Global well-posedness of the 3D incompressible porous media equation with critical dissipation in Triebel-Lizorkin spaces

Abstract

In this paper, we study the global well-posedness of the 3D incompressible critical dissipative porous media equation with small initial data in the Triebel-Lizorkin space $F^{s}_{p,q}(\mathbb{R}^{3})$. By a pointwise exponential decay estimate on the Poisson semigroup $e^{-t\nu\sqrt{-\Delta}}$ and the Fourier localization technique, we generalize the global well-posedness in the Sobolev spaces $H_{p}^{s}(\mathbb{R}^{3})=F^{s}_{p,2}(\mathbb{R}^{3})$ and $H^{s}(\mathbb {R}^{3})=F^{s}_{2,2}(\mathbb{R}^{3})$ into the general Triebel-Lizorkin spaces $F^{s}_{p,q}(\mathbb{R}^{3})$ with $s>\frac{3}{p}$, $p, q\in (1,\infty)$.

1 Introduction

The incompressible flow in porous media bears important significance in mathematical physics [1]. The Cauchy problem of the 3D incompressible dissipative porous media equation assumes the form

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l}\theta_{t}+u\cdot\nabla\theta+\nu\Lambda^{\alpha}\theta =0,& x\in\mathbb{R}^{3}, t>0,\\ u(t,x)=-k(\nabla p+g\gamma\theta),& x\in\mathbb{R}^{3}, t>0,\\ \operatorname{div}u(t,x)=0,\\ \theta(0,x)=\theta_{0}(x),& x\in\mathbb{R}^{3}, \end{array}\displaystyle \right . \end{aligned}$$

(1.1)

where $0\leq\alpha\leq2$, $\Lambda=\sqrt{-\Delta}$ is the Zygmund operator, $\nu>0$ is the dissipative coefficient, scalar function $\theta=\theta(t,x)$ is the liquid temperature, $u=u(t,x)$ is the liquid discharge (flux per unit area) to model the flow velocity by the Darcy law, k is the matrix of position-independent medium permeabilities in the different directions, respectively, divided by the viscosity, p is the pressure, g is the acceleration due to gravity and $\gamma\in\mathbb{R}^{3}$ is the last canonical vector $\mathbf{e}_{3}$. For simplicity, we set $g=1$ and $k=I$, the identity matrix.

The fractional Laplacian $\Lambda^{\alpha}$ is defined through the Fourier transform

$$\widehat{\Lambda^{\alpha}\theta}(\xi)=\vert \xi \vert ^{\alpha } \widehat{\theta}(\xi),\quad0< \alpha< 2. $$

The cases $0\leq\alpha<1$, $\alpha=1$, $1<\alpha\leq2$ are called supercritical, critical, subcritical, respectively. Roughly speaking, the critical and supercritical cases are mathematically harder to deal with than the subcritical case. For the fractional Laplacian $\Lambda ^{\alpha}$, we refer the reader to [1–4].

According to the Darcy law and the incompressibility condition, for $x\in\mathbb{R}^{3}$ one has [1]

$$\begin{aligned} -\Delta_{x} u(t,x) =&\operatorname{curl}\bigl(\operatorname{curl} u(t,x)\bigr) \\ =& \biggl(-\frac{\partial^{2}\theta}{\partial x_{1}\,\partial x_{3}},-\frac {\partial^{2}\theta}{\partial x_{2}\,\partial x_{3}}, \frac{\partial ^{2}\theta}{\partial x_{1}^{2}}+ \frac{\partial^{2}\theta}{\partial x_{2}^{2}} \biggr), \end{aligned}$$

using the Newton potential formula and integrating by parts, we have

$$\begin{aligned} u(t,x) =&-\frac{2}{3}\bigl(0,0,\theta(t,x)\bigr)+\frac{1}{4\pi} \mbox{P.V.} \int _{\mathbb{R}^{3}}K(x-y)\theta(t,y)\,dy \\ :=&\mathcal{C}(\theta)+\mathcal{S}(\theta), \end{aligned}$$

where the integral kernel is

$$K(x)= \biggl(\frac{3x_{1}x_{3}}{\vert x\vert ^{5}},\frac {3x_{2}x_{3}}{\vert x\vert ^{5}},\frac {2x_{3}^{2}-x_{1}^{2}-x_{2}^{2}}{\vert x\vert ^{5}} \biggr), $$

$x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$, and $\mathcal{C}=(\mathcal {C}_{k})$, $\mathcal{S}=(\mathcal{S}_{k})$, $1\leq k\leq3$, are all operators mapping scalar functions to vector-valued functions and $\mathcal{C}_{k}$ equals a constant multiplication operator whereas $\mathcal{S}_{k}$ means a Calderón-Zygmund singular integral operator.

The global and local well-posedness of the Cauchy problem (1.1) have been intensively investigated in the last few years.

Córdoba, Gancedo, and Orive [5] studied the analytical behavior of solutions with infinite energy in the case $\nu =0$ (without dissipation) in the two dimensional space, they obtained the local existence and uniqueness by the particle trajectory method in Hölder spaces $C^{s}(\mathbb{R}^{2})$ for $0< s<1$ and gave some blow-up criteria. Very recently, Córdoba, Faraco, and Gancedo [6] proved the non-uniqueness of solutions in $L^{\infty }(\mathbb{T}^{2})$ ($\mathbb{T}^{2}$ is the two dimensional flat torus) in space and time, Bae and Granero-Belinchón [7] studied transport equations with different nonlocal velocity fields and proved global weak solutions for very rough initial data (merely $L^{1+}$) for a one dimensional model of the incompressible porous media equation, and one dimensional and n dimensional models of the dissipative incompressible porous medium equation in the periodic domain.

Castro et al. [1] obtained the existence of strong solutions with regular initial data in the Sobolev space $\theta _{0}(x)\in H^{s}(\mathbb{R}^{N})$ ($s>0$) for the subcritical case $1<\alpha \leq2$. For the supercritical case $0\leq\alpha<1$, they also obtained the local well-posedness in the space $H^{s}(\mathbb{R}^{N})$, $s>\frac {N-\alpha}{2}+1$, and they extended it to be global under a smallness condition $\Vert \theta_{0}\Vert _{H^{s}}< c\nu$ on the initial data $\theta_{0}\in H^{s}$ with $s>\frac{N}{2}+1$. In the critical case $\alpha=1$, the existence of strong solutions was obtained. They also proved the global existence of weak solutions with $0\leq\alpha\leq2$.

Due to the method established by Hmidi and Keraani [8] for the quasi-geostrophic equation for $0<\alpha<1$, Xue [9] established the local well-posedness of the porous media equation (1.1) in the Besov space $B_{p,1}^{s}(\mathbb{R}^{N})$ ($1\leq p <\infty$, $s\geq1+\frac{N}{p}-\alpha$) and in $B_{\infty,1}^{s}(\mathbb {R}^{N})\cap\dot{B}_{\infty,1}^{0}(\mathbb{R}^{N})$ ($s\geq1-\alpha$), respectively. Furthermore, Xue [9] also obtained the global well-posedness with small initial data in $\dot{B}_{\infty,1}^{1-\alpha }(\mathbb{R}^{N})$.

For the critical case $\alpha=1$, by the method of modulus of continuity [10] and the Fourier localization technique, Yuan and Yuan [11] proved the global well-posedness in the critical Besov space $\dot{B}_{p,1}^{\frac{3}{p}}(\mathbb{R}^{3})$, $1\leq p \leq\infty$.

Based on Besov space techniques and the method of modulus of continuity, Yamazaki [12] studied the regularized IPM equation in the supercritical regime and the global well-posedness was established in the Sobolev space $H^{m}(\mathbb{R}^{3})$, $m \in\mathbb {Z}^{+}$, $m>\frac{5}{2}$.

We recall that the Triebel-Lizorkin space is a unification of most of the classical function spaces used in partial differential equations such as Lebesgue space $L^{p}(\mathbb{R}^{N})$, Sobolev space $H_{p}^{s}(\mathbb{R}^{N})$ and Hölder space $C^{s}(\mathbb {R}^{N})$ for $s>0$. Chae discussed the local well-posedness and blow-up criterion in the Triebel-Lizorkin space, respectively, for the Euler equation in [13] and for the quasi-geostrophic equation in [14]. Wang and Tang [15–17] studied the long time dynamics of 2D quasi-geostrophic equations.

In this paper, we focus on the critical case $\alpha=1$ in the Triebel-Lizorkin space $F^{s}_{p,q}(\mathbb{R}^{3})$ with $s>\frac {3}{p}$, $p, q \in(1,\infty)$. With the aid of the pointwise exponential decay estimate of the fractional heat semigroup $e^{-t\nu \Lambda^{\alpha}}$,

$$\begin{aligned} \bigl\vert e^{-t\nu\Lambda^{\alpha}}\dot{\Delta}_{j}f(x)\bigr\vert \leq Ce^{-ct2^{j\alpha}}\mathcal{M}(\dot{\Delta}_{j}f) (x),\quad j\in \mathbb{Z}, \end{aligned}$$

(1.2)

where $\dot{\Delta}_{j}$ is the Littlewood-Paley projection to the annulus $\{\vert \xi \vert \sim2^{j}\}$ and $\mathcal{M}$ is the Hardy-Littlewood maximal operator, if we work in a suitable space-time Triebel-Lizorkin space, after integrating in time we can get α derivatives from (1.2). Especially, for $\alpha=1$, we can obtain the first order derivative which exactly balances the nonlinear term.

Our main result reads as follows.

Theorem 1.1

Assume that $\alpha=1$ and $\theta_{0}\in F_{p,q}^{s}(\mathbb{R}^{3})$ with $s>\frac{3}{p}$, $p, q\in(1,\infty)$. If there exists a positive constant ϵ such that $\Vert \theta_{0}\Vert _{F_{p,q}^{s}}< \epsilon$, then the Cauchy problem (1.1) of 3D incompressible critical dissipative porous media equation possesses a unique global solution $\theta(t,x)$ such that

$$\theta(t,x)\in\mathbf{C}\bigl([0,+\infty);F_{p,q}^{s}\bigr) \cap\tilde {L}^{1}\bigl(0,+\infty;\dot{F}_{p,q}^{s+1} \bigr). $$

Remark 1.1

Since $F_{p,2}^{s}(\mathbb {R}^{N})=H_{p}^{s}(\mathbb{R}^{N})$, Theorem 1.1 implies that the Cauchy problem (1.1) has a global solution with small initial data in the Sobolev space $H_{p}^{s}(\mathbb{R}^{3})$, $s>\frac{3}{p}$, $1< p<\infty$.

Throughout this paper, C stands for a constant which may vary from line to line. We shall sometimes use the notation $A \lesssim B$ instead of $A \leq CB$, and $A\approx B$ means that $A \lesssim B$ and $B\lesssim A$.

2 Preliminaries

In this section, we provide a characterization of the Triebel-Lizorkin space based on the Littlewood-Paley decomposition. We follow [13, 14, 18–20].

We start with the dyadic partition of unity. Choose two nonnegative radial functions $\chi, \varphi\in\mathcal{S}(\mathbb{R}^{N})$, supported, respectively, in the ball $\mathcal{B}=\{\xi\in\mathbb {R}^{N}, \vert \xi \vert \leq\frac{4}{3}\}$ and in the ring $\mathcal{C}=\{\xi\in\mathbb{R}^{N}, \frac{3}{4}\leq \vert \xi \vert \leq\frac{8}{3}\}$, such that

$$\begin{aligned}& \chi(\xi)+\sum_{j\geq0}\varphi\bigl(2^{-j} \xi\bigr)=1, \quad\xi\in\mathbb {R}^{N}, \end{aligned}$$

(2.1)

$$\begin{aligned}& \sum_{j\in\mathbb{Z}}\varphi\bigl(2^{-j}\xi \bigr)=1, \quad\xi\in\mathbb {R}^{N}\setminus\{0\}. \end{aligned}$$

(2.2)

Then for $u \in\mathcal{S}'(\mathbb{R}^{N})$, the homogeneous dyadic block $\dot{\Delta}_{j}$, and the nonhomogeneous dyadic block $\Delta_{j}$ are defined as follows:

$$\begin{aligned}& \dot{\Delta}_{j}u(x)=\varphi\bigl(2^{-j}D\bigr)u(x)= \mathcal{F}^{-1}\bigl(\varphi \bigl(2^{-j}\xi\bigr)\hat{u}( \xi)\bigr) (x),\quad \forall j \in\mathbb{Z}; \end{aligned}$$

(2.3)

$$\begin{aligned}& \Delta_{j}u(x)=0,\quad j \leq-2;\qquad \Delta_{-1}u=\chi(D)u, \qquad \Delta_{j}u(x)=\varphi\bigl(2^{-j}D\bigr)u,\quad j\geq0. \end{aligned}$$

(2.4)

The homogeneous low-frequency cut-off operator $\dot{S}_{j}$ is defined by

$$\begin{aligned} \dot{S}_{j}u=\chi\bigl(2^{-j}D\bigr)u,\quad j \in \mathbb{Z}. \end{aligned}$$

(2.5)

It is easily checked that

$$\begin{aligned}& \dot{\Delta}_{j}\dot{\Delta}_{k}u=0, \quad \vert j-k \vert \geq2. \end{aligned}$$

(2.6)

$$\begin{aligned}& \dot{\Delta}_{j}(\dot{S}_{k-1}u\dot{\Delta}_{k}u)=0,\quad \vert j-k\vert \geq5. \end{aligned}$$

(2.7)

Using the notations $\dot{\Delta}_{j}$ and $\dot{S}_{j}$, the usual product uv of two distributions u and v can be decomposed into three parts in terms of the paraproduct operators introduced by Bony [19].

Formally, we can write the homogeneous Bony paraproduct decomposition

$$ uv=T_{u}v + T_{v}u + R(u, v), $$

(2.8)

where

$$\begin{aligned} T_{u}v=\sum_{j \in\mathbb{Z}}\dot{S}_{j-1}u \dot{\Delta}_{j}v, \qquad R(u, v)=\sum_{j \in\mathbb{Z}} \sum_{\vert \nu \vert \leq 1}\dot{\Delta}_{j-\nu}u\dot{ \Delta}_{j}v. \end{aligned}$$

Let us now introduce the Triebel-Lizorkin spaces.

Definition 2.1

Let $s \in\mathbb{R}$, $p, q \in [1,\infty]$. The homogeneous Triebel-Lizorkin space $\dot{F}_{p,q}^{s}$ is defined by

$$\dot{F}_{p,q}^{s}\bigl(\mathbb{R}^{N}\bigr)=\bigl\{ u \in\mathcal{S}'\bigl(\mathbb{R}^{N}\bigr)/ \mathcal{P} \bigl(\mathbb{R}^{N}\bigr),\Vert u\Vert _{\dot {F}_{p,q}^{s}}< \infty \bigr\} , $$

here $\mathcal{S}'/ \mathcal{P}$ denotes the space of tempered distributions modulus polynomials and

$$\Vert u\Vert _{\dot{F}_{p,q}^{s}}= \bigl\Vert \bigl\Vert \bigl(2^{js}\dot { \Delta}_{j}u\bigr)_{j \in\mathbb{Z}}\bigr\Vert _{l^{q}} \bigr\Vert _{{L^{p}}(\mathbb{R}^{N})}. $$

The nonhomogeneous Triebel-Lizorkin space ${F}_{p,q}^{s}(\mathbb{R}^{N})$ is defined by

$$F_{p,q}^{s}\bigl(\mathbb{R}^{N}\bigr)=\bigl\{ u \in \mathcal{S}'\bigl(\mathbb{R}^{N}\bigr),\Vert u\Vert _{{F}_{p,q}^{s}}< \infty \bigr\} $$

with

$$\Vert u\Vert _{F_{p,q}^{s}}\bigl(\mathbb{R}^{N}\bigr)= \big\Vert \big\Vert \bigl(2^{js}\Delta_{j}u\bigr)_{j \geq-1} \big\Vert _{l^{q}} \big\Vert _{{L^{p}}(\mathbb{R}^{N})}. $$

Remark 2.1

We point out that if $s>0$, we have $F_{p,q}^{s}(\mathbb{R}^{N})=\dot{F}_{p,q}^{s}(\mathbb{R}^{N})\cap L^{p}(\mathbb{R}^{N})$, then by the definition of the nonhomogeneous Triebel-Lizorkin space, the Minkowski inequality, and the fact that $\Vert \Delta_{-1}u\Vert _{L^{p}}\leq C\Vert u\Vert _{L^{p}}$, we get

$$\begin{aligned} \Vert u\Vert _{F_{p,q}^{s}} \approx&\Vert u\Vert _{\dot{F}_{p,q}^{s}}+ \Vert u\Vert _{L^{p}} \\ \approx& \Vert u\Vert _{\dot{F}_{p,q}^{s}}+\Vert \Delta _{-1}u \Vert _{L^{p}}. \end{aligned}$$

Remark 2.2

(Chae [13])

If $s>\frac {N}{p}$, $F_{p,q}^{s}(\mathbb{R}^{N})\hookrightarrow L^{\infty}(\mathbb {R}^{N})$, $p,q \in[1,\infty]$.

The following space-time Triebel-Lizorkin space will play an important role in the proof of Theorem 1.1.

Definition 2.2

Let $s\in\mathbb{R}$, $p, q, r \in [1,\infty]$, $I\subset\mathbb{R}$ be an interval. The homogeneous space-time Triebel-Lizorkin space $\tilde{L}^{r}(I; \dot {F}_{p,q}^{s}(\mathbb{R}^{N}))$ is the set of all distributions satisfying

$$\Vert u\Vert _{\tilde{L}^{r}(I; \dot{F}_{p,q}^{s})}= \big\Vert \big\Vert \bigl(2^{js}\Vert\dot{ \Delta}_{j}u\Vert_{L_{t}^{r}(I)}\bigr)_{j \in \mathbb{Z}} \big\Vert _{l^{q}} \big\Vert _{L^{p}}< \infty. $$

We can also define the inhomogeneous space-time Triebel-Lizorkin space $\tilde{L}^{r}(I; F_{p,q}^{s}(\mathbb{R}^{N}))$. By Remark 2.1, if $s>0$,

$$\begin{aligned} \Vert u\Vert _{\tilde{L}^{r}(I; F_{p,q}^{s})} \approx& \Vert u\Vert _{\tilde{L}^{r}(I;\dot{F}_{p,q}^{s})}+ \Vert u\Vert _{L^{r}_{t}(I;L^{p})} \\ \approx& \Vert u\Vert _{\tilde{L}^{r}(I;\dot {F}_{p,q}^{s})}+\Vert \Delta_{-1}u \Vert _{L^{r}_{t}(I;L^{p})}. \end{aligned}$$

For simplicity, we use $\tilde{L}^{r}_{t}\dot{F}_{p,q}^{s}$, $\tilde{L}^{r}\dot {F}_{p,q}^{s}$ to denote $\tilde{L}^{r}(0, t; \dot{F}_{p,q}^{s})$ and $\tilde {L}^{r}(0, \infty; \dot{F}_{p,q}^{s})$, respectively. For a locally integrable function f, the Hardy-Littlewood maximal function $\mathcal {M}f$ is defined by

$$\mathcal{M}f(x)=\sup_{r>0}\frac{1}{\vert \mathcal{B}(x,r)\vert } \int_{\mathcal{B}(x,r)}\bigl\vert f(y)\bigr\vert \,dy, $$

where $\vert \mathcal{B}(x,r)\vert$ denotes the volume of the ball $\mathcal{B}(x,r)$ with center x and radius r.

The following vector-valued maximal inequality, which can be found in [21], plays a fundamental tool in the proof of product estimate.

Lemma 2.1

(Vector-valued maximal inequality)

Let $1 < p < \infty$, $1\leq q \leq\infty$, and $\{u_{j}\}_{j \in\mathbb{Z}}$ be a sequence of functions in $L^{p}(l^{q})$. Then we have

$$\big\Vert \Vert\mathcal{M}u_{j}\Vert_{l^{q}} \big\Vert _{L^{p}} \lesssim \big\Vert \Vert u_{j}\Vert_{l^{q}} \big\Vert _{L^{p}}. $$

The following lemma is proved by Stein [22].

Lemma 2.2

Let ϕ be an integrable function on $\mathbb{R}^{N}$ and the least decreasing radial majorant of ϕ be integrable, i.e.,

$$\int_{\mathbb{R}^{N}}\sup_{\vert y\vert \geq \vert x\vert }\bigl\vert \phi(y) \bigr\vert \,dx = A < \infty. $$

Then for any $u \in L^{p}(\mathbb{R}^{N})$, $1\leq p \leq\infty$, we have

$$\sup_{\epsilon> 0}\epsilon^{N}\bigl\vert \bigl(u\ast\phi( \epsilon\cdot )\bigr) (x)\bigr\vert \leq A\mathcal{M}u(x). $$

Lemma 2.3

(Frazier-Torres-Weiss [23])

The Calderón-Zygmund singular integral operator is bounded from the Triebel-Lizorkin space $F_{p,q}^{s}$ into itself.

Using Lemma 2.3, we can control u constantly by θ modulus multiplication by a constant in the space $F_{p,q}^{s}$.

Finally, let us recall the maximum principle.

Lemma 2.4

Let θ be the smooth solution to the Cauchy problem (1.1) with $\alpha\in(0,2)$. Then we have

$$ \bigl\Vert \theta(t)\bigr\Vert _{L^{p}}\leq\bigl\Vert \theta(0)\bigr\Vert _{L^{p}}, \quad 1\leq p\leq\infty. $$

(2.9)

Proof

Hmidi and Keraani [8] established a maximum principle (2.9) for the quasi-geostrophic equation and the result does not depend on the space dimension. Following the idea of proof in [8] we can prove that (2.9) holds also for the 3D incompressible porous media equation similarly. Here we omit it. □

Remark 2.3

We can get an explicit decay estimates of the $L^{p}$ norm in θ using the methods developed by Córdoba and Córdoba [24], however, the boundedness is enough for the proof of Theorem 1.1.

3 Proof of Theorem 1.1

In this section, we prove Theorem 1.1. We divide the proof into four steps.

Step 1. A priori estimates. Firstly, we rewrite (1.1) in the following integral form:

$$ \theta(t)= e^{-\nu t\Lambda}\theta_{0} - \int_{0}^{t}e^{-(t-s)\nu\Lambda }\nabla\cdot(u\theta) (s)\,ds, $$

(3.1)

where we have used the fact that $\nabla\cdot u=0$.

Then localizing (3.1) through the Fourier localization operator $\dot{\Delta}_{j}$ and using estimates in Lemma A.2 in the Appendix, we get

$$\begin{aligned} \bigl\vert \dot{\Delta}_{j}\theta(t,x)\bigr\vert \lesssim e^{-c\nu t2^{j}}\mathcal{M}(\dot{\Delta}_{j} \theta_{0}) (x)+ \int_{0}^{t}e^{-c\nu (t-s)2^{j}}2^{j} \mathcal{M}\bigl(\dot{\Delta}_{j}(u\theta)\bigr) (s,x)\,ds. \end{aligned}$$

(3.2)

Taking $L^{\infty}$, $L^{1}$ norm, respectively, with respect to t on both sides of (3.2) and using Young’s inequality give

$$\begin{aligned}& \bigl\Vert \dot{\Delta}_{j}\theta(\cdot,x)\bigr\Vert _{L_{t}^{\infty}}\lesssim\mathcal{M}(\dot{\Delta}_{j} \theta_{0}) (x) +2^{j}\bigl\Vert \mathcal{M}\bigl(\dot{ \Delta}_{j}(u\theta)\bigr) (\cdot,x)\bigr\Vert _{L_{t}^{1}}, \end{aligned}$$

(3.3)

$$\begin{aligned}& \bigl\Vert \dot{\Delta}_{j}\theta(\cdot,x)\bigr\Vert _{L_{t}^{1}}\lesssim \frac{1}{\nu}2^{-j}\mathcal{M}(\dot{ \Delta}_{j}\theta_{0}) (x) +\frac{1}{\nu}\bigl\Vert \mathcal{M}\bigl(\dot{\Delta}_{j}(u\theta)\bigr) (\cdot ,x)\bigr\Vert _{L_{t}^{1}}. \end{aligned}$$

(3.4)

Multiplying $2^{js}$ on both sides of (3.3), taking $l^{q}(\mathbb{Z})$ norm, then taking $L^{p}$ norm, we get

$$\begin{aligned} \Vert \theta \Vert _{\tilde{L}^{\infty}\dot {F}_{p,q}^{s}} \lesssim& \Vert \theta_{0} \Vert _{\dot {F}_{p,q}^{s}}+\Vert u\theta \Vert _{\tilde{L}^{1}\dot{F}_{p,q}^{s+1}}, \end{aligned}$$

due to the Lemma 2.1, using the product estimate in Lemma A.1 we obtain

$$\begin{aligned} \Vert \theta \Vert _{\tilde{L}^{\infty}\dot {F}_{p,q}^{s}} \lesssim& \Vert \theta_{0} \Vert _{\dot{F}_{p,q}^{s}} +\Vert u\Vert _{\tilde{L}^{\infty}_{t}L^{\infty}_{x}}\Vert \theta \Vert _{\tilde{L}^{1}_{t} \dot{F}_{p,q}^{s+1}}+\Vert \theta \Vert _{\tilde{L}^{\infty}_{t}L^{\infty}_{x}}\Vert u\Vert _{\tilde{L}^{1}_{t} \dot{F}_{p,q}^{s+1}} \\ \lesssim& \Vert \theta_{0}\Vert _{\dot{F}_{p,q}^{s}} +\Vert u \Vert _{\tilde{L}^{\infty}_{t}F_{p,q}^{s}}\Vert \theta \Vert _{\tilde{L}^{1}_{t} \dot{F}_{p,q}^{s+1}}+\Vert \theta \Vert _{\tilde{L}^{\infty}_{t}F_{p,q}^{s}}\Vert u\Vert _{\tilde{L}^{1}_{t} \dot{F}_{p,q}^{s+1}}, \end{aligned}$$

according to the property of the Calderón-Zygmund singular integral operator in Lemma 2.3 we get

$$\begin{aligned} \Vert \theta \Vert _{\tilde{L}^{\infty}_{t} \dot {F}_{p,q}^{s}}\lesssim \Vert \theta_{0}\Vert _{\dot {F}_{p,q}^{s}}+\Vert \theta \Vert _{\tilde{L}^{1}_{t}\dot {F}_{p,q}^{s+1}}\Vert \theta \Vert _{\tilde{L}^{\infty}_{t} F_{p,q}^{s}}. \end{aligned}$$

(3.5)

Similarly to (3.5), we get

$$ \nu \Vert \theta \Vert _{\tilde{L}^{1}_{t} \dot {F}_{p,q}^{s+1}}\lesssim \Vert \theta_{0}\Vert _{\dot {F}_{p,q}^{s}}+\Vert \theta \Vert _{\tilde{L}^{1}_{t}\dot{F}_{p,q}^{s+1}} \Vert \theta \Vert _{\tilde{L}^{\infty}_{t} F_{p,q}^{s}}. $$

(3.6)

Combining (3.5), (3.6), and using Lemma 2.4, we get

$$ \Vert \theta \Vert _{\tilde{L}^{\infty}_{t} F_{p,q}^{s}} + \nu \Vert \theta \Vert _{\tilde{L}^{1}_{t} \dot {F}_{p,q}^{s+1}}\lesssim \Vert \theta_{0}\Vert _{F_{p,q}^{s}}+ \Vert \theta \Vert _{\tilde{L}^{1}_{t}\dot {F}_{p,q}^{s+1}}\Vert \theta \Vert _{\tilde{L}^{\infty}_{t} F_{p,q}^{s}}. $$

(3.7)

Step 2. Approximate solutions and uniform estimates. We construct the following successive approximate sequence $\{\theta^{n}\}$:

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l}\partial_{t}\theta^{n+1}+u^{n}\cdot\nabla\theta^{n+1}+\nu \Lambda\theta^{n+1}=0,& x\in\mathbb{R}^{3}, t>0,\\ u^{n}=\mathcal{C}(\theta^{n})+\mathcal{S}(\theta^{n}),& x\in\mathbb{R}^{3}, t>0,\\ \operatorname{div}u^{n} = 0,\\ \theta^{n+1}(0,x)=S_{n+2}\theta_{0}, \end{array}\displaystyle \right . \end{aligned}$$

(3.8)

where $S_{j}$ ($j\geq0$) are low-frequency cut-off operators which are defined similarly by (2.5):

$$\begin{aligned} S_{j}u=\chi\bigl(2^{-j}D\bigr)u,\quad j \in\mathbb{N}\cup{0}. \end{aligned}$$

Setting $(\theta^{0},u^{0})=(0,0)$, and solving the linear system, we can find $\{\theta^{n}, u^{n}\}_{n\in\mathbb{N}}$ for all $n\in\mathbb{N}$. As in Step 1, we can deduce that

$$\begin{aligned}& \bigl\Vert \theta^{n+1}\bigr\Vert _{\tilde{L}^{\infty}_{t} F_{p,q}^{s}} + \nu\bigl\Vert \theta^{n+1}\bigr\Vert _{\tilde{L}^{1}_{t} \dot {F}_{p,q}^{s+1}} \\& \quad\leq C \bigl(\bigl\Vert \theta_{0}^{n+1}\bigr\Vert _{F_{p,q}^{s}}+\bigl\Vert \theta^{n}\bigr\Vert _{\tilde{L}^{1}_{t}\dot{F}_{p,q}^{s+1}} \bigl\Vert \theta^{n+1}\bigr\Vert _{\tilde{L}^{\infty}_{t} F_{p,q}^{s}} +\bigl\Vert \theta^{n}\bigr\Vert _{\tilde{L}^{\infty}_{t} F_{p,q}^{s}}\bigl\Vert \theta^{n+1}\bigr\Vert _{\tilde{L}^{1}_{t}\dot{F}_{p,q}^{s+1}} \bigr). \end{aligned}$$

(3.9)

If we take $\epsilon>0$, such that $\Vert \theta_{0}\Vert _{F_{p,q}^{s}}\leq\epsilon$, $\epsilon\leq\frac{\nu}{4C^{2}}$, then we claim that, for all $n\in\mathbb{N}$,

$$ \bigl\Vert \theta^{n}\bigr\Vert _{\tilde{L}^{\infty}F_{p,q}^{s}} + \nu\bigl\Vert \theta^{n}\bigr\Vert _{\tilde{L}^{1} \dot{F}_{p,q}^{s+1}}\leq2C\Vert \theta_{0}\Vert _{F_{p,q}^{s}}. $$

(3.10)

In fact, assume that

$$\bigl\Vert \theta^{n}\bigr\Vert _{\tilde{L}^{\infty}F_{p,q}^{s}} + \nu\bigl\Vert \theta^{n}\bigr\Vert _{\tilde{L}^{1} \dot{F}_{p,q}^{s+1}}\leq2C\Vert \theta_{0}\Vert _{F_{p,q}^{s}}. $$

It follows from (3.9) that

$$\begin{aligned}& \bigl\Vert \theta^{n+1}\bigr\Vert _{\tilde{L}^{\infty}_{t} F_{p,q}^{s}} + \nu\bigl\Vert \theta^{n+1}\bigr\Vert _{\tilde{L}^{1}_{t} \dot {F}_{p,q}^{s+1}} \\& \quad\leq C\Vert \theta_{0}\Vert _{F_{p,q}^{s}} + \frac{C}{\nu} \bigl(\bigl\Vert \theta^{n}\bigr\Vert _{\tilde{L}^{\infty}_{t} F_{p,q}^{s}}+\nu \bigl\Vert \theta^{n}\bigr\Vert _{\tilde{L}^{1}_{t} \dot{F}_{p,q}^{s+1}} \bigr) \bigl(\bigl\Vert \theta^{n+1}\bigr\Vert _{\tilde{L}^{\infty}_{t} F_{p,q}^{s}}+\nu\bigl\Vert \theta^{n+1}\bigr\Vert _{\tilde{L}^{1}_{t} \dot {F}_{p,q}^{s+1}} \bigr) \\& \quad\leq C\Vert \theta_{0}\Vert _{F_{p,q}^{s}}+\frac{1}{2} \bigl(\bigl\Vert \theta^{n+1}\bigr\Vert _{\tilde{L}^{\infty}F_{p,q}^{s}} + \nu\bigl\Vert \theta^{n+1}\bigr\Vert _{\tilde{L}^{1} \dot{F}_{p,q}^{s+1}} \bigr), \end{aligned}$$

which implies (3.10).

Step 3. Compactness arguments and existence. We will show that the sequence $\{\theta^{n}\}$ has a subsequence converging to a solution of the Cauchy problem (1.1) in $\mathcal{D}'(\mathbb{R}^{+}\times\mathbb{R}^{3})$. The proof is based on compactness arguments.

Firstly, we show that $\partial_{t}\theta^{n}$ is uniformly bounded in space-time Triebel-Lizorkin $L_{t}^{\infty}(F_{p,q}^{s-1})$. By (3.8), we have

$$\begin{aligned} \bigl\Vert \partial_{t}\theta^{n+1}\bigr\Vert _{F_{p,q}^{s-1}} \leq& \bigl\Vert \nabla\cdot\bigl(u^{n} \theta^{n+1}\bigr)\bigr\Vert _{F_{p,q}^{s-1}}+\nu \bigl\Vert \Lambda \theta^{n+1}\bigr\Vert _{F_{p,q}^{s-1}} \\ \leq& \bigl\Vert u^{n}\theta^{n+1}\bigr\Vert _{F_{p,q}^{s}}+\nu\bigl\Vert \theta^{n+1}\bigr\Vert _{F_{p,q}^{s}}, \end{aligned}$$

due to the fact that $F_{p,q}^{s}$ ($s>\frac{3}{p}$) is a Banach algebra and the property of the Calderón-Zygmund singular integral operator in Lemma 2.3,

$$\begin{aligned} \bigl\Vert \partial_{t}\theta^{n+1}\bigr\Vert _{F_{p,q}^{s-1}} \leq& \bigl\Vert u^{n}\bigr\Vert _{F_{p,q}^{s}}\bigl\Vert \theta^{n+1}\bigr\Vert _{F_{p,q}^{s}}+ \nu\bigl\Vert \theta^{n+1}\bigr\Vert _{F_{p,q}^{s}} \\ \leq& C\bigl\Vert \theta^{n}\bigr\Vert _{F_{p,q}^{s}}\bigl\Vert \theta ^{n+1}\bigr\Vert _{F_{p,q}^{s}}+\nu\bigl\Vert \theta^{n+1}\bigr\Vert _{F_{p,q}^{s}}. \end{aligned}$$

(3.11)

On the other hand, since $\tilde{L}^{\infty}F_{p,q}^{s} \hookrightarrow L^{\infty}F_{p,q}^{s}$, (3.10) and (3.11) imply that

$$\begin{aligned} \bigl\Vert \partial_{t}\theta^{n+1}\bigr\Vert _{L^{\infty}F_{p,q}^{s-1}}< \infty. \end{aligned}$$

(3.12)

Now let us turn to the proof of the existence. We note that for any $\psi\in\mathcal{S}(\mathbb{R}^{3})$, the map $u\longmapsto\psi u$ is compact from $H_{p}^{s}$ into $H_{p}^{t}$ for $s>t$, $p< \infty$, together with the Sobolev embedding

$$F_{p,q}^{s} \hookrightarrow F_{p,2}^{s-\epsilon}=H_{p}^{s-\epsilon } \hookrightarrow L^{p},\quad\forall s>\epsilon, \epsilon>0, $$

we thus get the map $u\longmapsto\psi u$ is compact from $F_{p,q}^{s}(\mathbb{R}^{3})$ into $L^{p}(\mathbb{R}^{3})$. Thus by the Lions-Aubin compactness theorem, we can conclude that there exist a subsequence $\{\theta^{n_{k}}\}$ and a function θ such that

$$\begin{aligned} \lim_{k \rightarrow\infty}\theta^{n_{k}}= \theta \quad \mbox{in } L_{\mathrm{loc}}^{P}\bigl(R^{+} \times\mathbb{R}^{3}\bigr). \end{aligned}$$

Moreover, the uniform estimate (3.10) allows us to conclude that $\theta(t,x)\in\tilde{L}^{\infty}F_{p,q}^{s} \cap \tilde{L}^{1}\dot {F}_{p,q}^{s+1}$, and

$$ \Vert \theta \Vert _{\tilde{L}^{\infty}F_{p,q}^{s}}+\nu \Vert \theta \Vert _{\tilde{L}^{1} \dot{F}_{p,q}^{s+1}}\leq2C\Vert \theta_{0}\Vert _{F_{p,q}^{s}}. $$

(3.13)

Finally, by a standard limit argument, we can prove that the limit function $\theta(t,x)$ satisfies the Cauchy problem (1.1) in the sense of distribution.

We still have to prove that $\theta(t,x)\in\mathcal{C}(0,\infty; F_{p,q}^{s})$. From the definition of the Triebel-Lizorkin space and the Minkowski inequality we have

$$\begin{aligned} \bigl\Vert \theta(t)-\theta\bigl(t'\bigr)\bigr\Vert _{F_{p,q}^{s}} \leq&\biggl\Vert \biggl(\sum_{j< N}2^{jsq} \bigl\vert \Delta_{j}\theta(t)-\Delta_{j}\theta \bigl(t'\bigr)\bigr\vert ^{q}\biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}} \\ &{}+2\biggl\Vert \biggl(\sum_{j\geq N}2^{jsq} \Vert \Delta_{j}\theta\Vert _{L_{t}^{\infty}}^{q} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}}, \quad\forall N\in\mathbb{N}. \end{aligned}$$

From (3.13), for $\epsilon>0$, there exists a number $N_{0}\in \mathbb{N}$ such that

$$\begin{aligned} \biggl\Vert \biggl(\sum_{j\geq N_{0}}2^{jsq} \Vert \Delta_{j}\theta\Vert _{L_{t}^{\infty}}^{q} \biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}}\leq\frac{\epsilon}{4}, \end{aligned}$$

while

$$\begin{aligned} \biggl\Vert \biggl(\sum_{j< {N_{0}}}2^{jsq} \bigl\vert \Delta_{j}\theta(t)-\Delta _{j}\theta \bigl(t'\bigr)\bigr\vert ^{q}\biggr)^{\frac{1}{q}}\biggr\Vert _{L^{p}} \lesssim& \bigl\vert t-t'\bigr\vert \cdot\biggl\Vert \biggl(\sum_{j< {N_{0}}}2^{jsq} \Vert \partial_{t}\Delta_{j}\theta \Vert_{L_{\mathbb {R}^{+}}^{\infty}}^{q}\biggr)^{\frac{1}{q}} \biggr\Vert _{L^{p}} \\ \lesssim&\bigl\vert t-t'\bigr\vert 2^{N_{0}}\Vert \partial_{t}\theta \Vert _{\tilde{L}^{\infty}F_{p,q}^{s-1}}. \end{aligned}$$

Equation (3.12) allows us to finish the proof of continuity in time, that is, $\theta(t,x)\in\mathcal{C}(0,\infty; F_{p,q}^{s})$.

Step 4. Uniqueness. Assume that $\theta'(t,x)\in\tilde{L}^{\infty}F_{p,q}^{s} \cap\tilde {L}^{1}\dot{F}_{p,q}^{s+1}$ is another solution of the Cauchy problem (1.1) with the same initial data $\theta_{0}(x)$.

Let $\delta\theta= \theta-\theta'$ and $\delta u=u-u'$, then $(\delta \theta,\delta u)$ satisfies the following Cauchy problem:

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l}\partial_{t} \delta\theta+ u \cdot\nabla\delta\theta +\nu\Lambda\delta\theta= -\delta u \cdot\nabla\theta',& x\in\mathbb {R}^{3}, t>0,\\ \operatorname{div} u=0,\\ \delta\theta(0,x)=0. \end{array}\displaystyle \right . \end{aligned}$$

(3.14)

Following the same procedure of estimate leading to (3.7), by Lemma A.1 in the Appendix and Lemma 2.3 we can obtain

$$\begin{aligned}& \Vert \delta\theta \Vert _{\tilde{L}^{\infty}_{T}\dot {F}_{p,q}^{s}}+\nu \Vert \delta \theta \Vert _{\tilde{L}^{1}_{T}\dot {F}_{p,q}^{s+1}} \\& \quad\lesssim \Vert u\cdot\delta\theta \Vert _{\tilde{L}^{1}_{T}\dot {F}_{p,q}^{s+1}} + \bigl\Vert \delta u\cdot\theta'\bigr\Vert _{\tilde {L}^{1}_{T}\dot{F}_{p,q}^{s+1}} \\& \quad\lesssim \Vert u\Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}\Vert \delta\theta \Vert _{\tilde{L}^{1}_{T}\dot{F}_{p,q}^{s+1}}+ \Vert \delta\theta \Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}\Vert u\Vert _{\tilde{L}^{1}_{T}\dot{F}_{p,q}^{s+1}} \\& \qquad{}+\Vert \delta u\Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}\bigl\Vert \theta' \bigr\Vert _{\tilde{L}^{1}_{T}\dot{F}_{p,q}^{s+1}}+ \bigl\Vert \theta'\bigr\Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}\Vert \delta u\Vert _{\tilde{L}^{1}_{T}\dot{F}_{p,q}^{s+1}} \\& \quad\lesssim \Vert \delta\theta \Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}\bigl(\Vert \theta \Vert _{\tilde{L}^{1}_{T}\dot {F}_{p,q}^{s+1}}+\bigl\Vert \theta'\bigr\Vert _{\tilde{L}^{1}_{T}\dot {F}_{p,q}^{s+1}}\bigr) \\& \qquad{}+\Vert \delta\theta \Vert _{\tilde{L}^{1}_{T}\dot {F}_{p,q}^{s+1}}\bigl(\Vert \theta \Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}+\bigl\Vert \theta'\bigr\Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}\bigr). \end{aligned}$$

(3.15)

In order to get the estimates in the inhomogeneous mixed time-space Triebel-Lizorkin space, it remains to estimate $\Vert \Delta _{-1}(\delta\theta)\Vert _{L_{T}^{\infty}L_{x}^{p}}$. To do this, we apply the operator $\Delta_{-1}$ on both sides of (3.14). Modifying slightly the proof of Proposition 6.2 in [8], together with the maximum principle, Bernstein’s inequality [19], and Hölder’s inequality yield

$$\begin{aligned} \bigl\Vert \Delta_{-1}(\delta\theta)\bigr\Vert _{L_{T}^{\infty}L_{x}^{p}} \leq & \int_{0}^{T}\bigl\Vert \Delta_{-1} \bigl(\delta u\cdot\nabla\theta'+u\cdot\nabla \delta\theta\bigr) \bigr\Vert _{L^{p}}\,dt \\ \leq& \int_{0}^{T}\bigl\Vert \nabla\cdot \Delta_{-1}\bigl(\theta'\delta u +u\delta \theta\bigr) \bigr\Vert _{L^{p}}\,dt \\ \lesssim& \int_{0}^{T}\bigl\Vert \theta' \delta u +u\delta\theta\bigr\Vert _{L^{p}}\,dt, \end{aligned}$$

thus the property of Calderón-Zygmund singular integral operator in Lemma 2.3 and the Sobolev embedding $F_{p,q}^{s}\hookrightarrow L^{p}$ ($s>0$) imply that

$$\begin{aligned} \bigl\Vert \Delta_{-1}(\delta\theta)\bigr\Vert _{L_{T}^{\infty}L_{x}^{p}} \lesssim& \int_{0}^{T}\bigl(\Vert u\Vert _{L_{x}^{\infty}} \Vert \delta\theta \Vert _{L^{p}}+\Vert \delta u\Vert _{L^{p}}\bigl\Vert \theta'\bigr\Vert _{L^{\infty}_{x}} \bigr)\,dt \\ \lesssim& T\bigl(\Vert \theta \Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}+\bigl\Vert \theta'\bigr\Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}\bigr)\Vert \delta\theta \Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}. \end{aligned}$$

(3.16)

Combining (3.15) and (3.16), we get

$$\begin{aligned} \Vert \delta\theta \Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}+\Vert \delta\theta \Vert _{\tilde{L}^{1}_{T}\dot {F}_{p,q}^{s+1}} \leq& C \bigl(\Vert \delta\theta \Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}+ \Vert \delta\theta \Vert _{\tilde{L}^{1}_{T}\dot {F}_{p,q}^{s+1}} \bigr) \\ &{} \cdot\bigl(\Vert \theta \Vert _{\tilde{L}^{1}_{T}\dot {F}_{p,q}^{s+1}}+(1+T) \bigl(\Vert \theta \Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}+\bigl\Vert \theta'\bigr\Vert _{\tilde{L}^{\infty}_{T} F_{p,q}^{s}}\bigr)+\bigl\Vert \theta'\bigr\Vert _{\tilde{L}^{1}_{T}\dot {F}_{p,q}^{s+1}} \bigr). \end{aligned}$$

Since $\theta(t,x)$, $\theta'(t,x)$ satisfy (3.13), if ϵ and T have been chosen small enough then it follows that $\delta\theta (t,x)=0$ on $[0,T]\times\mathbb{R}^{3}$. By a standard continuation argument, we can show that $\delta\theta(t,x)=0$ in $[0,\infty)\times \mathbb{R}^{3}$, i.e. $\theta(t,x)=\theta'(t,x)$.

This completes the proof of Theorem 1.1.

Appendix

In this section, we provide the product estimate in the space-time Triebel-Lizorkin spaces which is used in the proof of Theorem 1.1. Chen and Zhang [18] established the estimate by the Littlewood-Paley trichotomy decomposition. Here we give a proof by the Bony paraproduct technique.

Lemma A.1

(Product estimate)

Assume that $u,v \in \tilde{L}^{\infty}_{T}(F_{p,q}^{s}(\mathbb{R}^{N}))\cap\tilde{L}^{1}_{T}(\dot {F}_{p,q}^{s+1}(\mathbb{R}^{N}))$, $s>\frac{N}{p}$, $1< p<\infty$, $1\leq q \leq\infty$. Then

$$ \Vert uv\Vert _{\tilde{L}^{1}_{T}\dot{F}_{p,q}^{s+1}}\leq C \bigl(\Vert u\Vert _{\tilde{L}^{\infty}_{T}F_{p,q}^{s}}\Vert v\Vert _{\tilde{L}^{1}_{T}\dot{F}_{p,q}^{s+1}} +\Vert u\Vert _{\tilde{L}^{1}_{T}\dot{F}_{p,q}^{s+1}}\Vert v\Vert _{\tilde{L}^{\infty}_{T}F_{p,q}^{s}} \bigr), $$

(A.1)

here C is a constant only depending on s.

Proof

Using Bony’s paraproduct decomposition and the property of quasi-orthogonality (2.6)-(2.7), $\forall j \in \mathbb{Z}$, we have

$$\begin{aligned} \dot{\Delta}_{j}(uv) =&\sum _{\vert k-j\vert \leq4}\dot{\Delta }_{j}(\dot{S}_{k-1}u \dot{\Delta}_{k}v)+\sum_{\vert k-j\vert \leq 4}\dot{ \Delta}_{j} (\dot{S}_{k-1}v\dot{\Delta}_{k}u)+\sum _{k\geq j-3}\sum_{\vert \nu \vert \leq1}\dot{ \Delta}_{j}(\dot{\Delta}_{k-\nu }u\dot{\Delta}_{k}v) \\ =& I + \mathit{II} + \mathit{III}. \end{aligned}$$

(A.2)

Thus we have

$$\begin{aligned} \Vert uv\Vert _{\tilde{L}^{1}_{T}\dot{F}_{p,q}^{s+1}} \leq& \big\Vert \big\Vert \bigl(2^{j(s+1)}\Vert I\Vert _{L_{T}^{1}}\bigr)_{j \in\mathbb {Z}} \big\Vert _{l^{q}} \big\Vert _{L^{p}} +\cdots+ \big\Vert \big\Vert \bigl(2^{j(s+1)}\Vert \mathit{III}\Vert _{L_{T}^{1}}\bigr)_{j \in\mathbb{Z}} \big\Vert _{l^{q}} \big\Vert _{L^{p}} \\ =& I' + \mathit{II}' +\mathit{III}'. \end{aligned}$$

(A.3)

We shall estimate the above three terms, respectively.

For the first term $I'$ on the RHS of (A.3), by using the Minkowski inequality, the Young inequality, Lemma 2.1, and Lemma 2.2, we get

$$\begin{aligned} I' \leq& \biggl\Vert \biggl\Vert \biggl(2^{j(s+1)}\sum _{\vert k-j\vert \leq4}\bigl\Vert \mathcal{M}(\dot{S}_{k-1}u\dot{ \Delta}_{k}v)\bigr\Vert _{L_{T}^{1}}\biggr)_{j \in\mathbb{Z}} \biggr\Vert _{l^{q}} \biggr\Vert _{L^{p}} \\ \lesssim& \biggl\Vert \biggl\Vert \biggl(2^{j(s+1)}\sum_{\vert k-j\vert \leq4} \Vert u\Vert_{L_{T}^{\infty}L_{x}^{\infty}}\bigl\Vert \mathcal{M}(\dot {\Delta}_{k}v) \bigr\Vert _{L_{T}^{1}}\biggr)_{j \in\mathbb{Z}}\biggr\Vert _{l^{q}} \biggr\Vert _{L^{p}} \\ \lesssim& \Vert u\Vert_{L_{T}^{\infty}L_{x}^{\infty}} \biggl\Vert \biggl\Vert \biggl(\sum _{\vert k-j\vert \leq4}2^{(j-k)(s+1)}\cdot2^{k(s+1)}\bigl\Vert \mathcal{M}( \dot{\Delta}_{k}v)\bigr\Vert _{L_{T}^{1}}\biggr)_{j \in\mathbb{Z}}\biggr\Vert _{l^{q}} \biggr\Vert _{L^{p}} \\ \lesssim& \Vert u\Vert_{L_{T}^{\infty}L_{x}^{\infty}} \bigl\Vert \bigl\Vert \bigl(2^{j(s+1)}\bigl\Vert \mathcal{M}(\dot{\Delta}_{j}v)\bigr\Vert _{L_{T}^{1}}\bigr)_{j \in \mathbb{Z}} \bigr\Vert _{l^{q}} \bigr\Vert _{L^{p}} \\ \lesssim& \Vert u\Vert_{L_{x}^{\infty}L_{T}^{\infty}} \bigl\Vert \bigl\Vert \bigl(2^{j(s+1)}\Vert \dot{\Delta}_{j}v\Vert_{L_{T}^{1}}\bigr)_{j \in\mathbb{Z}}\bigr\Vert _{l^{q}} \bigr\Vert _{L^{p}} \\ \lesssim& \Vert u\Vert_{\tilde{L}^{\infty}_{T}F_{p,q}^{s}}\Vert v\Vert _{\tilde{L}^{1}_{T}\dot{F}_{p,q}^{s+1}} . \end{aligned}$$

(A.4)

Similarly, for the second term $\mathit{II}'$ on the RHS of (A.3), we have

$$ \mathit{II}'\lesssim \Vert u\Vert _{\tilde{L}^{1}_{T}\dot {F}_{p,q}^{s+1}}\Vert v\Vert _{\tilde{L}^{\infty}_{T}F_{p,q}^{s}}. $$

(A.5)

Now, we turn to estimate the third term $\mathit{III}'$ on the RHS of (A.3). From the Young inequality, Lemma 2.1, and Lemma 2.2, we have

$$\begin{aligned} \mathit{III}' \lesssim& \biggl\Vert \biggl\Vert \biggl(2^{j(s+1)} \sum_{k\geq j-3}\sum_{\vert \nu \vert \leq1} \bigl\Vert \mathcal{M}(\dot{\Delta }_{k-\nu}u\dot{\Delta}_{k}v) \bigr\Vert _{L_{T}^{1}} \biggr)_{j \in\mathbb {Z}}\biggr\Vert _{l^{q}} \biggr\Vert _{L^{p}} \\ \lesssim& \Vert u\Vert _{L_{T}^{\infty}L_{x}^{\infty}} \biggl\Vert \biggl\Vert \biggl(2^{j(s+1)} \sum_{k\geq j-3}\bigl\Vert \mathcal{M}(\dot{\Delta }_{k}v)\bigr\Vert _{L_{T}^{1}}\biggr)_{j \in\mathbb{Z}} \biggr\Vert _{l^{q}} \biggr\Vert _{L^{p}} \\ \lesssim& \Vert u\Vert _{L_{T}^{\infty}L_{x}^{\infty}} \biggl\Vert \biggl\Vert \biggl(2^{(j-k)(s+1)} \sum_{k\geq j-3}2^{k(s+1)}\Vert \dot{\Delta }_{k}v\Vert _{L_{T}^{1}}\biggr)_{j \in\mathbb{Z}} \biggr\Vert _{l^{q}} \biggr\Vert _{L^{p}} \\ \lesssim& \Vert u\Vert _{\tilde{L}^{\infty}_{T}F_{p,q}^{s}}\Vert v\Vert _{\tilde{L}^{1}_{T}\dot{F}_{p,q}^{s+1}}. \end{aligned}$$

(A.6)

Summing up (A.4)-(A.6), we obtain the product estimate (A.1). □

Next we give a decay estimate of the fractional heat semigroup $e^{-t\nu (-\Delta)^{\frac{\alpha}{2}}}$ on distributions with Fourier transforms supported in the annulus. The result is due to Chen and Zhang [18] in the 2D case and was used in a crucial way for the proof of Theorem 1.1. For convenience of the reader, we will give a short proof.

Lemma A.2

Let $\alpha\in(0,2]$, $j \in\mathbb{Z}$. We have the following decay estimates:

$$\begin{aligned}& \bigl\vert e^{-t\nu\Lambda^{\alpha}}\dot{\Delta}_{j}f(x)\bigr\vert \leq Ce^{-ct2^{j\alpha}}\mathcal{M}(\dot{\Delta}_{j}f) (x), \quad \forall x \in\mathbb{R}^{3}, t > 0, \end{aligned}$$

(A.7)

$$\begin{aligned}& \bigl\vert e^{-t\nu\Lambda^{\alpha}}\nabla\cdot\dot{\Delta }_{j}f(x)\bigr\vert \leq C2^{j}e^{-ct2^{j\alpha}}\mathcal{M}(\dot{\Delta }_{j}f) (x), \quad\forall x \in\mathbb{R}^{3}, t > 0, \end{aligned}$$

(A.8)

where C and c are positive constants independent of t, j.

Proof

We consider a function $\phi\in\mathcal{D}(\mathbb{R}^{3} \setminus\mathcal{B}(0,\frac{1}{4}))$, the value of which is identically 1 near the annulus $\mathcal{C} = \{\xi\in\mathbb{R}^{3}, \frac{3}{4}\leq \vert \xi \vert \leq\frac{8}{3}\}$. Denote

$$\begin{aligned} e^{-t\nu\Lambda^{\alpha}}\dot{\Delta}_{j}f(x)= \mathcal{F}^{-1}\bigl(e^{-t\nu \vert \xi \vert ^{\alpha}}\phi\bigl(2^{-j}\xi \bigr)\widehat{\dot{\Delta }_{j}f}(\xi)\bigr) (x)=: K_{j,t} \ast\dot{\Delta}_{j}f(x), \end{aligned}$$

(A.9)

where $K_{j,t}(x)= (2\pi)^{-3}\int_{\mathbb{R}^{3}}e^{-t\nu \vert \xi \vert ^{\alpha}}\phi(2^{-j}\xi)e^{ix\cdot\xi}\,d\xi$. Since

$$K_{j,t}(x)= (2\pi)^{-3}2^{3j} \int_{\mathbb{R}^{3}}e^{-t\nu2^{j\alpha }\vert \xi \vert ^{\alpha}}\phi(\xi)e^{i2^{j}x\cdot\xi}\,d\xi =:2^{3j}\widetilde{K}_{j,t}\bigl(2^{j}x\bigr), $$

where $\widetilde{K}_{j,t}(x)= (2\pi)^{-3}\int_{\mathbb{R}^{3}}e^{-t\nu 2^{j\alpha} \vert \xi \vert ^{\alpha}}\phi(\xi)e^{ix\cdot\xi}\,d\xi$, we have

$$\begin{aligned} \bigl\vert \widetilde{K}_{j,t}(x)\bigr\vert \lesssim \int_{\operatorname {supp}\phi}e^{-t\nu2^{j\alpha} \vert \xi \vert ^{\alpha}}\,d\xi \lesssim e^{-ct\nu2^{j\alpha}}. \end{aligned}$$

(A.10)

Next, we study the asymptotic behavior of $\widetilde{K}_{j,t}(x)$ for large x. Let $L(x,D)=\frac{x\cdot\nabla_{\xi}}{i \vert x\vert ^{2}}$, then $L(x,D)e^{ix\cdot\xi}=e^{ix\cdot\xi}$. Consequently, $L^{*}=L^{*}(x,D)=-\frac{x\cdot\nabla_{\xi}}{i \vert x\vert ^{2}}$, then for any $N_{*} \in\mathbb{N}$, we have

$$\begin{aligned} \widetilde{K}_{j,t}(x) =&(2\pi)^{-3} \int_{\mathbb{R}^{3}}e^{-t\nu 2^{j\alpha} \vert \xi \vert ^{\alpha}}\phi(\xi )L^{N_{*}} \bigl(e^{ix\cdot\xi}\bigr)\,d\xi \\ =&(2\pi)^{-3} \int_{\mathbb{R}^{3}}\bigl(L^{*}\bigr)^{N_{*}}\bigl(\phi( \xi)e^{-t\nu 2^{j\alpha} \vert \xi \vert ^{\alpha}}\bigr)e^{ix\cdot\xi}\,d\xi. \end{aligned}$$

(A.11)

The integrand in (A.11) can be majorized by

$$\vert x\vert ^{-N_{*}}\operatorname{max} \bigl(1, \bigl(t2^{j\alpha} \vert \xi \vert ^{\alpha-1}\bigr)^{N_{*}}, t2^{j\alpha} \vert \xi \vert ^{\alpha-N_{*}} \bigr)e^{-t\nu2^{j\alpha} \vert \xi \vert ^{\alpha}}\chi_{\operatorname{supp}\phi}. $$

Thus, we have

$$\begin{aligned} \bigl\vert \widetilde{K}_{j,t}(x)\bigr\vert \leq& C_{N_{*}}\vert x\vert ^{-N_{*}}e^{-ct\nu2^{j\alpha}} \int_{\operatorname{supp}\phi }\bigl(1+t2^{j\alpha} \vert \xi \vert ^{\alpha}\bigr)^{N_{*}}e^{\frac {-t\nu2^{j\alpha}}{2}\vert \xi \vert ^{\alpha}}\,d\xi \\ \leq& C_{N}\vert x\vert ^{-N_{*}}e^{-ct\nu2^{j\alpha}}. \end{aligned}$$

(A.12)

Combining (A.10) and (A.12), we deduce that

$$\bigl\vert K_{j,t}(x)\bigr\vert \leq C_{N_{*}}2^{3j}e^{-ct\nu 2^{j\alpha}} \bigl(1+\bigl\vert 2^{j}x\bigr\vert \bigr)^{-N_{*}}. $$

Now, we turn to the proof of the inequality (A.7). For $N_{*}>3$, we have

$$\begin{aligned}& \bigl\vert e^{-t\nu(-\Delta)^{\frac{\alpha}{2}}}\dot{\Delta }_{j}f(x) \bigr\vert \\& \quad =\biggl\vert \int_{\mathbb{R}^{3}}K_{j,t}(x-y)\dot {\Delta}_{j}f(y)\,dy \biggr\vert \\& \quad\leq C_{N_{*}}2^{3j}e^{-ct\nu2^{j\alpha}} \int_{\mathbb{R}^{3}}\bigl(1+\bigl\vert 2^{j}(x-y)\bigr\vert \bigr)^{-N_{*}}\bigl\vert \dot{\Delta}_{j}f(y)\bigr\vert \,dy \\& \quad\leq C_{N_{*}}2^{3j}e^{-ct\nu2^{j\alpha}} \biggl[ \int_{\vert x-y\vert \leq2^{-j}}\bigl\vert \dot{\Delta}_{j}f(y)\bigr\vert \,dy \\& \qquad{}+\sum_{l\geq0} \int_{2^{-j+l}\leq \vert x-y\vert \leq 2^{-j+l+1}}\bigl(1+\bigl\vert 2^{j}(x-y)\bigr\vert \bigr)^{-N_{*}}\bigl\vert \dot {\Delta}_{j}f(y)\bigr\vert \,dy \biggr] \\& \quad\leq C_{N_{*}}2^{3j}e^{-ct\nu2^{j\alpha}} \biggl[\mathcal{M}(\dot{ \Delta }_{j}f) (x)2^{-3j}+\sum_{l\geq0} \int_{2^{-j+l}\leq \vert x-y\vert \leq2^{-j+l+1}}2^{-lN_{*}}\bigl\vert \dot{ \Delta}_{j}f(y)\bigr\vert \,dy \biggr] \\& \quad\leq C_{N_{*}}2^{3j}e^{-ct\nu2^{j\alpha}} \biggl[\mathcal{M}(\dot{ \Delta }_{j}f) (x)2^{-3j}+\sum_{l\geq0}2^{-lN_{*}}2^{-3j+3l} \mathcal{M}(\dot {\Delta}_{j}f) (x) \biggr] \\& \quad\leq C_{N_{*}}e^{-ct\nu2^{j\alpha}}\mathcal{M}(\dot{\Delta}_{j}f) (x). \end{aligned}$$

(A.13)

Thus we obtain the desired estimate (A.7).

Following the same procedure of estimate leading to (A.7), we can prove the estimate (A.8) similarly. □