The proof of
Theorem 3.9 is completed. ∎

The following theorem shows existence and uniqueness of very weak solutions in
$L_{d/(d+1-2\alpha),\infty}$ of (3.1) when $\mathbf{F}\in L_{1}$.

######  Theorem 3.10.

Let $1/2<\alpha<(d+1)/2$. Then for each $\mathbf{F}\in L_{1}$, there exists a
unique very weak solution $\bm{u}$ of (3.1) in $L_{d/(d+1-2\alpha),\infty}$.
Moreover, the solution $\bm{u}$ is given by
$\bm{u}=\mathbf{U}_{\alpha}*\mathbf{F}$ and satisfies

$\|{\bm{u}}\|_{{d/(d+1-2\alpha),\infty}}\apprle\|{{\mathbf{F}}}\|_{{1}}.$

###### Proof.

Given $\mathbf{F}\in L_{1}$, we define
$\bm{u}=\mathbf{U}_{\alpha}*\mathbf{F}$. Then it follows from Lemma 2.5 that
$\bm{u}\in L_{d/(d+1-2\alpha),\infty}$ and
$\|{\bm{u}}\|_{{d/(d+1-2\alpha),\infty}}\apprle\|{{\mathbf{F}}}\|_{{1}}$.
Hence by Lemma 3.8, it remains to show that $\bm{u}$ is a very weak solution
of (3.1).

Fixing any $r$ with $1<r<d/(2\alpha-1)$, we choose a sequence
$\\{\mathbf{G}_{l}\\}$ in $L_{1}\cap L_{r}$ such that
$\mathbf{G}_{l}\rightarrow\mathbf{F}$ in $L_{1}$. Then by Theorem 3.9,
$\bm{v}_{l}=\mathbf{U}_{\alpha}*\mathbf{G}_{l}$ satisfies

$\int_{\mathbb{R}^{d}}\bm{v}_{l}\cdot(\Lambda^{2\alpha}\Phi+\nabla\psi)\,d{x}=-\int_{\mathbb{R}^{d}}\mathbf{G}_{l}:\nabla\Phi\,d{x}$

for all $\Phi\in C_{c,\sigma}^{\infty}$ and $\psi\in C_{c}^{\infty}$. By Lemma
2.5 again, $\bm{v}_{l}\rightarrow\bm{u}$ in $L_{d/(d+1-2\alpha),\infty}$.
Hence for all $\Phi\in C_{c,\sigma}^{\infty}$ and $\psi\in C_{c}^{\infty}$, we
obtain

$\int_{\mathbb{R}^{d}}\bm{u}\cdot(\Lambda^{2\alpha}\Phi+\nabla\psi)\,d{x}=-\int_{\mathbb{R}^{d}}\mathbf{F}:\nabla\Phi\,d{x}.$

This completes the proof of Theorem 3.10. ∎

##  4\. Perturbed fractional heat equations

For $0<T<\infty$, we consider the following initial value problem for
perturbed fractional heat equations in $\mathbb{R}^{d},d\geq 2$:

(4.1) $\left\\{\begin{aligned}
\partial_{t}\bm{b}+\Lambda^{2\beta}\bm{b}+\operatorname{div}\hskip
1.00006pt(\bm{u}\otimes\bm{b})&=\operatorname{div}\mathbf{F}&&\quad\text{in
}\mathbb{R}^{d}\times(0,T),\\\ \bm{b}(\cdot,0)&=\bm{b}_{0}&&\quad\text{on
}\mathbb{R}^{d},\end{aligned}\right.$

where $\beta$ is a positive number and
$\bm{u}:\mathbb{R}^{d}\times(0,T)\rightarrow\mathbb{R}^{d}$ is a given vector
field with $\operatorname{div}\bm{u}=0$.

By a _weak solution_ of (4.1), we mean a vector field $\bm{b}\in
L_{\infty}(0,T;L_{2})\cap L_{2}(0,T;{H}^{\beta})$ satisfying

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{d}}\bm{b}\cdot(-\partial_{t}\Phi+\Lambda^{2\beta}\Phi)\,d{x}dt\,$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{R}^{d}}(\bm{u}\otimes\bm{b}):\nabla\Phi\,d{x}dt$
$\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{R}^{d}}\mathbf{F}:\nabla\Phi\,d{x}dt+\int_{\mathbb{R}^{d}}\bm{b}_{0}\cdot\Phi(0)\,dx$

for all $\Phi\in C_{c}^{\infty}(\mathbb{R}^{d}\times[0,T))$, provided that the
integrals are all well-defined. Note that if $\beta\geq 1$, then by
Proposition 2.6,

(4.2)
$\|{\nabla\bm{b}}\|_{{2}}\apprle\|{\bm{b}}\|_{{2}}^{1-1/\beta}\|{\Lambda^{\beta}\bm{b}}\|_{{2}}^{1/\beta}.$

Hence if $\beta\geq 1$, then every weak solution of (4.1) belongs to
$L_{2\beta}(0,T;\dot{H}^{1})$.

This section consists of two parts. We first show that if $\mathbf{F}\in
L_{2\beta/(2\beta-1)}(0,T;L_{2})$, $\beta\geq 1$, and $\bm{b}_{0}\in L_{2}$,
then there exists a unique weak solution of (4.1) with $\bm{u}=0$ satisfying
the energy identity. Next, extending this result to nonzero $\bm{u}$, we prove
that if $\beta\geq 1$ and $\bm{u}\in
L_{\infty}(0,T;L_{d/(d+1-2\alpha),\infty})\cap L_{2}(0,T;\dot{H}^{\alpha})$,
then for every $\mathbf{F}\in L_{2\beta/(2\beta-1)}(0,T;L_{2})$ and
$\bm{b}_{0}\in L_{2}$, there exists a unique weak solution $\bm{b}$ of (4.1).
Moreover, the solution $\bm{b}$ satisfies the energy identity, which will be
used to prove uniqueness of weak solutions of the original nonlinear problem
(1.1).

To prove existence and uniqueness of weak solutions of (4.1) when $\bm{u}=0$,
we introduce the Fourier truncation operators $\mathcal{S}_{R}$ ($R>0$)
defined by

$\widehat{\mathcal{S}_{R}f}(\xi)=\chi_{B_{R}}(\xi)\hat{f}(\xi)\quad(f\in
L_{2}),$

where $\chi_{B_{R}}$ is the characteristic function of $B_{R}$. The following
properties of $\mathcal{S}_{R}$ are quite elementary.

######  Lemma 4.1.

Let $0\leq s<t$.

  1. (1)

If $f\in L_{2}$, then $\mathcal{S}_{R}f\in H^{s}$ and
$\|{\mathcal{S}_{R}f}\|_{{H}^{s}}\apprle(1+R)^{s}\|{f}\|_{{2}}$.

  2. (2)

If $f\in{H}^{s}$, then $\mathcal{S}_{R}f\rightarrow f$ in ${H}^{s}$ as
$R\rightarrow\infty$

  3. (3)

If $f\in H^{t}$, then
$\|{\mathcal{S}_{R}f-f}\|_{{H}^{s}}\apprle(1+R)^{-(t-s)}\|{f}\|_{{H}^{t}}$.

For $R>0$, we denote by $V_{R}$ the space of all vector fields $\bm{b}\in
L_{2}$ such that $\operatorname{supp}\hat{\bm{b}}\subset\overline{B_{R}}$.
Note that $V_{R}$ is a closed subspace of $L_{2}$. Moreover, since
$\mathcal{S}_{R}\bm{b}=\bm{b}$ for any $\bm{b}\in V_{R}$, it follows from
Lemma 4.1 that $V_{R}$ is continuously embedded into ${H}^{s}$ for any $s\geq
0$.

######  Lemma 4.2.

Suppose that $\beta\geq 1$ and $\gamma=2\beta/(2\beta-1)$. Then for every
$\mathbf{F}\in L_{\gamma}(0,T;L_{2})$ and $\bm{b}_{0}\in L_{2}$, there exists
a unique weak solution $\bm{b}$ of (4.1) with $\bm{u}=0$. Moreover, $\bm{b}$
belongs to $C([0,T];L_{2})$ and satisfies the energy identity:

$\|{\bm{b}(t)}\|_{{2}}^{2}+2\int_{0}^{t}\|{\Lambda^{\beta}\bm{b}(s)}\|_{{2}}^{2}\,ds=-2\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathbf{F}:\nabla\bm{b}\,d{x}ds+\|{\bm{b}_{0}}\|_{2}^{2}$

for $0\leq t\leq T$.

###### Proof.

Choose a sequence $\\{\mathbf{F}_{k}\\}$ in $C([0,T];L_{2})$ such that
$\mathbf{F}_{k}\rightarrow\mathbf{F}$ in $L_{\gamma}(0,T;L_{2})$ and
$\|{\mathbf{F}_{k}(t)}\|_{{2}}\leq\|{\mathbf{F}(t)}\|_{{2}}$ for all $t$ and
$k$. For each $k\in\mathbb{N}$, we consider the following truncated problem
for $\bm{b}_{k}\in C^{1}([0,T);V_{k})$:

(4.3) $\left\\{\begin{aligned}
\partial_{t}\bm{b}_{k}+\Lambda^{2\beta}\bm{b}_{k}&=\operatorname{div}\mathcal{S}_{k}\mathbf{F}_{k}&&\quad\text{in
}\mathbb{R}^{d}\times(0,T)\\\
\bm{b}_{k}(\cdot,0)&=\mathcal{S}_{k}\bm{b}_{0}&&\quad\text{on
}\mathbb{R}^{d}.\end{aligned}\right.$

Since $\Lambda^{2\beta}:V_{k}\rightarrow V_{k}$ is a bounded linear operator,
it follows from a standard theory of linear differential equations (see e.g.
Cartan [10, Theorem 1.9.1]) in Banach spaces that the problem (4.3) has a
unique solution $\bm{b}_{k}\in C^{1}([0,T];V_{k})$.

Given $k,m\in\mathbb{N}$, write $\tilde{\bm{b}}=\bm{b}_{k}-\bm{b}_{m}$,
$\tilde{\mathbf{F}}=\mathcal{S}_{k}\mathbf{F}_{k}-\mathcal{S}_{m}\mathbf{F}_{m}$,
and $\tilde{\bm{b}}_{0}=\mathcal{S}_{k}\bm{b}_{0}-\mathcal{S}_{m}\bm{b}_{0}$.
Then $\tilde{\bm{b}}$ satisfies

(4.4) $\left\\{\begin{aligned}
\partial_{t}\tilde{\bm{b}}+\Lambda^{2\beta}\tilde{\bm{b}}&=\operatorname{div}\tilde{\mathbf{F}}&&\quad\text{in
}\mathbb{R}^{d}\times(0,T)\\\
\tilde{\bm{b}}(\cdot,0)&=\tilde{\bm{b}}_{0}&&\quad\text{on
}\mathbb{R}^{d}.\end{aligned}\right.$

Multiplying (4.4) by $\tilde{\bm{b}}$ and then integrating over
$\mathbb{R}^{d}\times[0,t]$, we have

$\frac{1}{2}\|{\tilde{\bm{b}}(t)}\|_{{2}}^{2}+\int_{0}^{t}\|{\Lambda^{\beta}\tilde{\bm{b}}(s)}\|_{{2}}^{2}\,\,d{s}=-\int_{0}^{t}\int_{\mathbb{R}^{d}}\tilde{\mathbf{F}}:\nabla\tilde{\bm{b}}\,\,d{x}ds+\frac{1}{2}\|{\tilde{\bm{b}}_{0}}\|_{{2}}^{2}$

for all $0\leq t\leq T$. By Lemma 4.1, (4.2), and Young’s inequality, we have

$\displaystyle-\int_{\mathbb{R}^{d}}\tilde{\mathbf{F}}:\nabla\tilde{\bm{b}}\,d{x}$
$\displaystyle\leq
C\|{\tilde{\mathbf{F}}}\|_{{2}}\|{\tilde{\bm{b}}}\|_{{2}}^{1-1/\beta}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{1/\beta}$
$\displaystyle\leq
C\|{\tilde{\mathbf{F}}}\|_{{2}}^{\gamma}\|{\tilde{\bm{b}}}\|_{{2}}^{2(\beta-1)/(2\beta-1)}+\frac{1}{2}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{2}.$

Since $0\leq{(\beta-1)}/{(2\beta-1)}<1$, it follows that

$\displaystyle\|{\tilde{\bm{b}}(t)}\|_{{2}}^{2}+\int_{0}^{t}\|{\Lambda^{\beta}\tilde{\bm{b}}(s)}\|_{{2}}^{2}\,d{s}$
$\displaystyle\leq\|{\tilde{\bm{b}}_{0}}\|_{{2}}^{2}+C\int_{0}^{t}\|{\tilde{\mathbf{F}}(s)}\|_{{2}}^{\gamma}\|{\tilde{\bm{b}}(s)}\|_{{2}}^{2(\beta-1)/(2\beta-1)}\,d{s}$
$\displaystyle\leq\|{\tilde{\bm{b}}_{0}}\|_{{2}}^{2}+C\int_{0}^{t}\|{\tilde{\mathbf{F}}(s)}\|_{{2}}^{\gamma}\left(1+\|{\tilde{\bm{b}}(s)}\|_{{2}}^{2}\right)ds$

for all $0\leq t\leq T$. Hence by Gronwall’s inequality, we get

$\displaystyle\sup_{0\leq t\leq T}\|{\bm{b}_{k}(t)-\bm{b}_{m}(t)}\|_{{2}}^{2}$
$\displaystyle+\int_{0}^{T}\|{\Lambda^{\beta}(\bm{b}_{k}-\bm{b}_{m})(t)}\|_{{2}}^{2}\,d{t}$
$\displaystyle\leq\left(\|\mathcal{S}_{k}\bm{b}_{0}-\mathcal{S}_{m}\bm{b}_{0}\|_{2}^{2}+C\int_{0}^{T}\|{\mathcal{S}_{k}\mathbf{F}_{k}(t)-\mathcal{S}_{m}\mathbf{F}_{m}(t)}\|_{{2}}^{\gamma}\,d{t}\right)$
$\displaystyle\quad\times\exp\left(C\int_{0}^{T}\|{\mathcal{S}_{k}\mathbf{F}_{k}(t)-\mathcal{S}_{m}\mathbf{F}_{m}(t)}\|_{{2}}^{\gamma}\,d{t}\right)$

for some constant $C$ depending only on $d$, $\beta$, and $T$. By the
dominated convergence theorem, we easily show that
$\mathcal{S}_{k}\mathbf{F}_{k}\rightarrow\mathbf{F}$ in
$L_{\gamma}(0,T;L_{2})$. Therefore, the sequence $\\{\bm{b}_{k}\\}$ converges
in $C([0,T];L_{2})\cap L_{2}(0,T;{H}^{\beta})$ to a function $\bm{b}$. It is
easy to check that $\bm{b}$ is a weak solution of (4.1) with $\bm{u}=0$. To
show that $\bm{b}$ satisfies the energy identity, we multiply (4.3) by
$\bm{b}_{k}$ to obtain

(4.5)
$\frac{1}{2}\|{{\bm{b}}_{k}(t)}\|_{{2}}^{2}+\int_{0}^{t}\|{\Lambda^{\beta}{\bm{b}}_{k}(s)}\|_{{2}}^{2}\,\,d{s}=-\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathcal{S}_{k}\mathbf{F}_{k}:\nabla{\bm{b}}_{k}\,d{x}ds+\frac{1}{2}\|{\mathcal{S}_{k}\bm{b}_{0}}\|_{{2}}^{2}.$

By the interpolation inequality (4.2), we get

$\displaystyle\int_{0}^{T}\|{\nabla(\bm{b}_{k}-\bm{b})}\|_{{2}}^{2\beta}\,d{t}\leq
C\int_{0}^{T}\|{\bm{b}_{k}-\bm{b}}\|_{{2}}^{2(\beta-1)}\|{\Lambda^{\beta}(\bm{b}_{k}-\bm{b})}\|_{{2}}^{2}\,d{t}\rightarrow
0.$

Hence the energy identity follows from (4.5) by letting $k\rightarrow\infty$.

Finally, to show the uniqueness, let $\bm{b}$ be a weak solution of (4.1) with
$(\bm{u},\mathbf{F})=(\mathbf{0},\mathbf{0})$. Then the truncations
$\bm{b}_{k}=\mathcal{S}_{k}\bm{b}$ satisfy

$\left\\{\begin{aligned}
\partial_{t}\bm{b}_{k}+\Lambda^{2\beta}\bm{b}_{k}&=\mathbf{0}&&\quad\text{in
}\mathbb{R}^{d}\times(0,T),\\\ \bm{b}_{k}(\cdot,0)&=\mathbf{0}&&\quad\text{on
}\mathbb{R}^{d}.\end{aligned}\right.$

It is easy to see that $\bm{b}_{k}$ is identically zero. Since
$\bm{b}_{k}\to\bm{b}$ in $L_{2}(0,T;L_{2})$, we conclude that $\bm{b}$ is
identically zero. This completes the proof of Lemma 4.2. ∎

######  Lemma 4.3.

Let $\alpha$ and $\beta$ satisfy

$\frac{1}{2}<\alpha<\frac{d+1}{2},\quad\beta\geq
1,\quad\text{and}\quad\alpha+\beta\geq\frac{d}{2}+1.$

  1. (1)

For all $\bm{u}\in L_{d/(d+1-2\alpha),\infty}\cap\dot{H}^{\alpha}$ and
$\bm{v},\bm{w}\in{H}^{\beta}$, we have

(4.6)
$\int_{\mathbb{R}^{d}}|\bm{u}|\,|\bm{v}|\,|\nabla\bm{w}|\,d{x}\apprle(\|{\bm{u}}\|_{{d/(d+1-2\alpha),\infty}}+\|{\bm{u}}\|_{\dot{H}^{\alpha}})\|{\bm{v}}\|_{{H}^{\beta}}\|{\bm{w}}\|_{{H}^{\beta}}.$

Moreover, if $\operatorname{div}\bm{u}=0$, then

(4.7)
$\int_{\mathbb{R}^{d}}\bm{u}\otimes\bm{v}:\nabla\bm{w}\,d{x}=-\int_{\mathbb{R}^{d}}(\bm{u}\cdot\nabla)\bm{v}\cdot\bm{w}\,d{x}.$

  2. (2)

If $\bm{u}\in L_{\infty}(0,T;L_{d/(d+1-2\alpha),\infty})\cap
L_{2}(0,T;\dot{H}^{\alpha})$ and $\bm{b}\in L_{\infty}(0,T;L_{2})\cap
L_{2}(0,T;\dot{H}^{\beta})$, then

$\bm{b}\in
L_{2\beta}(0,T;\dot{H}^{1})\quad\mbox{and}\quad\bm{u}\otimes\bm{b}\in
L_{2\beta/(2\beta-1)}(0,T;L_{2}).$

Moreover, if $\operatorname{div}\bm{u}=0$, then

(4.8)
$\int_{\mathbb{R}^{d}}\bm{u}(t)\otimes\bm{b}(t):\nabla\bm{b}(t)\,d{x}=0\quad\text{for
almost all }t\in(0,T)$

###### Proof.

(1) Let $\gamma=2\beta/(2\beta-1)$. Then by Lemma 2.8, there exist
$0<\theta_{1},\theta_{2}<1$ with $\theta_{1}+\theta_{2}\leq
2-1/\beta=2/\gamma$ such that

(4.9)
$\|{|\bm{u}|\,|\bm{v}|}\|_{{2}}\apprle\|{\bm{u}}\|_{{d/(d+1-2\alpha),\infty}}^{1-\theta_{1}}\|{\bm{u}}\|_{\dot{H}^{\alpha}}^{\theta_{1}}\|{\bm{v}}\|_{{2}}^{1-\theta_{2}}\|{\bm{v}}\|_{\dot{H}^{\beta}}^{\theta_{2}}.$

By Young’s inequality, we have

$\displaystyle\int_{\mathbb{R}^{d}}|\bm{u}|\,|\bm{v}|\,|\nabla\bm{w}|\,d{x}$
$\displaystyle\leq\|{|\bm{u}|\,|\bm{v}|}\|_{{2}}\|{\nabla\bm{w}}\|_{{2}}$
$\displaystyle\apprle(\|{\bm{u}}\|_{{d/(d+1-2\alpha),\infty}}+\|{\bm{u}}\|_{\dot{H}^{\alpha}})\|{\bm{v}}\|_{{H}^{\beta}}\|{\bm{w}}\|_{{H}^{\beta}},$

which proves (4.6).

To show (4.7), choose sequences $\\{\bm{v}_{k}\\}$ and $\\{\bm{w}_{k}\\}$ in
$C_{c}^{\infty}$ satisfying $\bm{v}_{k}\rightarrow\bm{v}$ and
$\bm{w}_{k}\rightarrow\bm{w}$ in ${H}^{\beta}$. Since
$\operatorname{div}\bm{u}=0$, it follows that

$0=\int_{\mathbb{R}^{d}}\bm{u}\cdot\nabla(\bm{v}_{k}\cdot\bm{w}_{k})\,d{x}=\int_{\mathbb{R}^{d}}(\bm{u}\cdot\nabla)\bm{v}_{k}\cdot\bm{w}_{k}\,d{x}+\int_{\mathbb{R}^{d}}\bm{u}\otimes\bm{v}_{k}:\nabla\bm{w}_{k}\,d{x}$

for all $k$. Hence the identity (4.7) is derived by letting
$k\rightarrow\infty$ and using (4.6).

(2) It immediately follows from (4.2) that $\bm{b}\in
L_{2\beta}(0,T;\dot{H}^{1})$. Let $1<r\leq\infty$ be defined by

$\frac{1}{r}=1-\frac{\gamma}{2}\left(\theta_{1}+\theta_{2}\right).$

Then by (4.9) and Hölder’s inequality,

$\displaystyle\int_{0}^{T}\|{\bm{u}\otimes\bm{b}}\|_{{2}}^{\gamma}\,d{t}$
$\displaystyle\apprle\int_{0}^{T}\|{\bm{u}}\|_{{d/(d+1-2\alpha),\infty}}^{\gamma(1-\theta_{1})}\|{\bm{u}}\|_{\dot{H}^{\alpha}}^{\gamma\theta_{1}}\|{\bm{b}}\|_{{2}}^{\gamma(1-\theta_{2})}\|{\bm{b}}\|_{\dot{H}^{\beta}}^{\gamma\theta_{2}}\,d{t}$
$\displaystyle\apprle\int_{0}^{T}\|{\bm{u}}\|_{\dot{H}^{\alpha}}^{\gamma\theta_{1}}\|{\bm{b}}\|_{\dot{H}^{\beta}}^{\gamma\theta_{2}}\,d{t}$
$\displaystyle\apprle
T^{1/r}\left(\int_{0}^{T}\|{\bm{u}}\|_{\dot{H}^{\alpha}}^{2}\,d{t}\right)^{\gamma\theta_{1}/2}\left(\int_{0}^{T}\|{\bm{b}}\|_{\dot{H}^{\beta}}^{2}\,d{t}\right)^{\gamma\theta_{2}/2}.$

Finally, the identity (4.8) is obtained by taking $\bm{v}=\bm{w}=\bm{b}$ in
(4.7). This completes the proof of Lemma 4.3. ∎

######  Theorem 4.4.

Let $\alpha$ and $\beta$ satisfy

$\frac{1}{2}<\alpha<\frac{d+1}{2},\quad\beta\geq
1,\quad\text{and}\quad\alpha+\beta\geq\frac{d}{2}+1.$

Suppose that $\bm{u}\in L_{\infty}(0,T;L_{d/(d+1-2\alpha),\infty})\cap
L_{2}(0,T;\dot{H}^{\alpha})$ and $\operatorname{div}\bm{u}=0$ on
$\mathbb{R}^{d}\times(0,T)$. Then for every $\mathbf{F}\in
L_{2\beta/(2\beta-1)}(0,T;L_{2})$ and $\bm{b}_{0}\in L_{2}$, there exists a
unique weak solution $\bm{b}$ of (4.1). Moreover, the solution $\bm{b}$
belongs to $C([0,T];L_{2})$ and satisfies the energy identity:

(4.10)
$\|{\bm{b}(t)}\|_{{2}}^{2}+2\int_{0}^{t}\|{\Lambda^{\beta}\bm{b}(s)}\|_{{2}}^{2}\,ds=-2\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathbf{F}:\nabla\bm{b}\,d{x}ds+\|{\bm{b}_{0}}\|_{{2}}^{2}$

for $0\leq t\leq T$.

###### Proof.

We first show that every weak solution $\bm{b}$ of (4.1) belongs to
$C([0,T];L_{2})$ and satisfies the energy identity (4.10). Let $\bm{b}$ be a
weak solution of (4.1) with $\mathbf{F}\in L_{2\beta/(2\beta-1)}(0,T;L_{2})$.
Then $\bm{b}$ satisfies

$\left\\{\begin{aligned}
\partial_{t}\bm{b}+\Lambda^{2\beta}\bm{b}&=\operatorname{div}\tilde{\mathbf{F}}&&\quad\text{in
}\mathbb{R}^{d}\times(0,T),\\\ \bm{b}(\cdot,0)&=\bm{b}_{0}&&\quad\text{on
}\mathbb{R}^{d},\end{aligned}\right.$

where $\tilde{\mathbf{F}}=\mathbf{F}-\bm{u}\otimes\bm{b}$. By Lemma 4.3, we
have

$\tilde{\mathbf{F}}\in
L_{2\beta/(2\beta-1)}(0,T;L_{2})\quad\text{and}\quad\int_{\mathbb{R}^{d}}\tilde{\mathbf{F}}:\nabla\bm{b}\,d{x}=\int_{\mathbb{R}^{d}}\mathbf{F}:\nabla\bm{b}\,d{x}.$

Hence it follows from Lemma 4.2 that $\bm{b}$ belongs to $C([0,T];L_{2})$ and
satisfies the energy identity (4.10).

Uniqueness of weak solutions of (4.1) is an immediate consequence of the
energy identity. Indeed, if $\bm{b}$ is a weak solution of (4.1) with
$(\mathbf{F},\bm{b}_{0})=(\mathbf{0},\mathbf{0})$, then since it satisfies the
energy identity, we have

$\|{\bm{b}(t)}\|_{{2}}^{2}+2\int_{0}^{t}\|{\Lambda^{\beta}\bm{b}(s)}\|_{{2}}^{2}\,\,d{s}=0$

for all $0\leq t\leq T$, which implies that $\bm{b}$ is identically zero.

To complete the proof, it remains to prove existence of a weak solution of
(4.1). To prove this, we may consider truncated problems as in the proof of
Lemma 4.2. The resulting truncated problems are still linear but non-
autonomous with coefficients being singular in time. To avoid this difficulty,
we apply the Galerkin approximation method. Choose a sequence $\\{\Phi_{k}\\}$
in $\mathscr{S}$ such that the span of $\\{\Phi_{k}\\}$ is dense in
${H}^{\beta}$ and $\\{\Phi_{k}\\}$ is orthonormal in $L_{2}$. Then for a fixed
$m\in\mathbb{N}$, we seek a function $\bm{b}_{m}$ of the form

$\bm{b}_{m}(t)=\sum_{k=1}^{m}c_{k}^{m}(t)\Phi_{k}$

that satisfies

$\bm{b}_{m}(0)=\sum_{k=1}^{m}\left(\bm{b}_{0},\Phi_{j}\right)$

and

(4.11)
$\left(\partial_{t}\bm{b}_{m}(t),\Phi_{j}\right)+\left(\Lambda^{\beta}\bm{b}_{m}(t),\Lambda^{\beta}\Phi_{j}\right)-\left(\bm{u}(t)\otimes\bm{b}_{m}(t),\nabla\Phi_{j}\right)=-\left(\mathbf{F}(t),\nabla\Phi_{j}\right)$

for $1\leq j\leq m$. Equivalently, since $\\{\Phi_{k}\\}$ is orthonormal in
$L_{2}$, we look for a vector function
$\bm{c}^{m}=(c_{1}^{m},\dots,c_{m}^{m}):[0,T]\to\mathbb{R}^{m}$ satisfying

(4.12)
$\frac{d}{dt}\bm{c}^{m}(t)+A(t)\bm{c}^{m}(t)=\bm{f}(t)\quad\mbox{and}\quad\bm{c}^{m}(0)=\bm{c}_{0},$

where $A(t)=[A_{jk}(t)]_{1\leq j,k\leq m}$,
$\bm{f}(t)=(f_{1}(t),\ldots,f_{m}(t))$, and
$\bm{c}_{0}=(c_{0,1},\ldots,c_{0,m})$ are defined by

$A_{jk}(t)=\left(\Lambda^{\beta}\Phi_{k},\Lambda^{\beta}\Phi_{j}\right)-\left(\bm{u}(t)\otimes\Phi_{k},\nabla\Phi_{j}\right),$
$f_{j}(t)=-\left(\mathbf{F}(t),\nabla\Phi_{j}\right),\quad\mbox{and}\quad
c_{0,j}=\left(\bm{b}_{0},\Phi_{j}\right)$

for $1\leq j,k\leq m$. By Lemma 4.3, we have

$|A(t)|\apprle\left(1+\|{\bm{u}(t)}\|_{{d/(d+1-2\alpha),\infty}}+\|{\bm{u}(t)}\|_{\dot{H}^{\alpha}}\right)\|{\Phi_{k}}\|_{H^{\beta}}\|{\Phi_{j}}\|_{H^{\beta}},$

which implies that $A\in L_{2}(0,T)$. Moreover, since $\bm{f}\in
L_{2\beta/(2\beta-1)}(0,T)$, it follows from a standard theory of ordinary
differential equations (ODEs) that there exists a unique absolutely continuous
function $\bm{c}^{m}:[0,T]\to\mathbb{R}^{m}$ satisfying (4.12).

Multiplying (4.11) by $c_{j}^{m}$ and taking summation over $1\leq j\leq m$,
we get

$\frac{1}{2}\frac{d}{dt}\|{\bm{b}_{m}}\|_{{2}}^{2}+\|{\Lambda^{\beta}\bm{b}_{m}}\|_{{2}}^{2}-\left(\bm{u}\otimes\bm{b}_{m},\nabla\bm{b}_{m}\right)=-\left(\mathbf{F},\nabla\bm{b}_{m}\right).$

By Lemma 4.3, (4.2), and Young’s inequality,

$\left(\bm{u}\otimes\bm{b}_{m},\nabla\bm{b}_{m}\right)=0$

and

$\displaystyle-\left(\mathbf{F},\nabla\bm{b}_{m}\right)$ $\displaystyle\leq
C\|{\mathbf{F}}\|_{{2}}\|{\bm{b}_{m}}\|_{{2}}^{1-1/\beta}\|{\Lambda^{\beta}\bm{b}_{m}}\|_{{2}}^{1/\beta}$
$\displaystyle\leq
C\|{\mathbf{F}}\|_{{2}}^{\gamma}\|{\bm{b}_{m}}\|_{{2}}^{2(\beta-1)/(2\beta-1)}+\frac{1}{2}\|{\Lambda^{\beta}\bm{b}_{m}}\|_{{2}}^{2}.$

Since $0\leq(\beta-1)/(2\beta-1)<1$, we have

$\displaystyle\frac{d}{dt}\|{\bm{b}_{m}}\|_{{2}}^{2}+\|{\Lambda^{\beta}\bm{b}_{m}}\|_{{2}}^{2}$
$\displaystyle\leq
2C\|{\mathbf{F}}\|_{{2}}^{\gamma}\left(1+\|{\bm{b}_{m}}\|_{{2}}^{2}\right).$

Hence, by Gronwall’s inequality,

$\displaystyle\sup_{0<t<T}\|{\bm{b}_{m}(t)}\|_{{2}}^{2}+\int_{0}^{T}\|{\Lambda^{\beta}\bm{b}_{m}(t)}\|_{{2}}^{2}\,\,d{t}\leq\|{\bm{b}_{0}}\|_{{2}}^{2}\,\exp\left(C\int_{0}^{T}\|{\mathbf{F}(t)}\|_{{2}}^{\gamma}\,\,d{t}\right)$

for some constant $C$ depending only on $d$, $\beta$, and $T$. This shows that
the sequence $\\{\bm{b}_{m}\\}$ is uniformly bounded in
$L_{\infty}(0,T;L_{2})\cap L_{2}(0,T;{H}^{\beta})$. Therefore, by weak-*
compactness, there exist a subsequence $\\{\bm{b}_{m_{k}}\\}$ and a function
$\bm{b}$ such that

$\bm{b}_{m_{k}}\rightarrow\bm{b}\quad\text{weakly-* in
}L_{\infty}(0,T;L_{2})\cap L_{2}(0,T;{H}^{\beta}).$

It is quite standard to show that $\bm{b}$ is a weak solution of (4.1). The
proof of Theorem 4.4 is complete. ∎

##  5\. Proof of Theorem 1.2

This section is fully devoted to proving Theorem 1.2. Throughout the remaining
sections, we assume that $\nu=\eta=1$ for simplicity.

###  5.1. Approximation of the problem by truncations

Let $\mathbb{P}$ be the Leray projection defined by

$\widehat{\mathbb{P}(\bm{u})}(\xi)=\left(I-\frac{\xi\otimes\xi}{|\xi|^{2}}\right)\widehat{\bm{u}}(\xi).$

It is well-known that the Leray projection $\mathbb{P}$ has the following
properties:

  1. (1)

$\operatorname{div}\mathbb{P}(\bm{u})=0$ for all $\bm{u}\in\mathscr{S}$;

  2. (2)

$\mathbb{P}(\nabla\phi)=0$ for all $\phi\in\mathscr{S}$;

  3. (3)

$\mathbb{P}(\bm{u})=\bm{u}$ for all $\bm{u}\in\mathscr{S}$ with
$\operatorname{div}\bm{u}=0$;

  4. (4)

$\mathbb{P}[\mathbb{P}(\bm{u})]=\mathbb{P}(\bm{u})$ for all
$\bm{u}\in\mathscr{S}$;

  5. (5)

$\mathbb{P}$ is bounded from $L_{q}$ to itself for $1<q<\infty$.

Note also that $\mathbb{P}\mathcal{S}_{R}(\bm{u})\in V_{R,\sigma}$ for all
$\bm{u}\in L_{2}$, where $V_{R,\sigma}$ is the space of all vector fields
$\bm{v}\in V_{R}$ such that $\operatorname{div}\bm{v}=0$.

The following lemma is necessary to construct approximate solutions of (1.1).

######  Lemma 5.1.

Let $1/2<\alpha<(d+1)/2$. Given $\bm{b}\in L_{2}$, we define

$\bm{u}=\mathbf{U}_{\alpha}*(\bm{b}\otimes\bm{b}),$

where $\mathbf{U}_{\alpha}=[U_{\alpha,j}^{kl}]_{1\leq j,k,l\leq d}$ is defined
by (3.5). Then

(5.1) $\bm{u}\in
L_{d/(d+1-2\alpha),\infty}\quad\mbox{and}\quad\|{\bm{u}}\|_{{d/(d+1-2\alpha),\infty}}\apprle\|{\bm{b}}\|_{{2}}^{2}.$

Moreover, if $\bm{b}\in V_{R}$, then for any $\gamma\in\mathbb{N}_{0}^{d}$ and
$d/(d+1-2\alpha)<q<\infty$,

(5.2) $D^{\gamma}\bm{u}\in
L_{q},\quad\|{D^{\gamma}\bm{u}}\|_{q}\apprle_{R}\|{\bm{b}}\|_{{2}}^{2},$

and

(5.3)
$\|{D^{\gamma}(\bm{u}\otimes\bm{b})}\|_{{2}}+\|{D^{\gamma}(\bm{b}\otimes\bm{u})}\|_{{2}}\apprle_{R}\|{\bm{b}}\|_{{2}}^{3}.$

###### Proof.

Set $\mathbf{F}=\bm{b}\otimes\bm{b}$. Then since $\mathbf{F}\in L_{1}$ and
$\mathbf{U}_{\alpha}\in L_{d/(d+1-2\alpha),\infty}$, the estimate (5.1)
immediately follows from Lemma 2.5. Suppose in addition that $\bm{b}\in
V_{R}$. Since $V_{R}\hookrightarrow{H}^{s}$ and ${H}^{d+s}\hookrightarrow
L_{\infty}$ for any $s>0$, it follows from Hölder’s inequality that

$\|D^{\gamma}\mathbf{F}\|_{L_{1}\cap
L_{\infty}}\apprle_{R}\|{\bm{b}}\|_{{2}}^{2}\quad\mbox{for
all}\,\,\gamma\in{\mathbb{N}}_{0}^{d}.$

Let $\gamma\in{\mathbb{N}}_{0}^{d}$ be fixed. Then since
$D^{\gamma}\bm{u}=\mathbf{U}_{\alpha}*D^{\gamma}\mathbf{F}$, it follows from
Lemma 2.5 that

$\|D^{\gamma}\bm{u}\|_{q}\apprle\|D^{\gamma}\mathbf{F}\|_{L_{1}\cap
L_{\infty}}\apprle_{R}\|{\bm{b}}\|_{{2}}^{2}$

for any $d/(d+1-2\alpha)<q<\infty$. Finally, since there exist $p$ and $q$
such that

$2<p<\infty,\quad\frac{d}{d+1-2\alpha}<q<\infty,\quad\mbox{and}\quad\frac{1}{2}=\frac{1}{p}+\frac{1}{q},$

the estimate (5.3) is easily deduced from (5.2) by using Hölder’s inequality.
This completes the proof of Lemma 5.1. ∎

By virtue of Lemma 5.1, we may consider the following truncated problem for
$\bm{b}_{R}\in C^{1}([0,\infty);V_{R,\sigma})$:

(5.4) $\left\\{\begin{aligned}
\partial_{t}\bm{b}_{R}+\Lambda^{2\beta}\bm{b}_{R}&=\mathbb{P}\mathcal{S}_{R}[\operatorname{div}\hskip
1.00006pt(\bm{b}_{R}\otimes\bm{u}_{R}-\bm{u}_{R}\otimes\bm{b}_{R})]&&\quad\text{in
}\mathbb{R}^{d}\times(0,\infty),\\\
\bm{u}_{R}&=\mathbf{U}_{\alpha}*(\bm{b}_{R}\otimes\bm{b}_{R})&&\quad\text{in
}\mathbb{R}^{d}\times(0,\infty),\\\
\bm{b}_{R}(\cdot,0)&=\mathbb{P}\mathcal{S}_{R}\bm{b}_{0}&&\quad\text{on
}\mathbb{R}^{d},\end{aligned}\right.$

where $\bm{b}_{0}\in L_{2}$ with ${\rm div}\,\bm{b}_{0}=0$ is given. Note here
that if $\bm{b}_{R}\in C^{1}([0,T);V_{R,\sigma})$, $0<T\leq\infty$, and
$\bm{u}_{R}=\mathbf{U}_{\alpha}*(\bm{b}_{R}\otimes\bm{b}_{R})$, then by Lemma
5.1,

$\bm{u}_{R}\in C([0,T);L_{d/(d+1-2\alpha),\infty})$

and

(5.5)
$\|{\bm{u}_{R}(t)}\|_{{d/(d+1-2\alpha),\infty}}\apprle\|{\bm{b}_{R}(t)}\|_{{2}}^{2}\quad\mbox{for}\,\,0\leq
t<T.$

Moreover, it follows from Theorem 3.10 that for each $t\in(0,T)$,
$\bm{u}_{R}(t)$ is a very weak solution of the fractional Stokes system

(5.6)
$\Lambda^{2\alpha}\bm{u}_{R}(t)+\nabla{p_{*}}(t)=\operatorname{div}\hskip
1.00006pt(\bm{b}_{R}(t)\otimes\bm{b}_{R}(t)),\quad\operatorname{div}\bm{u}_{R}(t)=0\quad\mbox{in}\,\,\mathbb{R}^{d};$

that is, $\bm{u}_{R}(t)$ satisfies

(5.7)
$\int_{\mathbb{R}^{d}}\bm{u}_{R}(t)\cdot\left(\Lambda^{2\alpha}\Phi+\nabla\psi\right)dx=-\int_{\mathbb{R}^{d}}\bm{b}_{R}(t)\otimes\bm{b}_{R}(t):\nabla\Phi\,d{x}$

for all $(\Phi,\psi)\in\mathscr{S}$ with $\operatorname{div}\Phi=0$. We shall
show that for each $R>0$, there exists a unique solution $\bm{b}_{R}\in
C^{1}([0,\infty);V_{R,\sigma})$ of the $R$-truncated problem (5.4) and then
that some sequence $\\{(\bm{u}_{R_{k}},\bm{b}_{R_{k}})\\}$ converges to a weak
solution of the original problem (1.1).

###  5.2. Existence of approximate solutions

Let $0<R<\infty$ be fixed. For $\bm{b}\in V_{R,\sigma}$, we define

$G_{R}(\bm{b})=-\Lambda^{2\beta}\bm{b}+\mathbb{P}\mathcal{S}_{R}[\operatorname{div}\hskip
1.00006pt(\bm{b}\otimes\bm{u}-\bm{u}\otimes\bm{b})],$

where $\bm{u}=\mathbf{U}_{\alpha}*(\bm{b}\otimes\bm{b})$. By Lemma 5.1,
$G_{R}$ is a well-defined mapping from $V_{R,\sigma}$ into $V_{R,\sigma}$.
Therefore, the $R$-truncated problem (5.4) can be rewritten as the following
Cauchy problem on an ODE in the Hilbert space $V_{R,\sigma}$:

(5.8) $\left\\{\begin{aligned}
\partial_{t}\bm{b}_{R}(t)&=G_{R}(\bm{b}_{R}(t))\quad(0<t<\infty),\\\
\bm{b}_{R}(0)&=\mathbb{P}\mathcal{S}_{R}\bm{b}_{0},\end{aligned}\right.$

where $\bm{b}_{0}\in L_{2}$ with ${\rm div}\,\bm{b}_{0}=0$ is given.

Suppose that $\bm{b}^{i}\in V_{R,\sigma}$ and
$\bm{u}^{i}=\mathbf{U}_{\alpha}*(\bm{b}^{i}\otimes\bm{b}^{i})$ for $i=1,2$.
Then since $V_{R,\sigma}\hookrightarrow{H}^{s}\hookrightarrow\dot{H}^{s}$ for
any $s\geq 0$, we have

$\|\Lambda^{2\beta}\bm{b}^{1}-\Lambda^{2\beta}\bm{b}^{2}\|_{2}\apprle_{R}\|\bm{b}^{1}-\bm{b}^{2}\|_{2}.$

Following the proof of Lemma 5.1, we also obtain

$\displaystyle\|{\operatorname{div}\hskip
1.00006pt(\bm{b}^{1}\otimes\bm{u}^{1}-\bm{b}^{2}\otimes\bm{u}^{2})}\|_{{2}}$
$\displaystyle\quad\leq\|{\operatorname{div}\hskip
1.00006pt((\bm{b}^{1}-\bm{b}^{2})\otimes\bm{u}^{1})}\|_{{2}}+\|{\operatorname{div}\hskip
1.00006pt(\bm{b}^{2}\otimes(\bm{u}^{1}-\bm{u}^{2}))}\|_{{2}}$
$\displaystyle\quad\apprle_{R}\|\bm{b}^{1}-\bm{b}^{2}\|_{2}\|\bm{b}^{1}\|_{2}^{2}+\|\bm{b}^{2}\|_{2}\left(\|\bm{b}^{1}\|_{2}+\|\bm{b}^{2}\|_{2}\right)\|\bm{b}^{1}-\bm{b}^{2}\|_{2}$

and

$\displaystyle\|{\operatorname{div}\hskip
1.00006pt(\bm{u}^{1}\otimes\bm{b}^{1}-\bm{u}^{2}\otimes\bm{b}^{2})}\|_{{2}}$
$\displaystyle\quad\leq\|{\operatorname{div}\hskip
1.00006pt((\bm{u}^{1}-\bm{u}^{2})\otimes\bm{b}^{1})}\|_{{2}}+\|{\operatorname{div}\hskip
1.00006pt(\bm{u}^{2}\otimes(\bm{b}^{1}-\bm{b}^{2}))}\|_{{2}}$
$\displaystyle\quad\apprle_{R}\left(\|\bm{b}^{1}\|_{2}+\|\bm{b}^{2}\|_{2}\right)\|\bm{b}^{1}-\bm{b}^{2}\|_{2}\|\bm{b}^{1}\|_{2}+\|\bm{b}^{2}\|_{2}^{2}\|\bm{b}^{1}-\bm{b}^{2}\|_{2}.$

Hence by the $L_{2}$-boundedness of $\mathbb{P}\mathcal{S}_{R}$, we conclude
that

$\displaystyle\|G_{R}(\bm{b}^{1})-G_{R}(\bm{b}^{2})\|_{2}$
$\displaystyle\apprle_{R}\left(1+\|\bm{b}^{1}\|_{2}+\|\bm{b}^{2}\|_{2}\right)^{2}\|\bm{b}^{1}-\bm{b}^{2}\|_{2},$

which implies that $G_{R}$ is locally Lipschitz continuous on $V_{R,\sigma}$.
Therefore it follows from Picard’s theorem for ODEs in infinite dimensional
spaces that for every $\bm{b}_{0}\in L_{2}$ with ${\rm div}\,\bm{b}_{0}=0$,
the $R$-truncated problem (5.4) has a unique local classical solution
$\bm{b}_{R}\in C^{1}([0,T_{R});V_{R,\sigma})$. Here $0<T_{R}\leq\infty$
denotes the maximal existence time of the local solution $\bm{b}_{R}$. Indeed,
$\bm{b}_{R}$ exists globally, that is, $T_{R}=\infty$. To show this, let
$\bm{u}_{R}=\mathbf{U}_{\alpha}*(\bm{b}_{R}\otimes\bm{b}_{R})$. Since
$V_{R,\sigma}\hookrightarrow{H}^{s}\hookrightarrow\dot{H}^{s}$ and
$(\dot{H}^{s})^{*}=\dot{H}^{-s}$ for any $s\geq 0$, it follows that
$\bm{b}_{R}\otimes\bm{b}_{R}\in C([0,T_{R});\dot{H}^{1-\alpha}).$ Hence by
Theorem 3.9, $\bm{u}_{R}$ belongs to $C([0,T_{R});\dot{H}^{\alpha})$ and
satisfies the energy identity

$\int_{\mathbb{R}^{d}}|\Lambda^{\alpha}\bm{u}_{R}(t)|^{2}\,d{x}=-\int_{\mathbb{R}^{d}}\bm{b}_{R}(t)\otimes\bm{b}_{R}(t):\nabla\bm{u}_{R}(t)\,dx$

for $0<t<T_{R}$. Multiply the $\bm{b}_{R}$-equation in (5.4) by $\bm{b}_{R}$
and using the divergence-free condition, we get

$\frac{d}{dt}\left(\frac{1}{2}\|{\bm{b}_{R}(t)}\|_{{2}}^{2}\right)+\|{\Lambda^{\beta}\bm{b}_{R}(t)}\|_{{2}}^{2}=-\int_{\mathbb{R}^{d}}\bm{b}_{R}(t)\otimes\bm{u}_{R}(t):\nabla\bm{b}_{R}(t)\,dx$

for $0<t<T_{R}$. Adding these two identities and using the divergence-free
condition, we have

$\frac{d}{dt}\left(\frac{1}{2}\|{\bm{b}_{R}(t)}\|_{{2}}^{2}\right)+\|{\Lambda^{\alpha}\bm{u}_{R}(t)}\|_{{2}}^{2}+\|{\Lambda^{\beta}\bm{b}_{R}(t)}\|_{{2}}^{2}=0,$

from which we derive the global energy estimate

(5.9)
$\|{\bm{b}_{R}(t)}\|_{{2}}^{2}+2\int_{0}^{t}\|{\Lambda^{\alpha}\bm{u}_{R}(\tau)}\|_{{2}}^{2}\,d\tau+2\int_{0}^{t}\|{\Lambda^{\beta}\bm{b}_{R}(\tau)}\|_{{2}}^{2}\,d\tau\leq\|{\bm{b}_{0}}\|_{{2}}^{2}$

for $0\leq t<T_{R}$. Therefore, by a standard ODE theory, we conclude that
$T_{R}=\infty$.

The following lemma will be used to construct weak solutions of (1.1) by
applying the Aubin-Lions compactness lemma.

######  Lemma 5.2.

There exist $r>1$ and $s\geq 1$ such that $\\{\partial_{t}\bm{b}_{R}\\}$ is
bounded in $L_{r}(0,T;H^{-s})$ for any $0<T<\infty$.

###### Proof.

To show this, let $s=\max\\{\beta,1\\}$. Then since $-s+1\leq 0$ and
$2\beta-s\leq\beta$, it follows that

$\displaystyle\left<\partial_{t}\bm{b}_{R},\Phi\right>$
$\displaystyle=\left<-\Lambda^{2\beta}\bm{b}_{R},\Phi\right>-\int_{\mathbb{R}^{d}}(\bm{b}_{R}\otimes\bm{u}_{R}-\bm{u}_{R}\otimes\bm{b}_{R}):\nabla(\mathbb{P}\mathcal{S}_{R}\Phi)\,d{x}$
$\displaystyle\apprle\|{\Lambda^{2\beta}\bm{b}_{R}}\|_{{H}^{-s}}\|{\Phi}\|_{{H}^{s}}+\|{|\bm{u}_{R}||\bm{b}_{R}|}\|_{{H}^{-s+1}}\|{\nabla(\mathbb{P}\mathcal{S}_{R}\Phi)}\|_{{H}^{s-1}}$
$\displaystyle\apprle\left(\|{\bm{b}_{R}}\|_{{H}^{2\beta-s}}+\|{|\bm{u}_{R}||\bm{b}_{R}|}\|_{{H}^{-s+1}}\right)\|{\Phi}\|_{{H}^{s}}$
$\displaystyle\apprle\left(\|{\bm{b}_{R}}\|_{{H}^{\beta}}+\|{|\bm{u}_{R}||\bm{b}_{R}|}\|_{{2}}\right)\|{\Phi}\|_{{H}^{s}}$

for all $\Phi\in\mathscr{S}$. Therefore,

(5.10)
$\|{\partial_{t}\bm{b}_{R}(t)}\|_{{H}^{-s}}\apprle\|{\bm{b}_{R}}\|_{{H}^{\beta}}+\|{|\bm{u}_{R}||\bm{b}_{R}|}\|_{{2}}.$

By Lemma 2.8, there exist $0<\theta_{1},\theta_{2}<1$ such that

$\|{|\bm{u}_{R}||\bm{b}_{R}|}\|_{2}\apprle\|{\bm{u}_{R}}\|_{{d/(d+1-2\alpha),\infty}}^{1-\theta_{1}}\|{\bm{u}_{R}}\|_{\dot{H}^{\alpha}}^{\theta_{1}}\|{\bm{b}_{R}}\|_{{2}}^{1-\theta_{2}}\|{\bm{b}_{R}}\|_{\dot{H}^{\beta}}^{\theta_{2}}.$

Setting $r_{0}=2/(\theta_{1}+\theta_{2})>1$ and using Young’s inequality, we
have

$\|{|\bm{u}_{R}||\bm{b}_{R}|}\|_{2}\apprle\left(\|{\bm{u}_{R}}\|_{{d/(d+1-2\alpha),\infty}}^{1-\theta_{1}}\|{\bm{u}_{R}}\|_{\dot{H}^{\alpha}}^{\theta_{1}}\right)^{2/r_{0}\theta_{1}}+\left(\|{\bm{b}_{R}}\|_{{2}}^{1-\theta_{2}}\|{\bm{b}_{R}}\|_{\dot{H}^{\beta}}^{\theta_{2}}\right)^{2/r_{0}\theta_{2}}$

and so

$\|{|\bm{u}_{R}||\bm{b}_{R}|}\|_{2}^{r_{0}}\apprle\|{\bm{u}_{R}}\|_{{d/(d+1-2\alpha),\infty}}^{2(1-\theta_{1})/\theta_{1}}\|{\Lambda^{\alpha}\bm{u}_{R}}\|_{{2}}^{2}+\|{\bm{b}_{R}}\|_{{2}}^{2(1-\theta_{2})/\theta_{2}}\|{\Lambda^{\beta}\bm{b}_{R}}\|_{{2}}^{2}.$

By (5.5) and (5.9), we thus obtain

$\int_{0}^{\infty}\|{|\bm{u}_{R}||\bm{b}_{R}|}\|_{2}^{r_{0}}\,dt\apprle\|{\bm{b}_{0}}\|_{{2}}^{4/\theta_{1}-2}+\|{\bm{b}_{0}}\|_{{2}}^{2/\theta_{2}}.$

Now, let $r=\min\\{2,r_{0}\\}$. Then by (5.9) and (5.10), we have

$\displaystyle\|{\partial_{t}\bm{b}_{R}}\|_{L_{r}(0,T;{H}^{-s})}$
$\displaystyle\apprle
T^{1/r-1/2}\|{\bm{b}_{R}}\|_{L_{2}(0,T;{H}^{\beta})}+T^{1/r-1/r_{0}}\|{\bm{u}_{R}\otimes\bm{b}_{R}}\|_{L_{r_{0}}(0,T;L_{2})}$
$\displaystyle\leq C(d,\alpha,\beta,T,\|{\bm{b}_{0}}\|_{{2}}),$

which proves that $\\{\partial_{t}\bm{b}_{R}\\}$ is bounded in
$L_{r}(0,T;{H}^{-s})$ for any $0<T<\infty$. ∎

###  5.3. Completion of the proof of Theorem 1.2

We have shown that for each $0<R<\infty$, the $R$-truncated problem (5.4) has
a unique global solution $\bm{b}_{R}\in C^{1}([0,\infty);V_{R,\sigma})$. By
(5.5) and (5.9), the families $\\{\bm{b}_{R}\\}_{R>0}$ and
$\\{\bm{u}_{R}\\}_{R>0}$ are bounded in $L_{\infty}(0,\infty;L_{2})\cap
L_{2}(0,\infty;\dot{H}^{\beta})$ and
$L_{\infty}(0,\infty;L_{d/(d+1-2\alpha),\infty})\cap
L_{2}(0,\infty;\dot{H}^{\alpha})$, respectively.

By the weak-* compactness result and (5.9), there exists a sequence
$\\{R_{k}\\}$ with $R_{k}\rightarrow\infty$ such that

$\bm{u}_{R_{k}}\rightarrow\bm{u}\quad\mbox{weakly-*
in}\,\,L_{\infty}(0,\infty;L_{d/(d+1-2\alpha),\infty})\cap
L_{2}(0,\infty;\dot{H}^{\alpha})$

and

$\bm{b}_{R_{k}}\rightarrow\bm{b}\quad\mbox{weakly-*
in}\,\,L_{\infty}(0,\infty;L_{2})\cap L_{2}(0,\infty;\dot{H}^{\beta})$

for some $(\bm{u},\bm{b})$ satisfying

(5.11)
$\|\bm{b}\|_{L_{\infty}(0,\infty;L_{2})}^{2}+\|\bm{b}\|_{L_{2}(0,\infty;\dot{H}^{\beta})}^{2}+\|\bm{u}\|_{L_{2}(0,\infty;\dot{H}^{\alpha})}^{2}\leq
2\|{\bm{b}_{0}}\|_{{2}}^{2}.$

By Lemma 5.2 and the Aubin-Lions compactness lemma, there exists a subsequence
of $\\{\bm{b}_{R_{k}}\\}$, denoted still by $\\{\bm{b}_{R_{k}}\\}$, such that

(5.12) $\bm{b}_{R_{k}}\rightarrow\bm{b}\quad\mbox{in}\,\,L_{2}(K\times(0,T))$

for any compact $K\subset\mathbb{R}^{d}$ and finite $T>0$. Moreover, by (5.9),
we have

$\displaystyle\|{\bm{b}_{R_{k}}\otimes\bm{b}_{R_{k}}-\bm{b}\otimes\bm{b}}\|_{{1};K}$
$\displaystyle\leq\|{\bm{b}_{R_{k}}\otimes(\bm{b}_{R_{k}}-\bm{b})}\|_{{1};K}+\|{(\bm{b}_{R_{k}}-\bm{b})\otimes\bm{b}}\|_{{1};K}$
$\displaystyle\leq 2\|{\bm{b}_{0}}\|_{{2}}\|{\bm{b}_{R_{k}}-\bm{b}}\|_{{2};K}$

and so

(5.13)
$\bm{b}_{R_{k}}\otimes\bm{b}_{R_{k}}\rightarrow\bm{b}\otimes\bm{b}\quad\mbox{in}\,\,L_{2}(0,T;L_{1}(K))$

for any compact $K\subset\mathbb{R}^{d}$ and finite $T>0$.

Next, we show that

(5.14)
$\bm{u}_{R_{k}}\rightarrow\mathbf{U}_{\alpha}*(\bm{b}\otimes\bm{b})\quad\mbox{in}\,\,L_{2}(0,T;L_{d/(d+1-2\alpha),\infty}(K))$

for any finite $T$ and compact $K\subset\mathbb{R}^{d}$, which automatically
implies that

(5.15) $\bm{u}=\mathbf{U}_{\alpha}*(\bm{b}\otimes\bm{b});$

hence $\bm{u}$ satisfies (1.6).

Let $\bm{v}=\mathbf{U}_{\alpha}*(\bm{b}\otimes\bm{b})$. Fixing $0<r,T<\infty$,
we write

$\displaystyle\bm{u}_{R_{k}}(x,t)-\bm{v}(x,t)$
$\displaystyle=\int_{B_{M+r}}\mathbf{U}_{\alpha}(x-y):(\bm{b}_{R_{k}}\otimes\bm{b}_{R_{k}}-\bm{b}\otimes\bm{b})(y,t)\,d{y}$
$\displaystyle\mathrel{\phantom{=}}+\int_{\mathbb{R}^{d}\setminus
B_{M+r}}\mathbf{U}_{\alpha}(x-y):(\bm{b}_{R_{k}}\otimes\bm{b}_{R_{k}}-\bm{b}\otimes\bm{b})(y,t)\,d{y}$
$\displaystyle=\bm{I}_{1}^{k}(x,t)+\bm{I}_{2}^{k}(x,t)$

for $(x,t)\in B_{r}\times(0,T)$, where $M>1$ is a large number to be
determined later. Recall from (3.5) that

$|\mathbf{U}_{\alpha}(x)|\apprle\frac{1}{|x|^{d+1-2\alpha}}.$

Hence by Lemma 2.5 and (5.13), we have

$\displaystyle\|{\bm{I}_{1}^{k}}\|_{L_{2}(0,T;L_{d/(d+1-2\alpha),\infty})}$
$\displaystyle\apprle\|{\mathbf{U}_{\alpha}}\|_{{d/(d+1-2\alpha),\infty}}\|{\bm{b}_{R_{k}}\otimes\bm{b}_{R_{k}}-\bm{b}\otimes\bm{b}}\|_{L_{2}(0,T;L_{1}(B_{M+r}))}$
$\displaystyle\rightarrow 0\quad\mbox{as}\,\,k\rightarrow\infty.$

To estimate $|\bm{I}_{2}^{k}|$, we choose $0<\theta\leq 1/2$ and
$2<p<{2d}/{(2\alpha-1)}$ such that

$\frac{1}{p}=\frac{1}{2}-\frac{\theta\beta}{d}.$

Then by Proposition 2.6,

$\|\bm{b}_{R_{k}}\|_{p}\apprle\|\bm{b}_{R_{k}}\|_{2}^{1-\theta}\|{\Lambda^{\beta}\bm{b}_{R_{k}}}\|_{{2}}^{\theta}.$

Hence setting $q=2/\theta$, we have

$\|{\bm{b}_{R_{k}}}\|_{L_{q}(0,\infty;L_{p})}\apprle\|{\bm{b}_{R_{k}}}\|_{L_{\infty}(0,\infty;L_{2})}^{1-\theta}\|{\bm{b}_{R_{k}}}\|_{L_{2}(0,\infty;\dot{H}^{\beta})}^{\theta}\apprle\|\bm{b}_{0}\|_{2}$

and similarly

$\|{\bm{b}}\|_{L_{q}(0,\infty;L_{p})}\apprle\|\bm{b}_{0}\|_{2}.$

On the other hand, since $d+1-2\alpha-{d(p-2)}/{p}>0$, it follows that if
$|x|\leq r$, then

$\displaystyle|\bm{I}_{2}^{k}(x,t)|$
$\displaystyle\leq\left(\int_{\mathbb{R}^{d}\setminus
B_{M}}|\mathbf{U}_{\alpha}(z)|^{(p/2)^{\prime}}dz\right)^{1/(p/2)^{\prime}}\|{[\bm{b}_{R_{k}}\otimes\bm{b}_{R_{k}}-\bm{b}\otimes\bm{b}](t)}\|_{{p/2}}$
$\displaystyle\apprle
M^{-[d+1-2\alpha-d(p-2)/p]}\left(\|{\bm{b}_{R_{k}}(t)}\|_{{p}}^{2}+\|{\bm{b}(t)}\|_{{p}}^{2}\right).$

Therefore, noting that $q\geq 4$, we deduce that

$\displaystyle\|{\bm{u}_{R_{k}}-\bm{v}}\|_{L_{2}(0,T;L_{d/(d+1-2\alpha),\infty}(B_{r}))}$
$\displaystyle\leq\|{\bm{I}_{1}^{k}}\|_{L_{2}(0,T;L_{d/(d+1-2\alpha),\infty}(B_{r}))}$
$\displaystyle\quad+C(r,T)M^{-[d+1-2\alpha-d(p-2)/p]}\|\bm{b}_{0}\|_{2}^{2}$

for some constant $C(r,T)$ independent of $k$ and $M$.

Now, let $\delta>0$ be given. Then choosing $M>1$ so large that

$C(r,T)M^{-[d+1-2\alpha-d(p-2)/p]}\|\bm{b}_{0}\|_{2}^{2}<\delta,$

and then letting $k\to\infty$, we have

$\limsup_{k\rightarrow\infty}\|{\bm{u}_{R_{k}}-\bm{v}}\|_{L_{2}(0,T;L_{d/(d+1-2\alpha),\infty}(B_{r}))}\leq\delta,$

which proves (5.14).

Finally, we show that

(5.16)
$\displaystyle\bm{u}_{R_{k}}\otimes\bm{b}_{R_{k}}\rightarrow\bm{u}\otimes\bm{b}\quad\mbox{in}\,\,L_{1}(K\times(0,T)),$
$\displaystyle\bm{b}_{R_{k}}\otimes\bm{u}_{R_{k}}\rightarrow\bm{b}\otimes\bm{u}\quad\mbox{in}\,\,L_{1}(K\times(0,T))$

for any compact $K\subset\mathbb{R}^{d}$ and finite $T>0$. By symmetry, it
suffices to show the first assertion. We write

$\bm{u}_{R_{k}}\otimes\bm{b}_{R_{k}}-\bm{u}\otimes\bm{b}=(\bm{u}_{R_{k}}-\bm{u})\otimes\bm{b}_{R_{k}}+\bm{u}\otimes(\bm{b}_{R_{k}}-\bm{b}).$

First, since $1/2<\alpha<(d+1)/2$, $\beta>0$, and $2\alpha+\beta>d/2+1$, there
exists $2<p_{1}<\infty$ such that

$\max\left\\{\frac{1}{2}-\frac{\beta}{d},0\right\\}<\frac{1}{p_{1}}<\min\left\\{\frac{1}{2},\frac{2\alpha-1}{d}\right\\}.$

Then by Proposition 2.7,

$\|\bm{b}_{R_{k}}\|_{{p_{1},1}}\apprle\|\bm{b}_{R_{k}}\|_{2}^{1-\theta_{1}}\|{\Lambda^{\beta}\bm{b}_{R_{k}}}\|_{{2}}^{\theta_{1}},$

where $0<\theta_{1}<1$ is defined by $1/p_{1}=1/2-(\theta_{1}\beta)/d$. Let
$K\subset\mathbb{R}^{d}$ be compact. Then since $1/p_{1}>1/2-\beta/d$ and
$d/(2\alpha-1)<p_{1}$, it follows from Lemma 2.3 and the Sobolev embedding
theorem that

$\displaystyle\|(\bm{u}_{R_{k}}-\bm{u})\otimes\bm{b}_{R_{k}}\|_{1;K}$
$\displaystyle\apprle\|\bm{u}_{R_{k}}-\bm{u}\|_{d/(d+1-2\alpha),\infty;K}\|\bm{b}_{R_{k}}\|_{d/(2\alpha-1),1;K}$
$\displaystyle\apprle_{K}\|\bm{u}_{R_{k}}-\bm{u}\|_{d/(d+1-2\alpha),\infty;K}\|\bm{b}_{R_{k}}\|_{H^{\beta}}.$

Hence by (5.9), (5.14) and (5.15), we have

$\displaystyle\|(\bm{u}_{R_{k}}-\bm{u})\otimes\bm{b}_{R_{k}}\|_{L_{1}(0,T;L_{1}(K))}$
$\displaystyle\apprle_{K,T}\|{\bm{u}_{R_{k}}-\bm{u}}\|_{L_{2}(0,T;L_{d/(d+1-2\alpha),\infty}(K))}\|\bm{b}_{0}\|_{2}$
$\displaystyle\rightarrow 0\quad\mbox{as}\,\,k\to\infty$

for any finite $T$. To estimate $\bm{u}\otimes(\bm{b}_{R_{k}}-\bm{b})$, we
choose $2<p_{2}<\infty$ such that

$\max\left\\{\frac{1}{2}-\frac{\alpha}{d},0\right\\}<\frac{1}{p_{2}}<\min\left\\{\frac{1}{2},\frac{d+1-2\alpha}{d}\right\\}.$

By Proposition 2.7 again,

$\|\bm{u}\|_{2;K}\apprle_{K}\|\bm{u}\|_{p_{2}}\apprle\|\bm{u}\|_{d/(d+1-2\alpha),\infty}^{1-\theta_{2}}\|\Lambda^{\alpha}\bm{u}\|_{2}^{\theta_{2}}$

for some $0<\theta_{2}<1$. Hence by (5.11), (5.12), and (5.15), we have

$\displaystyle\|\bm{u}\otimes(\bm{b}_{R_{k}}-\bm{b})\|_{L_{1}(0,T;L_{1}(K))}$
$\displaystyle\leq\|\bm{u}\|_{L_{2}(0,T;L_{2}(K))}\|\bm{b}_{R_{k}}-\bm{b}\|_{L_{2}(0,T;L_{2}(K))}$
$\displaystyle\apprle_{K,T}\left(\|\bm{b}_{0}\|_{2}+\|\bm{b}_{0}\|_{2}^{2}\right)\|{\bm{b}_{R_{k}}-\bm{b}}\|_{L_{2}(0,T;L_{2}(K))}$
$\displaystyle\rightarrow 0\quad\mbox{as}\,\,k\to\infty,$

which proves (5.16).

We have shown strong convergence for the nonlinear terms
$\bm{b}_{R_{k}}\otimes\bm{b}_{R_{k}}$, $\bm{u}_{R_{k}}\otimes\bm{b}_{R_{k}}$,
and $\bm{b}_{R_{k}}\otimes\bm{u}_{R_{k}}$. Hence by a standard argument, we
can show that $(\bm{u},\bm{b})$ satisfies (1.7). It is obvious that both
$\bm{u}$ and $\bm{b}$ are divergence-free. This completes the proof of Theorem
1.2. $\square$

##  6\. Proof of Theorem 1.3

This section is devoted to proving Theorem 1.3. By Theorem 1.2 and its proof,
there exists a global weak solution $(\bm{u},\bm{b})$ of (1.1), which also
satisfies $\bm{u}=\mathbf{U}_{\alpha}*(\bm{b}\otimes\bm{b})$ in
$\mathbb{R}^{d}\times[0,\infty)$. Since $\bm{b}$ is divergence-free, $\bm{b}$
is a weak solution of (4.1) with $\mathbf{F}=\bm{b}\otimes\bm{u}$. It follows
from Lemma 4.3 that $\bm{b}\otimes\bm{u}\in L_{2\beta/(2\beta-1)}(0,T;L_{2})$
for any $T<\infty$. Hence by Theorem 4.4, $\bm{b}$ belongs to
$C([0,\infty);L_{2})$ and satisfies

(6.1)
$\frac{1}{2}\frac{d}{dt}\|{\bm{b}(t)}\|_{{2}}^{2}+\|{\Lambda^{\beta}\bm{b}(t)}\|_{{2}}^{2}=-\int_{\mathbb{R}^{d}}\bm{b}(t)\otimes\bm{u}(t):\nabla\bm{b}(t)\,d{x}$

for almost all $t\in(0,\infty)$. Using Young’s inequality (Lemma 2.5), we
easily deduce that $\bm{u}\in C([0,\infty);L_{{d}/{(d+1-2\alpha)},\infty})$.
Let $1<p<d/(2\alpha-1)$ be chosen so that $H^{\beta}\hookrightarrow L_{2p}$.
Then it follows from Hölder’s inequality and Lemma 2.10 that

$\bm{b}(t)\otimes\bm{b}(t)\in
L_{p}\cap\dot{H}^{1-\alpha}\quad\mbox{and}\quad\operatorname{div}\hskip
1.00006pt(\bm{b}(t)\otimes\bm{b}(t))=(\bm{b}(t)\cdot\nabla)\bm{b}(t)\in
L_{2p/(p+1)}$

for almost all $t\in(0,\infty)$. Hence by Theorem 3.9 and Lemma 4.3, $\bm{u}$
satisfies

(6.2)
$\int_{\mathbb{R}^{d}}|\Lambda^{\alpha}\bm{u}(t)|^{2}\,d{x}=\int_{\mathbb{R}^{d}}(\bm{b}(t)\cdot\nabla)\bm{b}(t)\cdot\bm{u}(t)\,dx$

for almost all $t\in(0,\infty)$. Adding (6.1) and (6.2) and then integraing
over $(0,t)$, we get the energy identity (1.3) for any $t\geq 0$.

To complete the proof of Theorem 1.3 it remains to prove the uniqueness
assertion. Let $(\bm{u}^{1},\bm{b}^{1})$ and $(\bm{u}^{2},\bm{b}^{2})$ be two
weak solutions of (1.1) for the same initial data $\bm{b}_{0}\in L_{2}$ with
${\rm div}\,\bm{b}_{0}=0$. Write $\tilde{\bm{u}}=\bm{u}^{1}-\bm{u}^{2}$ and
$\tilde{\bm{b}}=\bm{b}^{1}-\bm{b}^{2}$. Then $(\tilde{\bm{u}},\tilde{\bm{b}})$
satisfies

$\partial_{t}\tilde{\bm{b}}+\Lambda^{2\beta}\tilde{\bm{b}}+(\bm{u}^{1}\cdot\nabla)\tilde{\bm{b}}=\operatorname{div}\hskip
1.00006pt(\tilde{\bm{b}}\otimes\bm{u}^{1}+\bm{b}^{2}\otimes\tilde{\bm{u}}-\tilde{\bm{u}}\otimes\bm{b}^{2})$

and

$\tilde{\bm{u}}=\mathbf{U}_{\alpha}*(\tilde{\bm{b}}\otimes\bm{b}^{1}+\bm{b}^{2}\otimes\tilde{\bm{b}}).$

Moreover, by Lemma 4.3,

$\bm{u}^{i}\otimes\bm{b}^{j}\in
L_{2\beta/(2\beta-1)}(0,T;L_{2})\quad\mbox{and}\quad\bm{b}^{i}\in
L_{2\beta}(0,T;\dot{H}^{1})$

for any $T<\infty$, where $i,j=1,2$. Hence by Theorem 4.4, we have

$\displaystyle\frac{1}{2}\frac{d}{dt}\|{\tilde{\bm{b}}}\|_{{2}}^{2}+\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{2}$
$\displaystyle=-\int_{\mathbb{R}^{d}}\tilde{\bm{b}}\otimes\bm{u}^{1}:\nabla\tilde{\bm{b}}\,d{x}+\int_{\mathbb{R}^{d}}\left(-\bm{b}^{2}\otimes\tilde{\bm{u}}+\tilde{\bm{u}}\otimes\bm{b}^{2}\right):\nabla\tilde{\bm{b}}\,d{x}$
$\displaystyle=I+II.$

Hence by Gronwall’s inequality, it suffices to show that

(6.3) $I+II\leq
C\phi(t)\|{\tilde{\bm{b}}}\|_{{2}}^{2}+\frac{1}{2}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{2}$

for some constant $C$, where $\phi$ is a locally integrable function defined
by

$\phi(t)=1+\|{\Lambda^{\alpha}\bm{u}^{1}(t)}\|_{{2}}^{2}+\|{\Lambda^{\alpha}\bm{u}^{2}(t)}\|_{{2}}^{2}+\|{\Lambda^{\beta}\bm{b}^{1}(t)}\|_{{2}}^{2}+\|{\Lambda^{\beta}\bm{b}^{2}(t)}\|_{{2}}^{2}.$

To do this, we make crucial use of Lemma 2.8; that is, we choose
$0<\theta_{1},\theta_{2}<1$ with $\theta_{1}+\theta_{2}+1/\beta\leq 2$ such
that

(6.4)
$\|{\bm{u}^{i}\otimes\bm{b}^{j}}\|_{{2}}\apprle\|{\bm{u}^{i}}\|_{{d/(d+1-2\alpha),\infty}}^{1-\theta_{1}}\|{\Lambda^{\alpha}\bm{u}^{i}}\|_{{2}}^{\theta_{1}}\|{\bm{b}^{j}}\|_{{2}}^{1-\theta_{2}}\|{\Lambda^{\beta}\bm{b}^{j}}\|_{{2}}^{\theta_{2}}$

for $i,j=1,2$. Note also that if $\alpha<1$ and $\mu={(1-\alpha)}/{\beta}$,
then since $\alpha+(1-\mu)\beta\geq d/2+1$, the numbers $\theta_{1}$ and
$\theta_{2}$ can be chosen so that $(1+\mu)\theta_{1}+\theta_{2}+1/\beta\leq
2$.

Now, by (6.4) and Proposition 2.6, we obtain

$\displaystyle I$
$\displaystyle\leq\|{\tilde{\bm{b}}\otimes\bm{u}^{1}}\|_{{2}}\|{\nabla\tilde{\bm{b}}}\|_{{2}}$
$\displaystyle\apprle\|{\bm{u}^{1}}\|_{{d/(d+1-2\alpha),\infty}}^{1-\theta_{1}}\|{\Lambda^{\alpha}\bm{u}^{1}}\|_{{2}}^{\theta_{1}}\|{\tilde{\bm{b}}}\|_{{2}}^{1-\theta_{2}}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{\theta_{2}}\|{\tilde{\bm{b}}}\|_{{2}}^{1-1/\beta}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{1/\beta}.$

and

(6.5)
$II\apprle\|{\tilde{\bm{u}}}\|_{{d/(d+1-2\alpha),\infty}}^{1-\theta_{1}}\|{\Lambda^{\alpha}\tilde{\bm{u}}}\|_{{2}}^{\theta_{1}}\|{\bm{b}^{2}}\|_{{2}}^{1-\theta_{2}}\|{\Lambda^{\beta}\bm{b}^{2}}\|_{{2}}^{\theta_{2}}\|{\tilde{\bm{b}}}\|_{{2}}^{1-1/\beta}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{1/\beta}.$

It is easy to estimate $I$. Indeed, by Lemma 5.1, the energy identity (1.3),
and Young’s inequality, we have

$\displaystyle I$
$\displaystyle\apprle\|{\bm{b}_{0}}\|_{{2}}^{2(1-\theta_{1})}\left(\|{\Lambda^{\alpha}{\bm{u}}^{1}}\|_{{2}}^{\theta_{1}}\|{\tilde{\bm{b}}}\|_{{2}}^{2-\theta_{2}-1/\beta}\right)\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{\theta_{2}+1/\beta},$
$\displaystyle\leq
C\|{\Lambda^{\alpha}\bm{u}^{1}}\|_{{2}}^{\frac{2\theta_{1}}{2-\theta_{2}-1/\beta}}\|{\tilde{\bm{b}}}\|_{{2}}^{2}+\frac{1}{4}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{2}$
$\displaystyle\leq
C\phi(t)^{\frac{\theta_{1}}{2-\theta_{2}-1/\beta}}\|{\tilde{\bm{b}}}\|_{{2}}^{2}+\frac{1}{4}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{2}$

for some $C=C(\alpha,\beta,d,\|{\bm{b}_{0}}\|_{{2}})$. Since
$\theta_{1}/(2-\theta_{2}-1/\beta)\leq 1$ and $\phi(t)\geq 1$, we get

$I\leq
C\phi(t)\|{\tilde{\bm{b}}}\|_{{2}}^{2}+\frac{1}{4}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{2}.$

To estimate $II$, we recall that

$\tilde{\bm{u}}=\mathbf{U}_{\alpha}*(\tilde{\bm{b}}\otimes\bm{b}^{1}+\bm{b}^{2}\otimes\tilde{\bm{b}}).$

By Theorems 3.9 and 3.10, we have

(6.6)
$\|{\tilde{\bm{u}}}\|_{{d/(d+1-2\alpha),\infty}}\apprle\left(\|{\bm{b}^{1}}\|_{{2}}+\|{\bm{b}^{2}}\|_{{2}}\right)\|{\tilde{\bm{b}}}\|_{{2}}\apprle\|{\tilde{\bm{b}}}\|_{{2}}$

and

(6.7)
$\|{\Lambda^{\alpha}\tilde{\bm{u}}}\|_{{2}}\apprle\|{\tilde{\bm{b}}\otimes\bm{b}^{1}+\bm{b}^{2}\otimes\tilde{\bm{b}}}\|_{\dot{H}^{1-\alpha}}.$

Suppose first that $\alpha\geq 1$. Then

$\dot{H}^{1-\alpha}=\left(\dot{H}^{\alpha-1}\right)^{*}\quad\text{and}\quad\dot{H}^{\alpha-1}\hookrightarrow
L_{r},$

where $1<r<\infty$ is defined by ${1}/{r}={1}/{2}-{(\alpha-1)}/{d}$. Fix any
$\Phi\in\dot{H}^{\alpha-1}$. Then by Hölder’s inequality and Proposition 2.6,
we have

$\displaystyle\int_{\mathbb{R}^{d}}\left(\tilde{\bm{b}}\otimes\bm{b}^{1}+\bm{b}^{2}\otimes\tilde{\bm{b}}\right):\Phi\,d{x}$
$\displaystyle\quad\leq\left(\|{\bm{b}^{1}}\|_{{2r^{\prime}}}+\|{\bm{b}^{2}}\|_{{2r^{\prime}}}\right)\|{\tilde{\bm{b}}}\|_{{2r^{\prime}}}\|{\Phi}\|_{{r}}$
$\displaystyle\quad\apprle\left(\|{\bm{b}^{1}}\|_{{2r^{\prime}}}+\|{\bm{b}^{2}}\|_{{2r^{\prime}}}\right)\|{\tilde{\bm{b}}}\|_{2r^{\prime}}\|{\Phi}\|_{\dot{H}^{\alpha-1}}$
$\displaystyle\quad\apprle\left(\|{\bm{b}^{1}}\|_{2}^{1-\lambda}\|{\bm{b}^{1}}\|_{\dot{H}^{\beta}}^{\lambda}+\|{\bm{b}^{2}}\|_{2}^{1-\lambda}\|{\bm{b}^{2}}\|_{\dot{H}^{\beta}}^{\lambda}\right)\|{\tilde{\bm{b}}}\|_{2}^{1-\lambda}\|{\tilde{\bm{b}}}\|_{\dot{H}^{\beta}}^{\lambda}\|{\Phi}\|_{\dot{H}^{\alpha-1}},$

where

$0<\lambda=\frac{d}{2\beta r}=\frac{d+2-2\alpha}{4\beta}\leq\frac{1}{2}.$

Hence by (6.7),

(6.8) $\displaystyle\|{\Lambda^{\alpha}\tilde{\bm{u}}}\|_{2}$
$\displaystyle\apprle\left(\|{\bm{b}^{1}}\|_{2}^{1-\lambda}\|{\bm{b}^{1}}\|_{\dot{H}^{\beta}}^{\lambda}+\|{\bm{b}^{2}}\|_{2}^{1-\lambda}\|{\bm{b}^{2}}\|_{\dot{H}^{\beta}}^{\lambda}\right)\|{\tilde{\bm{b}}}\|_{2}^{1-\lambda}\|{\tilde{\bm{b}}}\|_{\dot{H}^{\beta}}^{\lambda}$
$\displaystyle\apprle\phi(t)^{\lambda/2}\|{\tilde{\bm{b}}}\|_{2}^{1-\lambda}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{\lambda}.$

Combining (6.5), (6.6), and (6.8), we have

$\displaystyle II$ $\displaystyle\leq
C\|{\tilde{\bm{b}}}\|_{{2}}^{1-\theta_{1}}\left(\phi(t)^{\lambda/2}\|{\tilde{\bm{b}}}\|_{{2}}^{1-\lambda}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{\lambda}\right)^{\theta_{1}}\phi(t)^{\theta_{2}/2}\|{\tilde{\bm{b}}}\|_{{2}}^{1-1/\beta}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{1/\beta}$
$\displaystyle=C\phi(t)^{(\lambda\theta_{1}+\theta_{2})/2}\|{\tilde{\bm{b}}}\|_{2}^{2-\lambda\theta_{1}-1/\beta}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{\lambda\theta_{1}+1/\beta}$
$\displaystyle\leq
C\phi(t)^{\frac{\lambda\theta_{1}+\theta_{2}}{2-\lambda\theta_{1}-1/\beta}}\|{\tilde{\bm{b}}}\|_{{2}}^{2}+\frac{1}{4}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{2}.$

Since $\theta_{1}+\theta_{2}+1/\beta\leq 2$ and $2\lambda\leq 1$, it follows
that

$\frac{\lambda\theta_{1}+\theta_{2}}{2-\lambda\theta_{1}-1/\beta}\leq 1.$

Therefore,

$II\leq
C\phi(t)\|{\tilde{\bm{b}}}\|_{{2}}^{2}+\frac{1}{4}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{2}.$

Suppose next that $\alpha<1$. Then since
$\beta\geq\frac{d}{2}+1-\alpha>\frac{d}{2}$, it follows from (6.7) and Lemma
2.9 that

$\displaystyle\|{\Lambda^{\alpha}\tilde{\bm{u}}}\|_{{2}}\apprle\|{\tilde{\bm{b}}\otimes\bm{b}^{1}+\bm{b}^{2}\otimes\tilde{\bm{b}}}\|_{\dot{H}^{1-\alpha}}\apprle\left(\|{\bm{b}^{1}}\|_{{H}^{1-\alpha}}+\|{\bm{b}^{2}}\|_{{H}^{1-\alpha}}\right)\|{\tilde{\bm{b}}}\|_{{H}^{\beta}}.$

Moreover, by Proposition 2.6,

$\displaystyle\|{\bm{b}^{i}}\|_{{H}^{1-\alpha}}$
$\displaystyle\apprle\|{\bm{b}^{i}}\|_{{2}}+\|{\bm{b}^{i}}\|_{\dot{H}^{1-\alpha}}$
$\displaystyle\apprle\|{\bm{b}^{i}}\|_{{2}}+\|{\bm{b}^{i}}\|_{{2}}^{1-\mu}\|{\bm{b}^{i}}\|_{\dot{H}^{\beta}}^{\mu}\apprle\phi(t)^{\mu/2}$

for $i=1,2$, where $\mu=(1-\alpha)/\beta$. Hence

$\displaystyle II$ $\displaystyle\leq
C\|{\tilde{\bm{b}}}\|_{{2}}^{1-\theta_{1}}\left[\phi(t)^{\mu/2}\left(\|{\tilde{\bm{b}}}\|_{{2}}+\|{\tilde{\bm{b}}}\|_{\dot{H}^{\beta}}\right)\right]^{\theta_{1}}\phi(t)^{\theta_{2}/2}\|{\tilde{\bm{b}}}\|_{{2}}^{1-1/\beta}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{1/\beta}$
$\displaystyle\leq
C\phi(t)^{(\mu\theta_{1}+\theta_{2})/2}\|{\tilde{\bm{b}}}\|_{2}^{2-1/\beta}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{1/\beta}$
$\displaystyle\mathrel{\phantom{=}}+C\phi(t)^{(\mu\theta_{1}+\theta_{2})/2}\|{\tilde{\bm{b}}}\|_{2}^{2-\theta_{1}-1/\beta}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{\theta_{1}+1/\beta}$
$\displaystyle\leq
C\left[\phi(t)^{\frac{\mu\theta_{1}+\theta_{2}}{2-1/\beta}}+\phi(t)^{\frac{\mu\theta_{1}+\theta_{2}}{2-\theta_{1}-1/\beta}}\right]\|{\tilde{\bm{b}}}\|_{{2}}^{2}+\frac{1}{4}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{2}.$

Since $(1+\mu)\theta_{1}+\theta_{2}+1/\beta\leq 2$, it follows that

$\frac{\mu\theta_{1}+\theta_{2}}{2-1/\beta}\leq\frac{\mu\theta_{1}+\theta_{2}}{2-\theta_{1}-1/\beta}\leq
1.$

Therefore,

$II\leq
C\phi(t)\|{\tilde{\bm{b}}}\|_{{2}}^{2}+\frac{1}{4}\|{\Lambda^{\beta}\tilde{\bm{b}}}\|_{{2}}^{2}.$

This completes the proof of (6.3). By Gronwall’s inequality, we conclude that
$\bm{b}^{1}=\bm{b}^{2}$ on $[0,\infty)$. Moreover, it follows from (6.6) that
$\bm{u}^{1}=\bm{u}^{2}$ on $[0,\infty)$. This completes the proof of Theorem
1.3. ∎

##  Appendix A Completion of the proof of Lemma 2.2

To complete the proof of Lemma 2.2, it remains to prove the following result.

######  Lemma A.1.

$\mathscr{S}_{0}$ is dense in $L_{p}\cap\dot{H}^{s}$ for $1<p<\infty$ and
$s\in\mathbb{R}$.

Lemma A.1 will be proved by using Littlewood-Paley operators as in the proofs
of [3, Proposition 2.27] and [34, 5.1.5/Theorem]. Let $\psi$ and $\varphi$ be
radial $C^{\infty}$-functions on $\mathbb{R}^{d}$ such that

${\rm supp}\,\psi\subset B_{4/3},\quad{\rm
supp}\,\varphi\subset\\{\xi\in\mathbb{R}^{d}:3/4\leq|\xi|\leq 8/3\\},$
$\psi(\xi)+\sum_{j=0}^{\infty}\varphi(2^{-j}\xi)=1\quad\mbox{for
all}\,\,\xi\in\mathbb{R}^{d},$

and

$\sum_{j=-\infty}^{\infty}\varphi(2^{-j}\xi)=1\quad\mbox{for all}\,\,\xi\neq
0.$

For $j\in\mathbb{Z}$, let $\psi_{j}(\xi)=\psi(2^{-j}\xi)$ and
$\varphi_{j}(\xi)=\varphi(2^{-j}\xi)$. Then the Littlewood-Paley operators
$S_{j}$ and $\Delta_{j}$ are defined by

$S_{j}u=(\psi_{j}\hat{u})^{\vee}=\psi_{j}^{\vee}*u\quad\text{and}\quad\Delta_{j}u=(\varphi_{j}\hat{u})^{\vee}=\varphi_{j}^{\vee}*u$

for $u\in\mathscr{S}_{0}^{\prime}$. It immediately follows from Young’s
convolution inequality that

(A.1) $\|{S_{j}u}\|_{{p}}+\|{\Delta_{j}u}\|_{{p}}\leq
C\|{u}\|_{{p}}\quad(1\leq p\leq\infty)$

for some constant $C$ independent of $j$.

The following inequalities of Bernstein type are also easily proved by using
Young’s convolution inequality (see e.g. [3, Lemma 2.1] or [36, Lemma 2.4]).

######  Proposition A.2.

Let $1\leq p\leq q\leq\infty$, $k\in\mathbb{N}_{0}$, and $0<R<\infty$. Then
for all $u\in L_{p}$ with $\operatorname{supp}\hat{u}\subset B_{R}$, we have

$\|{D^{k}u}\|_{{q}}:=\max_{|\beta|=k}\|{D^{\beta}u}\|_{{q}}\apprle
R^{k+d(1/p-1/q)}\|{u}\|_{{p}}.$

Moreover, if $u\in L_{p}$ and $\operatorname{supp}\hat{u}\subset
B_{R}\setminus B_{R/2}$, then

$\|{D^{k}u}\|_{{p}}\approx R^{k}\|{u}\|_{{p}}.$

In particular, for all $u\in L_{p}$, we have

(A.2) $\displaystyle\|{\Delta_{j}u}\|_{{q}}$ $\displaystyle\apprle
2^{jd(1/p-1/q)}\|{\Delta_{j}u}\|_{{p}},$ (A.3)
$\displaystyle\|{S_{j}u}\|_{{q}}$ $\displaystyle\apprle
2^{jd(1/p-1/q)}\|{S_{j}u}\|_{{p}},$ (A.4)
$\displaystyle\|{D^{k}(\Delta_{j}u)}\|_{{p}}$ $\displaystyle\approx
2^{jk}\|{\Delta_{j}u}\|_{{p}}.$

The following is the classical Littlewood-Paley characterization of the
homogeneous Sobolev space $\dot{H}^{s}_{p}$ (see e.g. [19, Theorem 6.2.7]).

######  Proposition A.3.

Let $s\in\mathbb{R}$ and $1<p<\infty$. For all $u\in\dot{H}^{s}_{p}$, we have

$\|{u}\|_{\dot{H}^{s}_{p}}\approx\left\|\left(\sum_{j\in\mathbb{Z}}2^{2js}|{\Delta}_{j}u|^{2}\right)^{1/2}\right\|_{p}.$

###### Proof of Lemma A.1.

Suppose that $u\in L_{p}\cap\dot{H}^{s}$. Let $\varepsilon>0$ be given. Then
by Proposition A.3 with $s=0$ and $p=2$, respectively, there exists
$N\in\mathbb{N}$ such that

$\|{u_{N}-u}\|_{{p}}+\|{u_{N}-u}\|_{\dot{H}^{s}}<\frac{\varepsilon}{2},$

where $u_{N}=\sum_{|j|\leq N}\Delta_{j}u$. Note that $u_{N}$ belongs to
$L_{p}\cap L_{2}$ and has support in an annulus. Hence it follows from
Proposition A.2 that $u_{N}\in H_{2}^{k}\cap H_{p}^{k}$ for all
$k\in\mathbb{N}$.

Choose $\theta\in C_{c}^{\infty}(B_{2})$ with $\theta=1$ in $B_{1}$, and let
$\theta_{R}(x)=\theta(x/R)$ for $R>0$. Fixing an integer $M>N$ (say, $M=N+1$),
we define

$u_{N}^{R}=(I-S_{-M})(\theta_{R}u_{N}).$

Note then that

(A.5) $u_{N}^{R}\in\mathscr{S}_{0}\quad\mbox{and}\quad
u_{N}^{R}-u_{N}=(I-S_{-M})((\theta_{R}-1)u_{N}).$

By (A.1), we have

$\|{u_{N}^{R}-u_{N}}\|_{{p}}\apprle\|{(\theta_{R}-1)u_{N}}\|_{{p}}.$

Since $u_{N}\in L_{p}$ and $(\theta_{R}-1)u_{N}\rightarrow 0$ as
$R\rightarrow\infty$, it follows from the dominated convergence theorem that

$\|{u_{N}^{R}-u_{N}}\|_{{p}}\rightarrow
0\quad\mbox{as}\,\,R\rightarrow\infty.$

Let $k=\max\,\\{0,\lfloor s\rfloor+2\\}$, where $\lfloor s\rfloor$ is the
greatest integer less than or equal to $s$. If $j\geq 0$, then by (A.5),
(A.4), and (A.1), we have

$\displaystyle 2^{js}\|{\Delta_{j}(u_{N}^{R}-u_{N})}\|_{{2}}$
$\displaystyle\leq
2^{j(k-1)}\|{\Delta_{j}(I-S_{-M})((\theta_{R}-1)u_{N})}\|_{{2}}$
$\displaystyle\approx
2^{-j}\|{D^{k}[\Delta_{j}[(I-S_{-M})((\theta_{R}-1)u_{N})]]}\|_{{2}}$
$\displaystyle\apprle 2^{-j}\|{D^{k}((\theta_{R}-1)u_{N})}\|_{{2}}.$

If $-M-1\leq j\leq-1$, then by (A.1) and (A.5),

$2^{js}\|{\Delta_{j}(u_{N}^{R}-u_{N})}\|_{{2}}\apprle
2^{js}\|{(\theta_{R}-1)u_{N}}\|_{{2}}.$

Finally, noting that

$\Delta_{j}(u_{N}^{R}-u_{N})=0\quad\mbox{for all}\,\,j\leq-M-2,$

we obtain

$\displaystyle\|{u_{N}^{R}-u_{N}}\|_{\dot{H}^{s}}^{2}$
$\displaystyle\approx\int_{\mathbb{R}^{d}}\left(\sum_{j\in\mathbb{Z}}2^{2js}|\Delta_{j}(u_{N}^{R}-u)|^{2}\right)\,d{x}$
$\displaystyle\apprle\sum_{j=-M-1}^{-1}2^{2js}\|{(\theta_{R}-1)u_{N}}\|_{{2}}^{2}+\sum_{j=0}^{\infty}2^{-j}\|{D^{k}((\theta_{R}-1)u_{N})}\|_{{2}}^{2}$
$\displaystyle\apprle\|{(\theta_{R}-1)u_{N}}\|_{{H}^{k}}^{2}.$

Hence by the dominated convergence theorem again, we have

$\|{u_{N}^{R}-u_{N}}\|_{\dot{H}^{s}}^{2}\to
0\quad\mbox{as}\,\,R\rightarrow\infty.$

This completes the proof of Lemma A.1. ∎

##  Appendix B Proof of Lemma 3.4

Our proof of Lemma 3.4 is based on the following well-known result (see e.g.
[31, Chapter V, Lemma 2] and [18, Theorem 2.4.6]).

######  Lemma B.1.

If $0<\operatorname{Re}\lambda<d$, then

(B.1)
$\int_{\mathbb{R}^{d}}\frac{1}{|\xi|^{\lambda}}\phi^{\vee}(\xi)\,d\xi=\frac{\Gamma((d-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\frac{1}{|x|^{d-\lambda}}\phi(x)\,dx$

for all $\phi\in\mathscr{S}$.

We will also use the method of analytic continuation. The following result is
useful to show the analyticity of some functions defined by integrals.

######  Lemma B.2.

Fix $\phi\in\mathscr{S}$ and $\beta\in{\mathbb{N}}_{0}^{d}$. Then the function

$f(z)=\int_{\mathbb{R}^{d}}x^{\beta}|x|^{z}\phi(x)\,d{x}$

is holomorphic in the region
$H=\\{z\in\mathbb{C}:\operatorname{Re}z>-d-|\beta|\\}$.

###### Proof.

Let $z\in H$ be fixed. Choose $\varepsilon>0$ such that
$\\{w\in\mathbb{C}:|w-z|\leq\varepsilon\\}\subset H$.

Suppose that $x\neq 0$. Then the function $|x|^{w}=\exp(w\ln|x|)$ is entire
and

$\frac{d}{dw}\left[|x|^{w}\right]=(\log|x|)|x|^{w}.$

Hence for all $w\in\overline{B_{\varepsilon}(z)}$ with $w\neq z$,

$\left|\frac{|x|^{w}-|x|^{z}}{w-z}\right|\leq|\log|x||\max_{|\eta-z|\leq\varepsilon}||x|^{\eta}|=|\log|x|||x|^{\operatorname{Re}z+\varepsilon}.$

Moreover, since $\operatorname{Re}z+\varepsilon>-d-|\beta|$, we have

$\int_{\mathbb{R}^{d}}\left|x^{\beta}\log|x||x|^{\operatorname{Re}z+\varepsilon}\phi(x)\right|\,d{x}=\int_{\mathbb{R}^{d}}|\log|x|||x|^{\operatorname{Re}z+\varepsilon+|\beta|}|\phi(x)|\,d{x}<\infty.$

Hence it follows from the dominated convergence theorem that

$\lim_{w\rightarrow
z}\frac{f(w)-f(z)}{w-z}=\int_{\mathbb{R}^{d}}x^{\beta}(\log|x|)|x|^{z}\phi(x)\,dx.$

This completes the proof of Lemma B.2. ∎

Recall that

$(D^{\beta}\phi)^{\vee}(\xi)=(-2\pi i\xi)^{\beta}{\phi}^{\vee}(\xi),\quad
D^{\beta}\widehat{\phi}(\xi)=\left((-2\pi
ix)^{\beta}\phi\right)^{\wedge}(\xi)$

for $\phi\in\mathscr{S}$ and $\beta\in\mathbb{N}_{0}^{d}$.

###### Proof of Lemma 3.4 (1).

Assume that $0<\operatorname{Re}\lambda<d$. Set $\phi=D_{j}\psi$. Then since
${\phi}^{\vee}(\xi)=(-2\pi i\xi_{j}){\psi}^{\vee}(\xi)$, it follows from Lemma
B.1 that

$\int_{\mathbb{R}^{d}}\frac{-2\pi
i\xi_{j}}{|\xi|^{\lambda}}{\psi}^{\vee}(\xi)\,d\xi=\frac{\Gamma((d-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\frac{1}{|x|^{d-\lambda}}D_{j}\psi(x)\,dx.$

By the fundamental property of the Gamma function,

$(d-\lambda)\int_{\mathbb{R}^{d}}\frac{-2\pi
i\xi_{j}}{|\xi|^{\lambda}}{\psi}^{\vee}(\xi)\,d\xi=\frac{2\Gamma((d+2-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\frac{1}{|x|^{d-\lambda}}D_{j}\psi(x)\,dx,$

which holds for $0<\operatorname{Re}\lambda<d+1$, by analytic continuation
based on Lemma B.2.

Assume next that $1<\operatorname{Re}\lambda<d+1$. Then performing the
integration by parts, we easily obtain

$\int_{\mathbb{R}^{d}}\frac{1}{|x|^{d-\lambda}}D_{j}\psi(x)\,dx=(d-\lambda)\int_{\mathbb{R}^{d}}\frac{x_{j}}{|x|^{d+2-\lambda}}\psi(x)\,dx.$

Hence, if $d<\operatorname{Re}\lambda<d+1$, then

$\int_{\mathbb{R}^{d}}\frac{-2\pi
i\xi_{j}}{|\xi|^{\lambda}}{\psi}^{\vee}(\xi)\,d\xi=\frac{2\Gamma((d+2-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\frac{x_{j}}{|x|^{d+2-\lambda}}\psi(x)\,dx.$

This identity holds for $1<\operatorname{Re}\lambda<d+1$, by Lemma B.2 and
analytic continuation again. This completes the proof of Lemma 3.4 (1). ∎

###### Proof of Lemma 3.4 (2).

Assume that $1<\operatorname{Re}\lambda<d+1$. Then by Lemma 3.4 (1),

$\int_{\mathbb{R}^{d}}\frac{-2\pi
i\xi_{j}}{|\xi|^{\lambda}}{\phi}^{\vee}(\xi)\,d\xi=\frac{2\Gamma((d+2-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\frac{x_{j}}{|x|^{d+2-\lambda}}\phi(x)\,dx.$

for all $\phi\in\mathscr{S}$. Setting $\phi=D_{k}\psi$, we have

$\int_{\mathbb{R}^{d}}\frac{(-2\pi i\xi_{j})(-2\pi
i\xi_{k})}{|\xi|^{\lambda}}{\psi}^{\vee}(\xi)\,d\xi=\frac{2\Gamma((d+2-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\frac{x_{j}}{|x|^{d+2-\lambda}}D_{k}\psi(x)\,dx,$

which holds for $1<\operatorname{Re}\lambda<d+2$, by analytic continuation.
Moreover, if $d+1<\operatorname{Re}\lambda<d+2$, then

$\int_{\mathbb{R}^{d}}\frac{x_{j}}{|x|^{d+2-\lambda}}D_{k}\psi(x)\,dx=\int_{\mathbb{R}^{d}}\frac{(d+2-\lambda)x_{j}x_{k}-\delta_{jk}|x|^{2}}{|x|^{d+4-\lambda}}{\psi}(x)\,d{x}.$

Therefore, the proof of Lemma 3.4 (2) is completed by analytic continuation. ∎

###### Proof of Lemma 3.4 (3).

Assume that $2<\operatorname{Re}\lambda<d+2$. Then by Lemma 3.4 (2),

$\displaystyle\int_{\mathbb{R}^{d}}\frac{(-2\pi i\xi_{j})(-2\pi
i\xi_{k})}{|\xi|^{\lambda}}{\phi}^{\vee}(\xi)\,d\xi$
$\displaystyle\quad=\frac{2\Gamma((d+2-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\frac{(d+2-\lambda)x_{j}x_{k}-\delta_{jk}|x|^{2}}{|x|^{d+4-\lambda}}{\phi}(x)\,dx$

for all $\phi\in\mathscr{S}$. Setting $\phi=D_{l}\psi$, we have

$\displaystyle\int_{\mathbb{R}^{d}}\frac{(-2\pi i\xi_{j})(-2\pi
i\xi_{k})(-2\pi i\xi_{l})}{|\xi|^{\lambda}}{\psi}^{\vee}(\xi)\,d\xi$
$\displaystyle\quad=\frac{2\Gamma((d+2-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\frac{(d+2-\lambda)x_{j}x_{k}-\delta_{jk}|x|^{2}}{|x|^{d+4-\lambda}}D_{l}{\psi}(x)\,dx$
$\displaystyle\quad=\frac{4\Gamma((d+4-\lambda)/2)}{(d+2-\lambda)\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\frac{(d+2-\lambda)x_{j}x_{k}-\delta_{jk}|x|^{2}}{|x|^{d+4-\lambda}}D_{l}{\psi}(x)\,dx,$

which holds for $2<\operatorname{Re}\lambda<d+3$ with $\lambda\neq d+2$, by
the analytic continuation. Moreover, if $3<\operatorname{Re}\lambda<d+3$, then

$\int_{\mathbb{R}^{d}}\frac{1}{|x|^{d+2-\lambda}}D_{l}{\psi}(x)\,d{x}=(d+2-\lambda)\int_{\mathbb{R}^{d}}\frac{x_{l}}{|x|^{d+4-\lambda}}\psi(x)\,dx.$

Hence for $d+2<\operatorname{Re}\lambda<d+3$, we have

$\displaystyle\int_{\mathbb{R}^{d}}\frac{(-2\pi i\xi_{j})(-2\pi
i\xi_{k})(-2\pi i\xi_{l})}{|\xi|^{\lambda}}{\psi}^{\vee}(\xi)\,d\xi$
$\displaystyle\quad=\frac{4\Gamma((d+4-\lambda)/2)}{\Gamma(\lambda/2)\pi^{d/2-\lambda}}\int_{\mathbb{R}^{d}}\left[-D_{l}\left(\frac{x_{j}x_{k}}{|x|^{d+4-\lambda}}\right)-\frac{\delta_{jk}x_{l}}{|x|^{d+4-\lambda}}\right]{\psi}(x)\,d{x}.$

The proof of Lemma 3.4 (3) is completed by analytic continuation. ∎

##  Appendix C Proof of Lemma 3.6

This section is devoted to proving Lemma 3.6. It is a classical result due to
de Rham [14] that if a distribution $\bm{u}$ satisfies

(C.1) $\left<\bm{u},\Phi\right>=0\quad\text{ for all }\Phi\in
C_{c,\sigma}^{\infty},$

then $\bm{u}=\nabla p$ for some distribution $p$. We prove that if $\bm{u}$ is
a tempered distribution, then $p$ is also a tempered distribution by following
the idea of Wang [35]. It will be first shown (see Lemma C.2 below) that for
each $g\in\mathscr{S}(\mathbb{R}^{d})$ with $\int_{\mathbb{R}^{d}}g\,d{x}=0$,
there exists $\bm{w}_{g}\in\mathscr{S}(\mathbb{R}^{d};\mathbb{R}^{d})$ such
that

$\operatorname{div}\bm{w}_{g}=g\quad\text{in }\mathbb{R}^{d}.$

The following lemma is necessary to show that
$\bm{w}_{g}\in\mathscr{S}(\mathbb{R}^{d};\mathbb{R}^{d})$ and the mapping
$g\mapsto\bm{w}_{g}$ is linear and continuous.

######  Lemma C.1.

For a fixed $1\leq j\leq d$, let $\mathscr{S}_{\\#}^{(j)}$ be the subspace of
$\mathscr{S}$ that consists of all $u\in\mathscr{S}$ satisfying

$\int_{-\infty}^{\infty}u(x_{1},\dots,x_{j-1},t,x_{j+1},\dots,x_{d})\,d{t}=0$

for all $(x_{1},...,x_{j-1},x_{j+1},...,x_{d})\in\mathbb{R}^{d-1}$. For each
$u\in\mathscr{S}_{\\#}^{(j)}$, we define
$T^{(j)}(u):\mathbb{R}^{d}\rightarrow\mathbb{R}$ by

$T^{(j)}(u)(x)=\int_{-\infty}^{x_{j}}u(x_{1},\dots,x_{j-1},t,x_{j+1},\dots,x_{d})\,d{t}\quad\text{for
all }x\in\mathbb{R}^{d}.$

Then $T^{(j)}$ is a linear operator from $\mathscr{S}_{\\#}^{(j)}$ into
$\mathscr{S}$. Moreover, if $u_{k}\in\mathscr{S}_{\\#}^{(j)}$ for each
$k\in\mathbb{N}$ and $u_{k}\rightarrow 0$ in $\mathscr{S}$, then
$T^{(j)}(u_{k})\rightarrow 0$ in $\mathscr{S}$.

###### Proof.

We prove the lemma only for the case when $d\geq 2$ and $j=1$.

Suppose that $u\in\mathscr{S}_{\\#}^{(1)}$ and $v=T^{(1)}(u)$. Obviously, $v$
is smooth on $\mathbb{R}^{d}$. Moreover for all
$\beta=(\beta_{1},\beta^{\prime})\in\mathbb{N}_{0}^{d}$, we have

$D^{\beta}v(x)=\begin{dcases}\int_{-\infty}^{x_{1}}D^{\beta^{\prime}}_{x^{\prime}}u(t,x^{\prime})\,d{t}&\quad\text{if
}\beta_{1}=0,\\\
D^{\beta_{1}-1}_{1}D^{\beta^{\prime}}_{x^{\prime}}u(x_{1},x^{\prime})&\quad\text{if
}\beta_{1}>0.\end{dcases}$

Hence to show that $v\in\mathscr{S}$, it remains to show that

$\sup_{x\in\mathbb{R}^{d}}\left|x^{\alpha}D^{\beta^{\prime}}_{x^{\prime}}v(x)\right|<\infty$

for every $\beta^{\prime}\in\mathbb{N}_{0}^{d-1}$ and
$\alpha\in\mathbb{N}_{0}^{d}$.

Let $\beta^{\prime}\in\mathbb{N}_{0}^{d-1}$ and
$\alpha=(\alpha_{1},\alpha^{\prime})\in\mathbb{N}_{0}^{d}$ be fixed. Then
since

$\int_{-\infty}^{\infty}D^{\beta^{\prime}}_{x^{\prime}}u(t,x^{\prime})\,d{t}=D^{\beta^{\prime}}_{x^{\prime}}\int_{-\infty}^{\infty}u(t,x^{\prime})\,d{t}=0,$

it follows that

$x^{\alpha}D^{\beta^{\prime}}_{x^{\prime}}v(x)=x_{1}^{\alpha_{1}}\int_{-\infty}^{x_{1}}(x^{\prime})^{\alpha^{\prime}}D^{\beta^{\prime}}_{x^{\prime}}u(t,x^{\prime})\,d{t}=-x_{1}^{\alpha_{1}}\int_{x_{1}}^{\infty}(x^{\prime})^{\alpha^{\prime}}D^{\beta}_{x^{\prime}}u(t,x^{\prime})\,d{t}$

for all $x=(x_{1},x^{\prime})\in\mathbb{R}^{d}$. On the other hand, since
$u\in\mathscr{S}$, there exists a constant $C$ such that

$\sup_{(t,x^{\prime})\in\mathbb{R}^{d}}|(1+|t|)^{\alpha_{1}+2}(x^{\prime})^{\alpha^{\prime}}D^{\beta^{\prime}}_{x^{\prime}}u(t,x^{\prime})|\leq
C.$

Hence for all $x=(x_{1},x^{\prime})\in\mathbb{R}^{d}$ with $x_{1}\geq 0$, we
have

$\displaystyle\left|x^{\alpha}D^{\beta^{\prime}}_{x^{\prime}}v(x)\right|$
$\displaystyle\leq
x_{1}^{\alpha_{1}}\int_{x_{1}}^{\infty}\left|(x^{\prime})^{\alpha^{\prime}}D^{\beta^{\prime}}_{x^{\prime}}u(t,x^{\prime})\right|\,d{t}$
$\displaystyle\leq
x_{1}^{\alpha_{1}}\int_{x_{1}}^{\infty}\frac{C}{(1+|t|)^{\alpha_{1}+2}}dt\leq
C(\alpha,\beta^{\prime}).$

Similarly, if $x=(x_{1},x^{\prime})\in\mathbb{R}^{d}$ and $x_{1}\leq 0$, then

$\left|x^{\alpha}D_{x^{\prime}}^{\beta^{\prime}}v(x)\right|\leq|x_{1}|^{\alpha_{1}}\int_{-\infty}^{x_{1}}\left|(x^{\prime})^{\alpha^{\prime}}D_{x^{\prime}}^{\beta^{\prime}}u(t,x^{\prime})\right|dt\leq
C(\alpha,\beta^{\prime}).$

This proves that $v\in\mathscr{S}$.

We have shown that $T^{(1)}$ maps $\mathscr{S}_{\\#}^{(1)}$ to $\mathscr{S}$.
Linearity of $T^{(1)}$ is obvious from the very definition of $T^{(1)}$. Hence
to complete the proof, it remains to prove the continuity property of
$T^{(1)}$.

Suppose that $u_{k}\in\mathscr{S}_{\\#}^{(1)}$ and $u_{k}\rightarrow 0$ in
$\mathscr{S}$. Let $\alpha,\beta\in\mathbb{N}^{d}_{0}$ be fixed. Then since
$u_{k}\rightarrow 0$ in $\mathscr{S}$, it follows that if $\beta_{1}>0$, then

$x^{\alpha}D^{\beta}T^{(1)}(u_{k})(x)=x^{\alpha}D^{\beta_{1}-1}_{1}D^{\beta^{\prime}}_{x^{\prime}}u_{k}(x)\rightarrow
0$

uniformly on $\mathbb{R}^{d}$. Moreover, for any $\varepsilon>0$, there exists
$N\in\mathbb{N}$ such that

$\sup_{x\in\mathbb{R}^{d}}\left|(1+|x_{1}|)^{\alpha_{1}+2}(x^{\prime})^{\alpha^{\prime}}D^{\beta^{\prime}}_{x^{\prime}}u_{k}(x)\right|<\varepsilon$

for all $k\geq N$. If $x=(x_{1},x^{\prime})\in\mathbb{R}^{d}$ and $x_{1}\geq
0$, then for all $k\geq N$,

$\left|x^{\alpha}D^{\beta^{\prime}}_{x^{\prime}}T^{(1)}(u_{k})(x)\right|\leq\varepsilon
x_{1}^{\alpha_{1}}\int_{x_{1}}^{\infty}\frac{1}{(1+|t|)^{\alpha_{1}+2}}dt<C\varepsilon.$

Similar estimates hold for $x_{1}\leq 0$. This implies that
$T^{(1)}(u_{k})\rightarrow 0$ in $\mathscr{S}$, which completes the proof of
Lemma C.1. ∎

######  Lemma C.2.

Fix $\phi_{i}\in C_{c}^{\infty}(\mathbb{R})$ with
$\int_{\mathbb{R}}\phi_{i}\,d{x}=1$ ($i=1,2,\dots,d$) and define

$\phi(x)=\phi_{1}(x_{1})\cdots\phi_{d}(x_{d})\quad\mbox{for
all}\,\,x=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}.$

Then there exists a continuous linear operator $\mathcal{B}$ from
$\mathscr{S}(\mathbb{R}^{d})$ to $\mathscr{S}(\mathbb{R}^{d};\mathbb{R}^{d})$
such that

(C.2)
$\operatorname{div}\mathcal{B}(g)=g-\phi\int_{\mathbb{R}^{d}}g\,d{x}\quad\mbox{in}\,\,\mathbb{R}^{d}$

for all $g\in\mathscr{S}(\mathbb{R}^{d})$.

###### Proof.

For the sake of notational simplicity, we mainly focus on the case $d=3$. The
general case can be proved in a similar way.

For a fixed $g\in\mathscr{S}(\mathbb{R}^{3})$, we define

$\overline{g}=g-\phi\int_{\mathbb{R}^{3}}g\,d{x}.$

Also for $1\leq j\leq 3$, let $S^{(j)}(g):\mathbb{R}^{3}\to\mathbb{R}$ be
defined by

$\displaystyle S^{(1)}(g)(x)$
$\displaystyle=\overline{g}(x)-\phi_{1}(x_{1})\int_{\mathbb{R}}\overline{g}(s_{1},x_{2},x_{3})\,d{s_{1}},$
$\displaystyle S^{(2)}(g)(x)$
$\displaystyle=\phi_{1}(x_{1})\left[\int_{\mathbb{R}}\overline{g}(s_{1},x_{2},x_{3})\,d{s_{1}}-\phi_{2}(x_{2})\int_{\mathbb{R}^{2}}\overline{g}(s_{1},s_{2},x_{3})\,d{s_{1}}ds_{2}\right],$
$\displaystyle S^{(3)}(g)(x)$
$\displaystyle=\phi_{1}(x_{1})\phi_{2}(x_{2})\left[\int_{\mathbb{R}^{2}}\overline{g}(s_{1},s_{2},x_{3})\,d{s_{1}}ds_{2}\right].$

Then it is easy to check that $S^{(j)}$ is a linear operator from
$\mathscr{S}(\mathbb{R}^{3})$ into $\mathscr{S}_{\\#}^{(j)}$ and that if
$g_{k}\rightarrow 0$ in $\mathscr{S}$, then $S^{(j)}(g_{k})\rightarrow 0$ in
$\mathscr{S}$.

Now, for each $g\in\mathscr{S}$, we define

$\mathcal{B}(g)=\left(T^{(1)}S^{(1)}(g),T^{(2)}S^{(2)}(g),T^{(3)}S^{(3)}(g)\right).$

Then by Lemma C.1, $\mathcal{B}$ is a continuous linear operator from
$\mathscr{S}(\mathbb{R}^{3})$ to $\mathscr{S}(\mathbb{R}^{3};\mathbb{R}^{3})$.
Moreover, by a direct computation,

$\operatorname{div}\mathcal{B}(g)=S^{(1)}(g)+S^{(2)}(g)+S^{(3)}(g)=\overline{g}\quad\mbox{in}\,\,\mathbb{R}^{3}.$

This completes the proof of Lemma C.2. ∎

###### Proof of Lemma 3.6.

Suppose that $\bm{u}\in\mathscr{S}^{\prime}$ satisfies
$\left<\bm{u},\Phi\right>=0$ for all $\Phi\in C_{c,\sigma}^{\infty}$. Then by
a simple density argument, we have

(C.3) $\left<\bm{u},\Phi\right>=0\quad\mbox{for all
}\Phi\in\mathscr{S}\,\,\mbox{with}\,\,\operatorname{div}\Phi=0.$

Let
$\mathcal{B}:\mathscr{S}(\mathbb{R}^{d})\to\mathscr{S}(\mathbb{R}^{d};\mathbb{R}^{d})$
be the operator defined in Lemma C.2. Then since $\mathcal{B}$ is linear and
continuous, the mapping

$g\mapsto-\left<\bm{u},\mathcal{B}(g)\right>$

defines a continuous linear functional $p$ on $\mathscr{S}$.

Given $\Phi\in\mathscr{S}$, we define $g=\operatorname{div}\Phi$ and
$\Psi=\mathcal{B}(g)-\Phi$. Then $\Psi\in\mathscr{S}$. Moreover, since
$\int_{\mathbb{R}^{d}}g\,d{x}=0$, we have

$\operatorname{div}\Psi=\operatorname{div}\mathcal{B}(g)-\operatorname{div}\Phi=0\quad\mbox{in}\,\,\mathbb{R}^{d}.$

Hence it follows from (C.3) that $\left<\bm{u},\Psi\right>=0$. Therefore,

$\left<\bm{u},\Phi\right>=\left<\bm{u},\mathcal{B}(g)\right>=-\left<p,g\right>=-\left<p,\operatorname{div}\Phi\right>.$

Since $\Phi\in\mathscr{S}$ is arbitrary, we deduce that $\bm{u}=\nabla p$.
This completes the proof of Lemma 3.6. ∎

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