Global well-posedness for the 2-D inhomogeneous incompressible Navier-Stokes system with large initial data in critical spaces

TL;DR

This paper proves the global existence and uniqueness of solutions for the 2-D inhomogeneous incompressible Navier-Stokes equations with large initial data in critical Besov spaces, without smallness assumptions.

Contribution

It establishes the first global well-posedness result for large initial data in critical spaces for this system, extending previous results that required small data.

Findings

Global unique solvability without smallness condition

Initial data in critical Besov space suffices for well-posedness

Results apply to nearly energy space in 2-D

Abstract

Without any smallness assumption, we prove the global unique solvability of the 2-D incompressible inhomogeneous Navier-Stokes equations with initial data in the critical Besov space, which is almost the energy space in the sense that they have the same scaling in terms of this 2-D system.

Equations

$$\left\{\begin{array}[]{l}\displaystyle\partial_{t}\rho+\mbox{div}(\rho u)=0,\qquad(t,x)\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{+}\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{d},\\ \displaystyle\partial_{t}(\rho u)+\mbox{div}(\rho u\otimes u)-\mbox{div}(\mu\mathbb{D}(u))+\nabla\Pi=0,\\ \displaystyle\mbox{div}\,u=0,\\ \displaystyle\rho|_{t=0}=\rho_{0},\quad\rho u|_{t=0}=m_{0},\end{array}\right.$$

$$\left\{\begin{array}[]{l}\displaystyle\partial_{t}\rho+\mbox{div}(\rho u)=0,\qquad(t,x)\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{+}\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{d},\\ \displaystyle\rho\partial_{t}u+\rho(u\cdot\nabla)u-\Delta u+\nabla\Pi=0,\\ \displaystyle\mbox{div}\,u=0,\\ \displaystyle(\rho,u)|_{t=0}=(\rho_{0},u_{0}).\end{array}\right.$$

$$\quad\left\{\begin{array}[]{l}\displaystyle\partial_{t}a+u\cdot\nabla a=0,\qquad(t,x)\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{+}\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{d},\\ \displaystyle\partial_{t}u+u\cdot\nabla u+(1+a)(\nabla\Pi-\Delta\,u)=0,\\ \displaystyle\mbox{div}\,u=0,\\ \displaystyle(a,u)|_{t=0}=(a_{0},u_{0}).\end{array}\right.$$

$$\frac{(a,u{)}_{\ell }(t,x)\text{buildrel}def}{=(a({\ell }^2\cdot ,\ell \cdot ),\ell u({\ell }^2\cdot ,\ell \cdot ))}$$

$$m\le {\rho }_0\le M.$$

$$\frac{1}{\rho }-1\in C(R_{+};{\dot{B}}_{\frac{2}{\epsilon },1}^{\epsilon }(R^2))$$

$$u\in C(R_{+};{\dot{B}}_{2,1}^0(R^2))\cap L_{loc}^1(R_{+};{\dot{B}}_{2,1}^2(R^2))and$$

$${\partial }_{t}u,\nabla \Pi \in L_{loc}^1(R_{+};{\dot{B}}_{2,1}^0(R^2)).$$

$${\partial }_{t}u-(1+a)(\Delta u-\nabla \Pi )=f,$$

$${\partial }_{t}u-(1+{\dot{S}}_{m}a)(\Delta u-\nabla \Pi )=(a-{\dot{S}}_{m}a)(\Delta u-\nabla \Pi )+f,$$

$$\Vert [{\dot{\Delta }}_q,v\cdot \nabla ]u{\Vert }_{L^p}\lesssim c_{q,r}{2}^{-q\alpha }\Vert \nabla v{\Vert }_{L^{\infty }}\Vert u{\Vert }_{{\dot{B}}_{p,r}^{\alpha }}.$$

$$[\dot{\Delta}_{q},v\cdot\nabla]u=\dot{\Delta}_{q}\bigl{(}\partial_{j}{\mathcal{R}}(u,v^{j}))+\dot{\Delta}_{q}\bigl{(}T_{\partial_{j}u}v^{j}\bigr{)}-R(v^{j},\dot{\Delta}_{q}\partial_{j}u)-[T_{v^{j}},\dot{\Delta}_{q}]\partial_{j}u\buildrel\hbox{\footnotesize def}\over{=}\sum_{i=1}^{4}\mathcal{R}^{i}_{q}.$$

$$\begin{array}{rl}& \Vert {\mathcal{R}}_q^1{\Vert }_{L^p}\lesssim 2^q\sum _{k\ge q-3}\Vert {\dot{\Delta }}_{k}u{\tilde{\dot{\Delta }}}_k{v}^j{\Vert }_{L^p}\lesssim 2^q\sum _{k\ge q-3}\Vert {\dot{\Delta }}_{k}u{\Vert }_{L^p}\Vert {\tilde{\dot{\Delta }}}_k{v}^j{\Vert }_{L^{\infty }}\\ & \lesssim 2^q\Vert \nabla v{\Vert }_{L^{\infty }}\sum _{k\ge q-3}\Vert {\dot{\Delta }}_{k}u{\Vert }_{L^p}{2}^{-k}\lesssim c_{q,r}{2}^{-q\alpha }\Vert \nabla v{\Vert }_{L^{\infty }}\Vert u{\Vert }_{{\dot{B}}_{p,r}^{\alpha }}.\end{array}$$

$$\begin{array}{rl}& \Vert {\dot{S}}_{k-1}\nabla u{\Vert }_{L^p}\lesssim \sum _{\ell \le k-2}{2}^{\ell }\Vert {\dot{\Delta }}_{\ell }u{\Vert }_{L^p}\lesssim c_{k,r}{2}^{k(1-\alpha )}\Vert u{\Vert }_{{\dot{B}}_{p,r}^{\alpha }},\end{array}$$

$$\begin{array}{r}\Vert {\mathcal{R}}_q^2{\Vert }_{L^p}\lesssim c_{q,r}{2}^{-q\alpha }\Vert \nabla v{\Vert }_{L^{\infty }}\Vert u{\Vert }_{{\dot{B}}_{p,r}^{\alpha }}.\end{array}$$

$${\mathcal{R}}_q^3=-R(v^j,{\dot{\Delta }}_q{\partial }_{j}u)=-\sum _{k\ge q-2}{\dot{S}}_{k+2}{\dot{\Delta }}_q{\partial }_{j}u{\dot{\Delta }}_k{v}^j,$$

$$\begin{array}{rl}\Vert {\mathcal{R}}_q^3{\Vert }_{L^p}& \lesssim \Vert {\dot{\Delta }}_{q}u{\Vert }_{L^p}\sum _{k\ge q-2}{2}^{q-k}\Vert {\dot{\Delta }}_k\nabla v{\Vert }_{L^{\infty }}\\ & \lesssim c_{q,r}{2}^{-q\alpha }\Vert u{\Vert }_{{\dot{B}}_{p,r}^{\alpha }}\Vert \nabla v{\Vert }_{L^{\infty }}.\end{array}$$

$${\mathcal{R}}_q^4=[{\dot{\Delta }}_q,T_{v^j}]{\partial }_{j}u=\sum _{|k-q|\le 4}[{\dot{\Delta }}_q,S_{k-1}{v}^j]{\dot{\Delta }}_k{\partial }_{j}u,$$

$$\Vert [{\dot{S}}_{k-1}{u}^j,{\dot{\Delta }}_q]{\dot{\Delta }}_k{\partial }_{j}u{\Vert }_{L^p}$$

$$\lesssim 2^{k-q}\Vert \nabla v{\Vert }_{L^{\infty }}\Vert {\dot{\Delta }}_{k}u{\Vert }_{L^p},$$

$$\Vert {\mathcal{R}}_q^4{\Vert }_{L^p}\lesssim c_{q,r}{2}^{-q\alpha }\Vert u{\Vert }_{{\dot{B}}_{p,r}^{\alpha }}\Vert \nabla v{\Vert }_{L^{\infty }}.$$

$$\{\begin{array}{ll}& {\partial }_{t}a+u\cdot \nabla a=0,(t,x)\in [0,T]\times {\mathbb{R}}^2,\\ & a{|}_{t=0}=a_{0}x\in {\mathbb{R}}^2.\end{array}$$

$$\|a-\dot{S}_{k}a\|_{\widetilde{L}^{\infty}_{t}(\dot{B}^{\alpha}_{p,r})}\leq\Bigl{(}\sum_{q\geq k}2^{rq\alpha}\|\dot{\Delta}_{q}a_{0}\|_{L^{p}}^{r}\Bigr{)}^{\frac{1}{r}}+\|a_{0}\|_{\dot{B}^{\alpha}_{p,r}}(e^{CU(t)}-1)$$

$${\partial }_t{\dot{\Delta }}_{q}a+u\cdot \nabla {\dot{\Delta }}_{q}a=[u\cdot \nabla ,{\dot{\Delta }}_q]a$$

$$\Vert {\dot{\Delta }}_{q}a(t){\Vert }_{L^p}\le \Vert {\dot{\Delta }}_q{a}_0{\Vert }_{L^p}+{\int }_0^t\Vert [u\cdot \nabla ,{\dot{\Delta }}_q]a{\Vert }_{L^p}(\tau )d\tau .$$

$$\Vert [u\cdot \nabla ,{\dot{\Delta }}_q]a{\Vert }_{L^p}\lesssim c_{q,r}{2}^{-q\alpha }\Vert \nabla u{\Vert }_{L^{\infty }}\Vert a{\Vert }_{{\dot{B}}_{p,r}^{\alpha }},$$

$$\Vert a(t){\Vert }_{{\dot{B}}_{p,r}^{\alpha }}\le \Vert a_0{\Vert }_{{\dot{B}}_{p,r}^{\alpha }}+C{\int }_0^t\Vert \nabla u{\Vert }_{L_{\tau }^1(L^{\infty })}\Vert a(\tau ){\Vert }_{{\dot{B}}_{p,r}^{\alpha }}d\tau .$$

$$\Vert a(t){\Vert }_{{\dot{B}}_{p,r}^{\alpha }}\le C\Vert a_0{\Vert }_{{\dot{B}}_{p,r}^{\alpha }}{e}^{C\Vert \nabla u{\Vert }_{L_t^1(L^{\infty })}},$$

$$2^{q\alpha }\Vert {\dot{\Delta }}_{q}a(t){\Vert }_{L_t^{\infty }(L^p)}\le 2^{q\alpha }\Vert {\dot{\Delta }}_q{a}_0{\Vert }_{L^p}+C\Vert a_0{\Vert }_{{\dot{B}}_{p,r}^{\alpha }}{\int }_0^t{c}_{q,r}(\tau )\Vert \nabla u(\tau ){\Vert }_{L^{\infty }}{e}^{C\Vert \nabla u{\Vert }_{L_{\tau }^1(L^{\infty })}}d\tau .$$

$$\|a-\dot{S}_{k}a\|_{\widetilde{L}^{\infty}_{t}(\dot{B}^{\alpha}_{p,r})}\leq\Bigl{(}\sum_{q\geq k}2^{rq\alpha}\|\dot{\Delta}_{q}a_{0}\|_{L^{p}}^{r}\Bigr{)}^{\frac{1}{r}}+\|a_{0}\|_{\dot{B}^{\alpha}_{p,r}}\int_{0}^{t}C\,U^{\prime}(\tau)e^{C\,U(\tau)}\,d\tau,$$

$$\{\begin{array}{l}{\partial }_{t}u+v\cdot \nabla u-(1+a)(\Delta u-\nabla \Pi )=f,(t,x)\in R_{+}\times R^2,\\ \mathrm{div}u=0,\\ u_{|t=0}=u_0.\end{array}$$

$$\Vert u{\Vert }_{L_T^{\infty }(B_{2,\infty }^{-1})}+\Vert u{\Vert }_{{\tilde{L}}_T^1(B_{2,\infty }^1)}\le$$

Full text

Keywords: Inhomogeneous Navier-Stokes equations, Well-posedness, Critical spaces

AMS Subject Classification: 35Q30, 76D05

1 Introduction

In this paper, we consider the Cauchy problem of the following $2$-D incompressible inhomogeneous Navier-Stokes equations

[TABLE]

where $\rho,u=(u_{1},u_{2},...,u_{d})$ stand for the density and velocity of the fluid respectively, $d=2,\,3$, $\mathbb{D}(u)=\frac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i}),$ $\Pi$ is a scalar pressure function, and in general, the viscous coefficient $\mu(\rho)$ is a smooth, positive function on $[0,\infty).$ Such system describes a fluid which is obtained by mixing two miscible fluids that are incompressible and that have different densities. It may also describe a fluid containing a melted substance. One may check [19] for the detailed derivation.

When $\mu(\rho)$ is independent of $\rho,$ that is, $\mu$ is a positive constant (taking $\mu=1$ for simplicity), and $\rho_{0}$ is bounded away from [math], the system is rewritten as the form

[TABLE]

Kazhikov [5] proved the global existence of strong solutions to the system (1.2) for small smooth data in three space dimensions and all smooth data in two dimensions, also proved that the $d$-dimensional system (1.1) (with $d=2,3$) has at least one global weak solutions in the energy space. However, the uniqueness of both type weak solutions has not be solved. Considering the case of the bounded domain $\Omega$ with homogeneous Dirichlet boundary condition for the fluid velocity, Ladyvzenskaja and Solonnikov [18] first addressed the question of unique resolvability of (1.1). In particular, under the assumptions that $u_{0}\in W^{2-\frac{2}{p},p}(\Omega)$ $(p>2)$ is divergence free and vanishes on $\partial\Omega$ and that $\rho_{0}\in C^{1}(\Omega)$ is bounded away from zero, then they [18] proved global well-posedness of (1.1) in dimension $d=2$. Similar results were obtained by Danchin [12] in $\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}$ with initial data in the almost critical Sobolev spaces.

In general, DiPerna and Lions [16, 19] proved the global existence of weak solutions to (1.1) in energy space in any space dimensions. Yet the uniqueness and regularities of such weak solutions are big open questions even in two space dimension, as was mentioned by Lions in [19].

On the other hand, if the density $\rho$ is away from zero, we denote by $a\buildrel\hbox{\footnotesize def}\over{=}{\rho^{-1}}-1$, then the system (1.2) can be equivalently reformulated as

[TABLE]

Just as the classical Navier-Stokes system, which is the case when $a=0$ in (1.3), the system (1.3) also has a scaling-invariant transformation. Indeed if $(a,u)$ solves (1.3) with initial data $(a_{0},u_{0})$, then for $\forall\,\ell>0$,

[TABLE]

is also a solution of (1.3) with initial data $(a_{0}(\ell\cdot),\ell u_{0}(\ell\cdot))$. Some results about global existence and uniqueness of the solutions in critical spaces for small data were proved in [1, 3, 15]. Recently, we [2] first investigated the well-posedness of the 3-D incompressible inhomogeneous Navier-Stokes equation (1.2) with initial data $(a_{0},u_{0})$ in the critical spaces and without size restriction on $a_{0}.$

For the two-dimensional case, when the density and the velocity have more regularity, R. Danchin [12] proved the global well-poedness result of the system (1.2). More precisely, if $0<m\leq\rho_{0}\leq M,$ $\frac{1}{\rho_{0}}-1\in H^{1+\alpha}$ and $u_{0}\in H^{\beta}$ with $\alpha,\beta>0$, the system (1.2) is globally well-posed. Recently, some improvements of this result have been achieved. Paicu, Zhang, and Zhang [20] investigated the unique solvability of the global solution of the 2-D system (1.2) if $0<m\leq\rho_{0}\leq M$ and $u_{0}\in H^{s}$ with $s>0$, and the first author in the paper and Zhang [4] proved the global existence and uniqueness of the solution to the 2-D system (1.2) if $0<m\leq\rho_{0}\leq M,$ $\frac{1}{\rho_{0}}-1\in\dot{B}^{1}_{2,1}\cap\dot{B}^{\alpha}_{\infty,\infty}$ with $\alpha>0$ and $u_{0}\in\dot{B}^{0}_{2,1}$, and Danchin and Mucha [14] studied the existence and uniqueness of global solution to (1.2) if $0\leq\rho_{0}\leq M,$ $\int_{\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}}\rho_{0}>0$ and $u_{0}\in H^{1}.$

In summary, all the well-posedness results of the $2$-D system (1.2) obtained so far are under the additional assumption that the density or the velocity has more regularity compared to the critical spaces.

In this paper, we investigate the global well-posedness of the $2$-D inhomogeneous incompressible Navier-Stokes system (1.2) with large initial data in the critical space, which is almost the energy space in the sense that they have the same scaling in terms of the system (1.1) (see Remark 1.1).

The main theorem of the paper is stated as follows.

Theorem 1.1**.**

Let $m,M$ be two positive constants and $\varepsilon\in(0,1).$ Let $u_{0}\in\dot{B}^{0}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ be a solenoidal vector field and $\frac{1}{\rho_{0}}-1\in\dot{B}^{\varepsilon}_{\frac{2}{\varepsilon},1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ satisfy

[TABLE]

Then the system (1.2) has a global solution $(\rho,u,\nabla\Pi)$ with

[TABLE]

Furthermore, if, in addition, $\frac{1}{\rho_{0}}-1\in B^{1}_{2,1},$ then the solution is unique.

Remark 1.1**.**

Compared to the theorem of global weak solutions in the energy space ([5, 16, 19]), where $\rho_{0}-1\in L^{\infty}(\mathbb{R}^{2})$ and $u_{0}\in L^{2}(\mathbb{R}^{2})$, especially in the non-vacuum case, in the assumptions of Theorem 1.1, the initial density $\rho_{0}-1\in B^{1}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}))$ has the same scaling as $\rho_{0}-1\in L^{\infty}(\mathbb{R}^{2})$ in terms of the scaling-invariant transformation (1.4) of the system (1.2), and the initial velocity $u_{0}\in\dot{B}^{0}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ has the same scaling and regularity as $u_{0}\in L^{2}(\mathbb{R}^{2})$ in the energy space.

The proof of Theorem 1.1 is completed in Sections 2-4. We now present a summary of the principal difficulties we encounter in our analysis as well as a sketch of the key ideas used in our proof.

The first difficulty to the proof of Theorem 1.1 lies in the fact that when $a$ is not small, we can not use the classical arguments in [1, 3] to deal with the following linearized momentum equations of (1.3):

[TABLE]

Motivated by [11] and [2], for some large enough integer $m,$ we shall rewrite (1.7) as

[TABLE]

with $\dot{S}_{m}a$ being partial sum of $a$ defined in (A.1) in Appendix. Then the basic energy method can be used to solve (1.8) when we deal with the global existence of the solution to (1.3).

The other difficulty in the proof of Theorem 1.1 is how to deal with the uniqueness issue of the solution. In order to solve this problem, the crucial part is, roughly speaking, to control the $Lip$ norm of the velocity $u$, which will conserve all the regularities of the density and the velocity in the critical spaces, as well as the smallness of $a-\dot{S}_{m}a$ with $m$ being large enough.

Usually, if the density or the velocity has more regularity than the critical space, the losing estimates for transport equations and the theory of transport-diffusion equations [6] provide the boundness of the $Lip$ norm of the velocity, which will in turn close the estimates in the proof of global well-posedness of (1.2) (see [20, 4, 14]).

In our critical case, there is no more regularity of the density or the velocity to rescue their losing regularity when we solve the transport equation of the density or the transport-diffusion equations in terms of the velocity.

For this reason, we need first to get, at least in a small time interval, the $L^{1}([0,T];\dot{B}^{2}_{2,1})$ estimate for the velocity field (see Proposition 3.2), which relies on more elaborate application of Littlewood-Paley theory, as well as the basic energy and the estimate of $\|\nabla\Pi\|_{L^{1}_{t}(L^{2})}$. Based on this, together with Osgood’ lemma applied, we solve the uniqueness issue of the solution to (1.2) in the critical space.

The rest of the paper is organized as follows. In Section 2, we derive some qualitative and analytic properties of the flow, as well as the necessary commutator estimates. We prove the $L^{1}([0,T];\dot{B}^{2}_{2,1})$ estimate for the velocity field in Section 3. The proof of Theorem 1.1 is completed in Section 4. Finally, we recall some basic ingredients of Littlwood-Paley theory in Appendix.

Notations: Let $A,B$ be two operators, we denote $[A,B]=AB-BA,$ the commutator between $A$ and $B$. For $a\lesssim b$, we mean that there is a uniform constant $C,$ which may be different on different lines, such that $a\leq Cb$ and $C_{0}$ denotes a positive constant depending only on the initial data.

For $X$ a Banach space and $I$ an interval of $\mathop{\mathbb{R}\kern 0.0pt}\nolimits,$ we denote by ${\mathcal{C}}(I;\,X)$ the set of continuous functions on $I$ with values in $X,$ and by ${\mathcal{C}}_{b}(I;\,X)$ the subset of bounded functions of ${\mathcal{C}}(I;\,X).$ For $q\in[1,+\infty],$ the notation $L^{q}(I;\,X)$ stands for the set of measurable functions on $I$ with values in $X,$ such that $t\longmapsto\|f(t)\|_{X}$ belongs to $L^{q}(I).$

2 Preliminaries

In this section, we shall first derive the following commutator’s estimate which will be frequently used throughout the succeeding sections.

Lemma 2.1**.**

Let $\alpha\in(-1,1),$ $(p,r)\in[1,\infty]^{2},$ $u\in\dot{B}^{\alpha}_{p,r}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ and $\nabla v\in L^{\infty}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ with $\mathop{\rm div}\nolimits v=0.$ Then there holds

[TABLE]

Proof.

Thanks to Bony’s decomposition (A.4) and the divergence free condition of $v,$ we write

[TABLE]

For $\mathcal{R}^{1}_{q}=\sum_{k\geq q-3}\dot{\Delta}_{q}\partial_{j}(\dot{\Delta}_{k}u\widetilde{\dot{\Delta}}_{k}v^{j})$, it follows from Lemma A.1 and the condition $\alpha>-1$ that

[TABLE]

Notice that for $\alpha<1$,

[TABLE]

then for \mathcal{R}^{2}_{q}=\dot{\Delta}_{q}\bigl{(}T_{\partial_{j}u}v^{j}\bigr{)}=\sum_{|q-k|\leq 4}\dot{\Delta}_{q}(\dot{S}_{k-1}\partial_{j}u\dot{\Delta}_{k}v^{j}), we get

[TABLE]

Whereas thanks to the properties to the support of Fourier transform to $\dot{S}_{k+2}\dot{\Delta}_{q}\partial_{j}u,$ one has

[TABLE]

which implies that for $\alpha\in\mathbb{R}$

[TABLE]

For the last term in (2.2), we write

[TABLE]

which along with the classical estimate (see [6] for example)

[TABLE]

yields that for $\alpha\in\mathbb{R}$

[TABLE]

This achieves the proof of Lemma 2.1. ∎

Applying Lemma 2.1 to the transport equation, we have

Proposition 2.1**.**

Let $k\in{\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}$, $\alpha\in[0,1),$ $(p,r)\in[1,\infty]^{2},$ $a_{0}\in\dot{B}^{\alpha}_{p,r}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$, $\nabla u\in L^{1}_{T}(L^{\infty})$ with $\mathop{\rm div}\nolimits\,u=0$, and the function $a\in{\mathcal{C}}([0,{T}];\,\dot{B}^{\alpha}_{p,r}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})))$ solves

[TABLE]

Then there holds that for $\forall\,t\in(0,T]$

[TABLE]

with $U(t)=\|\nabla u\|_{L^{1}_{t}(L^{\infty})}$.

Proof.

Applying $\dot{\Delta}_{q}$ to (2.3) yields

[TABLE]

From the maximum principle we deduce

[TABLE]

While thanks to (2.1), we have

[TABLE]

which along with (2.6) implies that

[TABLE]

Hence, it follows from Gronwall’s inequality that

[TABLE]

from which, we use (2.6) again to deduce that

[TABLE]

Then, taking the $\ell^{r}$ norm in terms of $q\in\{q\geq k\}$ leads to

[TABLE]

which implies (2.4). ∎

To prove the uniqueness part of Theorem 1.1, we need the following Propositions.

Proposition 2.2**.**

Let $u_{0}\in B^{-1}_{2,\infty}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ and $v$ be a divergence free vector field satisfying $v\in L^{1}_{T}(B^{1}_{\infty,1})$. Let $f\in\widetilde{L}^{1}_{T}(B^{-1}_{2,\infty}),$ and $a\in\widetilde{L}^{\infty}_{T}(B^{2}_{2,1})$ with $1+a\geq c_{1}>0,$ we assume that $u\in L^{\infty}_{T}(B^{-1}_{2,\infty})\cap\widetilde{L}^{1}_{T}(B^{1}_{2,\infty})$ and $\nabla\Pi\in\widetilde{L}^{1}_{T}(B^{-1}_{2,\infty}),$ which solves

[TABLE]

Then there holds:

[TABLE]

Proof.

Applying $\Delta_{q}\mathbb{P}$ to (2.9), then a standard commutator process gives

[TABLE]

Thanks to the fact that $\mathop{\rm div}\nolimits u=\mathop{\rm div}\nolimits v=0$ and $1+a\geq c_{1},$ we get by taking the $L^{2}$ inner product of (2.11) with $\Delta_{q}u$ that

[TABLE]

We get, by using integration by parts and Lemma A.5 of [11], that for $0\leq q$

[TABLE]

This leads to

[TABLE]

First as $\mbox{div}\,v=0,$ we deduce by similar proof to inequality (3.12)

[TABLE]

Thanks to (A.4) in the inhomogeneous context, we write

[TABLE]

Whereas applying Lemma A.1 gives

[TABLE]

The same estimate holds for $T^{\prime}_{\Delta_{q}\nabla u}a.$ Note that

[TABLE]

this along with Lemma A.1 leads to

[TABLE]

Finally notice that

[TABLE]

one has

[TABLE]

As a consequence, we obtain

[TABLE]

On the other hand, it follows from (A.4) and Lemma A.1, that

[TABLE]

Similar, we have

[TABLE]

And a similar argument gives the same estimate for $\Delta_{q}\mathbb{P}T_{\nabla a}\Pi.$

Plugging (2.13 - 2.19) and into (2.12), we arrive at

[TABLE]

or by interpolation, we have

[TABLE]

Then by Young inequality Gronwall Lemma, we deduce

[TABLE]

which completes the proof of this Proposition. ∎

Proposition 2.3**.**

Let $a\in B^{1}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ such that $0<{\underline{b}}\leq 1+a\leq{\bar{b}}$, and

[TABLE]

for some sufficiently small positive constant $c$ and some integer $k\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits.$ Let $F\in B^{-1}_{2,\infty}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ and $\nabla\Pi\buildrel\hbox{\footnotesize def}\over{=}\mathcal{H}_{b}({F})\in B^{-1}_{2,\infty}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ solves

[TABLE]

Then there holds

[TABLE]

Proof.

We first deduce from (3.22) and ${\underline{b}}\leq 1+a$ that

[TABLE]

Motivated by [2, 13], we shall use a duality argument to prove (3.24). For the sake of simplicity, we just prove (3.24) for sufficiently smooth function $F$. In order to make the following computation rigorous, one has to use a density argument, which we omit here.

For this, we first estimate $\|\nabla\Pi\|_{B^{1}_{2,1}}$ under the assumption that $F\in B^{1}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$. Indeed, we write

[TABLE]

applying $\Delta_{q}$ to the above equation gives

[TABLE]

Taking the $L^{2}$ inner product of this equation with $\Delta_{q}\Pi,$ we obtain by a similar estimated of inequality (2.17)

[TABLE]

and by classical elliptic estimate, we have

[TABLE]

This along with (3.22) and interpolation, leads to

[TABLE]

Now we use a duality argument (see Corollary 6.2.8 [7]) to estimate $\|\nabla\Pi\|_{B^{-1}_{2,\infty}}$ in the case when $F\in B^{-1}_{2,\infty}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}).$ Notice that

[TABLE]

where \bigl{\langle}g,\ \nabla\Pi\bigr{\rangle} denotes the duality bracket between ${\mathcal{S}}^{\prime}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ and ${\mathcal{S}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}).$ Whereas (2.24) ensures that for any $g\in B^{1}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$

[TABLE]

has a unique solution $\nabla h_{g}\in B^{1}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ satisfying

[TABLE]

which along with (2.25) yields

[TABLE]

Hence, it follows that

[TABLE]

Whence thanks to (2.26), we obtain

[TABLE]

which completes the proof of this proposition. ∎

In order to get the uniqueness of the solution in the critical case in Theorem 1.1, we need to recall the following Osgood’s lemma [17].

Lemma 2.2** ([17], Osgood’s lemma).**

Let $f\geq 0$ be a measurable function, $\gamma$ be a locally integrable function and $\mu$ be a positive, continuous and nondecreasing function which verifies the following condition

[TABLE]

Let also $a$ be a positive real number and let $f$ satisfy the inequality

[TABLE]

Then if $a$ is equal to zero, the function $f$ vanishes. If $a$ is not zero, then we have

[TABLE]

3 The $L^{1}([0,T];\dot{B}^{2}_{2,1})$ estimate for the velocity

field

In this section, we want to get, at least in the small time interval, the $L^{1}([0,T];\dot{B}^{2}_{2,1})$ estimate for the velocity field, which plays a crucial role in the study of the uniqueness of the solution to (1.1). For this, we first investigate some a priori estimates about the basic energy and the pressure.

Proposition 3.1**.**

Let $\varepsilon\in(0,1),$ $a_{0}:=\frac{1}{\rho_{0}}-1\in\dot{B}^{0}_{\infty,1}\cap\dot{B}^{\varepsilon}_{\frac{2}{\varepsilon},\infty}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ and $u_{0}\in L^{2}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$, and (1.5) holds. Let $(\rho,u,\nabla\Pi)$ be a smooth enough solution of (1.1) on $[0,T^{\ast}[,$ then for any $t\in]0,T^{\ast}[$, there hold that

[TABLE]

and

[TABLE]

for any positive constant $\eta$.

Proof.

We first get, by using standard energy estimate to (1.2), that

[TABLE]

On the other hand, let $a\buildrel\hbox{\footnotesize def}\over{=}\frac{1}{\rho}-1,$ the system (1.2) can be equivalently reformulated as

[TABLE]

Applying the div operator to the momentum equation of (3.3) yields that

[TABLE]

for some large enough integer $k$ we shall rewrite the above equality as

[TABLE]

By taking $L^{2}$ inner product of the above equation with $\Pi,$ we get from the fact $1+a=\frac{1}{\rho}\geq\frac{1}{M}$ that

[TABLE]

Thanks to the product law in Besov spaces, one can see

[TABLE]

from which and (3.5), we infer

[TABLE]

From the maximum principle and Lemma 2.1, we deduce

[TABLE]

Thus by the Young inequality, we deduce that for any positive constant $\eta$

[TABLE]

Thanks to Proposition 2.1, we get (3.2), which completes the proof of Proposition 3.1. ∎

With Proposition 3.1 in hand, we are in a position to prove the following proposition about the estimate $\|u\|_{L^{1}_{t}(\dot{B}^{2}_{2,1})}$.

Proposition 3.2**.**

Let $\varepsilon\in(0,1),$ $u_{0}\in\dot{B}^{0}_{2,1}$ and $a_{0}\in\dot{B}^{\varepsilon}_{\frac{2}{\varepsilon},1}.$ Let $(a,u,\nabla\Pi)$ be a smooth enough solution of (3.3) on $[0,T^{\ast}[,$ then there is small positive time $T_{1}<T^{\ast}$ such that, for all $t\leq T_{1}$, there holds

[TABLE]

Proof.

Let $\mathbb{P}\buildrel\hbox{\footnotesize def}\over{=}I+\nabla(-\Delta)^{-1}\mbox{div}$ be the Leray projection operator. We get, by first dividing the momentum equation of (1.2) by $\rho$ and then applying the resulting equation by the operator $\mathbb{P},$ that

[TABLE]

Applying $\dot{\Delta}_{j}$ to the above equation and using a standard commutator’s process, we write

[TABLE]

Taking $L^{2}$ inner product of (3.8) with $\dot{\Delta}_{j}u,$ we obtain

[TABLE]

We get, by using integration by parts and Lemma A.1, that

[TABLE]

Thus, we deduce from (3.9) that

[TABLE]

where c=\bar{c}\bigl{/}M. This gives rise to

[TABLE]

As a consequence, by virtue of Definition A.2, we infer

[TABLE]

In what follows, we shall deal with term by term the right-hand side of (3.11). Firstly applying Bony’s decomposition (A.4) yields

[TABLE]

Hence, due to Lemma 1 of [22], we achieve

[TABLE]

which follows that

[TABLE]

While by using Lemma A.1, one has

[TABLE]

which along with the interpolation inequality $\|u\|_{\dot{B}^{1}_{2,1}}\lesssim\|u\|_{L^{2}}^{\frac{1}{2}}\|u\|_{\dot{B}^{2}_{2,1}}^{\frac{1}{2}},$ implies that

[TABLE]

for any positive constant $\eta$. The same estimates holds for $T^{\prime}_{\nabla\dot{\Delta}_{j}u}u.$ Thus we obtain

[TABLE]

Exactly along the same line to the proof of (3.12), we get, by applying Bony’s decomposition (A.4), that (for $a:=\frac{1}{\rho}-1$)

[TABLE]

It follows again from Lemma 1 of [22] that

[TABLE]

and

[TABLE]

The same estimate holds for \bigl{\|}T^{\prime}_{\dot{\Delta}_{j}\mathbb{P}f}a\bigr{\|}_{L^{2}}. Therefore, we obtain

[TABLE]

from which, we deduce that

[TABLE]

On the other hand, note that

[TABLE]

Thanks to the inequality (3.13), we deduce

[TABLE]

For $[\dot{\Delta}_{j}\mathbb{P},\dot{S}_{k}a]\Delta u$, Bony’s decomposition implies

[TABLE]

It follows again from Lemma 1 of [22] that

[TABLE]

for any positive constant $\gamma$, and

[TABLE]

The same estimate holds for \bigl{\|}T^{\prime}_{\dot{\Delta}_{j}\Delta u}\dot{S}_{m}a\bigr{\|}_{L^{2}}. Hence, we obtain

[TABLE]

which along with (3.15) ensures that

[TABLE]

Substituting (3.12), (3.14) and (3.16) into (3.11), and taking $\eta$ and $\gamma$ small enough, we obtain

[TABLE]

Thanks to (2.7) and (2.4), we get

[TABLE]

Therefore, thanks to (3.2) and (3.18), we obtain from (3.19) that

[TABLE]

By using (3.10) again, we deduce from the fact $\ell^{1}\hookrightarrow\ell^{2}$ that

[TABLE]

Similar to the proof of(3.12), we may get

[TABLE]

Hence, from (3.14), (3.16), and (3.2), it follows that

[TABLE]

Combining (3.19) and (3.20), we achieve

[TABLE]

Consequently, taking $\eta$ small enough, $k$ large enough, and then $t$ sufficiently small in (3.21), we deduce (3.7), which completes the proof of of Proposition 3.2. ∎

Based on this, we may get further estimate about the pressure.

Proposition 3.3**.**

Let $\varepsilon\in(0,1)$ and $a\in\dot{B}^{\varepsilon}_{\frac{2}{\varepsilon},1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ such that $0<{\underline{b}}\leq 1+a\leq{\bar{b}}$, and

[TABLE]

for some sufficiently small positive constant $c$ and some integer $k\in{\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}.$ Let $F\in\dot{B}^{0}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ and $\nabla\Pi\buildrel\hbox{\footnotesize def}\over{=}\mathcal{H}_{b}({F})\in\dot{B}^{0}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ solves

[TABLE]

Then there holds

[TABLE]

Proof.

We first deduce from (3.22) and ${\underline{b}}\leq 1+a$ that

[TABLE]

We rewrite (3.23) in the following form

[TABLE]

and applying $\dot{\Delta}_{q}$ to the above equation gives

[TABLE]

Taking the $L^{2}$ inner product of this equation with $\dot{\Delta}_{q}\Pi,$ we obtain by a similar estimate of (3.14) that

[TABLE]

which along with (3.22) leads to (3.24). This completes the proof of the Proposition. ∎

Consequently, we may derive the main result of the section as follows.

Proposition 3.4**.**

Under the assumptions of Proposition 3.2, there holds that for any $t\in[0,T_{1}]$

[TABLE]

where the constant $C_{0}$ depends only on the initial data $(\rho_{0},\,u_{0})$, and the positive time $T_{1}$ is determined by Proposition 3.2.

Proof.

Back to the proof of Proposition 3.2, according to (3.4), we have

[TABLE]

Combining Proposition 2.1 with Proposition 3.2, we know that the inequality (3.22) holds for any $t\in[0,T_{1}]$. Then applying Proposition 3.3 to (3.27) yields that

[TABLE]

and then

[TABLE]

Due to the product law in Besov spaces and the interpolation inequality, we get

[TABLE]

[TABLE]

Hence, thanks to Propositions 3.1, 3.2, and (2.7), we achieve

[TABLE]

Inserting (3.29) into (3.28) ensures that

[TABLE]

On the other hand, thanks to (3.10), (3.12), (3.14) and (3.16), we readily deduce that

[TABLE]

While from the momentum equations in (1.2) and (3.29), one has

[TABLE]

which follows (3.26). This ends the proof of Proposition 3.4. ∎

4 The proof of Theorem 1.1

We now turn to the proof of Theorem 1.1.

Proof of Theorem 1.1.

We divide the proof into two steps.

Step 1. Existence of strong solutions.

Given $\rho_{0}$ with $a_{0}:=\frac{1}{\rho_{0}}-1\in\dot{B}^{\varepsilon}_{\frac{2}{\varepsilon},1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ and satisfying (1.5), $u_{0}\in\dot{B}^{0}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}),$ we first mollify the initial data to be

[TABLE]

where $j_{n}(|x|)=n^{2}j(|x|/n)$ is the standard Friedrich’s mollifier. Then we deduce from the standard well-posedness theory of inhomogeneous Navier-Stokes system (see [12] for instance) that (1.2) has a unique solution $(\rho_{n},u_{n},\nabla\Pi_{n})$ on $[0,T^{\ast}_{n}[$ for some positive time $T^{\ast}_{n}.$ It is easy to observe from (4.1) that

[TABLE]

so, under the assumptions of Theorem 1.1, we infer from Lemma 3.2 that there holds

[TABLE]

for $t<T^{\ast}_{n}.$ Without loss of generality, we may assume $T^{\ast}_{n}$ is the lifespan of the approximate solutions $(\rho_{n},u_{n},\nabla\Pi_{n}).$ Then by virtue of [12] and (4.2), we conclude that $T_{n}^{\ast}\geq\,T_{1}$ for some positive constant $T_{1}$. With (4.2), we get, by using a standard compactness argument, that (1.2) has a solution $(\rho,u,\nabla\Pi)$ so that $a\in C([0,T_{1}[;\,\dot{B}^{\varepsilon}_{\frac{2}{\varepsilon},1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})),$ $u\in C([0,T_{1}[;\,\dot{B}^{0}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}))\cap L^{1}([0,T_{1}[;\,\dot{B}^{2}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})),$ $\partial_{t}u,\nabla\Pi\in L^{1}_{loc}([0,T_{1}[;\dot{B}^{0}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})).$ Furthermore, we can find some $t_{0}\in(0,T_{1})$ such that $u(t_{0})\in H^{1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$. Based on the initial data $a(t_{0})\in\dot{B}^{\varepsilon}_{\frac{2}{\varepsilon},1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ and $u(t_{0})\in H^{1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$, we may deduce the global existence of the solution to (1.2) according to [14, 20]. This completes the proof of the existence of the global solution to (1.2).

Step 2. Uniqueness of strong solutions.

Let’s first say from (2.7) that for any $t>0$

[TABLE]

Let $(\rho^{i},u^{i},\nabla\Pi^{i})$ with $i=1,2$ be two solutions of (1.2) which satisfies (1.6) and $\rho=\frac{1}{1+a}.$ We denote

[TABLE]

Then the system for $(\delta a,\delta u,\nabla\delta\Pi)$ reads

[TABLE]

where $\delta F$ is determined by

[TABLE]

For $\delta u,$ we first write the momentum equation of (4.4) as

[TABLE]

with

[TABLE]

Applying Proposition 2.2 to (4.5) yields that for $\forall\,0<t\leq T$

[TABLE]

On the other hand, applying div to the momentum equation of (4.4) yields

[TABLE]

with

[TABLE]

Thanks to Propositions 3.2 and 2.1, we get that, for any small constant $c_{0}>0$, there exist sufficiently large $j_{0}\in\mathbb{N}$ and a positive existence time $T_{1}$ such that $\|a^{2}-S_{j}a^{2}\|_{\widetilde{L}^{\infty}_{t}(B^{1}_{2,1})}<c_{0}$ for any $j\geq\,j_{0}$ and $t\in[0,T_{1}]$. Then applying Proposition 3.3 to (4.7) leads to

[TABLE]

While by Lemma A.1 and Bony’s decomposition, one can see

[TABLE]

[TABLE]

Similarly, one has

[TABLE]

and

[TABLE]

Thus, we obtain

[TABLE]

Toward the estimate of $\|H\|_{\widetilde{L}^{1}_{t}(B^{-1}_{2,\infty})}$, by using Bony’s decomposition again, we get

[TABLE]

Taking $k_{0}$ sufficiently large and $0<T_{2}(\leq T_{1})$ small enough, one may achieve, due to (4.3), that, for any $k\geq\,k_{0}$ and $t\in(0,T_{2}]$

[TABLE]

Therefore, thanks to (4.6), (4.8) and (4.9), we prove

[TABLE]

Taking $m_{0}$ sufficiently large and the positive time $T_{3}(\leq\,T_{2})$ small enough, we obtain that, for any $m\geq\,m_{0}$ and $t\in(0,T_{3}]$

[TABLE]

On the other hand, by a classical estimate of the transport equation, we get from the first equation in (4.4) that

[TABLE]

While thanks to the interpolation inequality, one may prove that

[TABLE]

As a result, we get

[TABLE]

then for $t$ small enough, we obtain

[TABLE]

Let $N$ be an arbitrary positive integer which will be determined later on, then

[TABLE]

Hence, due to Bernstein’s inequality, we infer

[TABLE]

If we choose $N$ such that

[TABLE]

then there holds

[TABLE]

and then

[TABLE]

Notice that for $\alpha\geq 0$ and $x\in(0,1],$ there holds

[TABLE]

Thus, plugging (4.13) into (4.12) leads to

[TABLE]

As $\int_{0}^{1}\frac{dx}{x(1-\ln x)}=+\infty,$ and $\|\Delta u^{1}\|_{L^{2}}+\|\nabla\Pi^{1}\|_{L^{2}}$ is locally integral in $\mathbb{R}^{+}$, then by Osgood’s lemma (Lemma 2.2), we obtain that $\delta u(t)=0$, which together with (4.11) and (4.8) implies that $\delta a(t)=\delta\nabla\Pi(t)=0$ for all $t\in[0,T]$ with $T$ small. Applying an inductive argument implies that $\delta u(t)=\delta a(t)=\delta\nabla\Pi(t)=0$ for all $t>0$.

Furthermore, applying (4.14) (up to a slight modification) to the system (1.1), we may readily prove that the solution $(a,\,u)\in C(\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{+};\,B^{1}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}))\times C(\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{+};\,\dot{B}^{0}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}))$ depends continuously on the initial data $(a_{0},\,u_{0})\in B^{1}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})\times\dot{B}^{0}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$. This completes the proof of Theorem 1.1. ∎

Appendix A Littlewood-Paley analysis

The proof of Theorem 1.1 requires a dyadic decomposition of the Fourier variables, which is called the Littlewood-Paley decomposition. Let us briefly explain how it may be built in the case $x\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}$ (see e.g. [6]). Let $\varphi$ be a smooth function supported in the ring $\mathcal{C}\buildrel\hbox{\footnotesize def}\over{=}\{\xi\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2},\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\}$ and $\chi(\xi)$ be a smooth function supported in the ball $\mathcal{B}\buildrel\hbox{\footnotesize def}\over{=}\{\xi\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2},\ |\xi|\leq\frac{4}{3}\}$ such that

[TABLE]

Now, for $u\in{\mathcal{S}}^{\prime}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}),$ we set

[TABLE]

We have the formal decomposition

[TABLE]

where ${\mathcal{P}}[\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}]$ is the set of polynomials (see [21]). Moreover, the Littlewood-Paley decomposition satisfies the property of almost orthogonality:

[TABLE]

We recall now the definition of nonhomogeneous and homogeneous Besov spaces from [23].

Definition A.1**.**

Let $(p,r)\in[1,+\infty]^{2},$ $s\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits$ and $u\in{\mathcal{S}}^{\prime}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}),$ we set

[TABLE]

with the usual modification if $r=\infty.$

• •

For $s\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits,$ we define B^{s}_{p,r}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})\buildrel\hbox{\footnotesize def}\over{=}\big{\{}u\in{\mathcal{S}}^{\prime}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})\;\big{|}\;\|u\|_{B^{s}_{p,r}}<\infty\big{\}}.

• •

For $s<\frac{2}{p}$ (or $s=\frac{2}{p}$ if $r=1$), we define \dot{B}^{s}_{p,r}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})\buildrel\hbox{\footnotesize def}\over{=}\big{\{}u\in{\mathcal{S}}^{\prime}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})\;\big{|}\;\|u\|_{\dot{B}^{s}_{p,r}}<\infty\big{\}}.

• •

If $k\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$ and $\frac{2}{p}+k\leq s<\frac{2}{p}+k+1$ (or $s=\frac{2}{p}+k+1$ if $r=1$), then $\dot{B}^{s}_{p,r}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ is defined as the subset of distributions $u\in{\mathcal{S}}^{\prime}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ such that $\partial^{\beta}u\in\dot{B}^{s-k}_{p,r}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ whenever $|\beta|=k.$

Remark A.1**.**

We point out that if $s>0$ then $B^{s}_{p,r}=\dot{B}^{s}_{p,r}\cap L^{p}$ and

[TABLE] 2. 2.

It is easy to verify that the homogeneous Besov space $\dot{B}^{s}_{2,2}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ (resp. ${B}^{s}_{2,2}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$) coincides with the classical homogeneous Sobolev space $\dot{H}^{s}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ (resp. $H^{s}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$) and $\dot{B}^{s}_{\infty,\infty}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ coincides with the classical homogeneous Hölder space $\dot{C}^{s}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ when $s$ is not positive integer, in case $s$ is a nonnegative integer, $\dot{B}^{s}_{\infty,\infty}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ coincides with the classical homogeneous Zygmund space $\dot{C}^{s}_{\ast}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}).$ 3. 3.

Let $s\in\mathbb{R},1\leq p,r\leq\infty$, and $u\in{\mathcal{S}}^{\prime}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}).$ Then $u$ belongs to $\dot{B}^{s}_{p,r}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$ if and only if there exists $\{c_{j,r}\}_{j\in\mathbb{Z}}$ such that $\|c_{j,r}\|_{\ell^{r}}=1$ and

[TABLE]

For the convenience of the reader, in what follows, we recall some basic facts on Littlewood-Paley theory, one may check [6, 23] for more details.

Lemma A.1**.**

Let ${\mathcal{B}}$ be a ball and ${\mathcal{C}}$ a ring of $\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}.$ A constant $C$ exists so that for any positive real number $\lambda,$ any non negative integer $k,$ any smooth homogeneous function $\sigma$ of degree $m,$ and any couple of real numbers $(a,\;b)$ with $b\geq a\geq 1,$ there hold

[TABLE]

In the rest of the paper, we shall frequently use homogeneous Bony’s decomposition [8]:

[TABLE]

where

[TABLE]

In order to obtain a better description of the regularizing effect of the transport-diffusion equation, we will use Chemin-Lerner type spaces from [9, 10].

Definition A.2**.**

Let $s\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits,$ $(r,\lambda,p)\in[1,\,+\infty]^{3},$ $T\in]0,\,+\infty]$. and $u\in{\mathcal{S}}^{\prime}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}),$ we set

[TABLE]

with the usual modification if $r=\infty.$

• •

For $s\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits,$ we define \widetilde{L}^{\lambda}_{T}(B^{s}_{p,r})\buildrel\hbox{\footnotesize def}\over{=}\big{\{}u\in{\mathcal{S}}^{\prime}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})\;\big{|}\;\|u\|_{\widetilde{L}^{\lambda}_{T}(B^{s}_{p,r})}<\infty\big{\}}.

• •

For $s\leq\frac{2}{p}$ (resp. $s\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits$), we define $\widetilde{L}^{\lambda}_{T}(\dot{B}^{s}_{p\,r}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}))$ as the completion of $C([0,T],{\mathcal{S}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}))$ by norm $\|\cdot\|_{\widetilde{L}^{\lambda}_{T}(\dot{B}^{s}_{p,r})}.$

In the particular case when $p=r=2,$ we denote $\widetilde{L}^{\lambda}_{T}(B^{s}_{2,2})$ (resp. $\widetilde{L}^{\lambda}_{T}(\dot{B}^{s}_{2,2})$) by $\widetilde{L}^{\lambda}_{T}({H}^{s})$ (resp. $\widetilde{L}^{\lambda}_{T}(\dot{H}^{s}).$ **

Remark A.2**.**

It is easy to observe that for $\theta\in[0,1],$ we have

[TABLE]

with $\frac{1}{\lambda}=\frac{\theta}{\lambda_{1}}+\frac{1-\theta}{\lambda_{2}}$ and $s=\theta s_{1}+(1-\theta)s_{2}.$ Moreover, Minkowski inequality implies that

[TABLE]

Acknowledgments. G. Gui is supported in part by the National Natural Science Foundation of China under Grants 11571279 and 11331005.