Optimal time-decay estimates for the compressible Navier-Stokes-Poisson equations without additional smallness assumptions

TL;DR

This paper establishes optimal time-decay estimates for solutions to the compressible Navier-Stokes-Poisson equations without requiring small initial data, using energy methods instead of spectral analysis.

Contribution

It introduces a new decay framework for critical Besov space norms and removes the smallness assumption on low-frequency initial data.

Findings

Proves large-time decay rates for global strong solutions

Develops a decay framework based on energy methods

Removes the smallness condition on initial data in decay analysis

Abstract

The present paper is dedicated to the large time asymptotic behavior of global strong solutions near constant equilibrium (away from vacuum) to the compressible Navier-Stokes-Poisson equations. Precisely, we present that under the same regularity assumptions as in \cite{SX2}, a \textit{different} time-decay framework of the $\dot{B}_{p,1}^{s}$ norm of the critical global solutions is established. The proof mainly depends on the pure energy argument \textit{without the spectral analysis}, which allows us to remove \textit{the usual smallness assumption of low frequencies of initial data}.

Equations

$$\left\{\begin{array}[]{l}\partial_{t}\varrho+\mathrm{div}\big{(}\varrho u\big{)}=0,\\[2.84526pt] \partial_{t}(\varrho u)+\mathrm{div}\big{(}\varrho u\otimes u\big{)}+\nabla P\big{(}\varrho\big{)}=\mathrm{div}\big{(}2\mu\,D(u)+\lambda\,\mathrm{div}\,u\,\mathrm{Id}\big{)}-\varrho\nabla\psi,\\[2.84526pt] -\Delta\psi=\varrho-\varrho_{\infty},\end{array}\right.$$

$$\mu >0and\lambda +2\mu >0.$$

$$(\varrho ,u){|}_{t=0}=({\varrho }_0(x),u_0(x)),x\in {\mathbb{R}}^d.$$

$$\varrho(t,x)\rightsquigarrow\varrho\big{(}l^{2}t,lx\big{)},\ \ u(t,x)\rightsquigarrow lu\big{(}l^{2}t,lx\big{)},\ \ \psi(t,x)\rightsquigarrow l^{-2}\psi\big{(}l^{2}t,lx\big{)}.$$

$$2\leq p\leq\min\big{(}4,2d/(d-2)\big{)}\ \hbox{and},\ \hbox{additionally},\ p\neq 4\ \hbox{if}\ d=2.$$

$${\mathcal{E}}_{p,0}\triangleq \Vert (a_0,\nabla u_0){\Vert }_{{\dot{B}}_{2,1}^{\frac{d}{2}-2}}^{\ell }+\Vert (\nabla a_0,u_0){\Vert }_{{\dot{B}}_{p,1}^{\frac{d}{p}-1}}^h\le c,$$

$$a^{\ell }\in \widetilde{{\mathcal{C}}_b}({\mathbb{R}}_{+};{\dot{B}}_{2,1}^{\frac{d}{2}-2})\cap L^1({\mathbb{R}}_{+};{\dot{B}}_{2,1}^{\frac{d}{2}}),u^{\ell }\in \widetilde{{\mathcal{C}}_b}({\mathbb{R}}_{+};{\dot{B}}_{2,1}^{\frac{d}{2}-1})\cap L^1({\mathbb{R}}_{+};{\dot{B}}_{2,1}^{\frac{d}{2}+1}),$$

$$a^h\in \widetilde{{\mathcal{C}}_b}({\mathbb{R}}_{+};{\dot{B}}_{p,1}^{\frac{d}{p}})\cap L^1({\mathbb{R}}_{+};{\dot{B}}_{p,1}^{\frac{d}{p}}),u^h\in \widetilde{{\mathcal{C}}_b}({\mathbb{R}}_{+};{\dot{B}}_{p,1}^{\frac{d}{p}-1})\cap L^1({\mathbb{R}}_{+};{\dot{B}}_{p,1}^{\frac{d}{p}+1}).$$

$${\mathcal{E}}_p(t)\le C{\mathcal{E}}_{p,0},$$

$${\mathcal{E}}_p(t)\triangleq \Vert (a,\nabla u){\Vert }_{{\tilde{L}}_t^{\infty }({\dot{B}}_{2,1}^{\frac{d}{2}-2})}^{\ell }+\Vert (a,\nabla u){\Vert }_{L_t^1({\dot{B}}_{2,1}^{\frac{d}{2}})}^{\ell }+\Vert (\nabla a,u){\Vert }_{{\tilde{L}}_t^{\infty }({\dot{B}}_{p,1}^{\frac{d}{p}-1})}^h+\Vert (a,\nabla u){\Vert }_{L_t^1({\dot{B}}_{p,1}^{\frac{d}{p}})}^h.$$

$$\Vert (\varrho -{\varrho }_{\infty })(t){\Vert }_{L^2}\le C(1+t{)}^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{2})}and\Vert u(t){\Vert }_{L^2}\le C(1+t{)}^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{2})+\frac{1}{2}}.$$

$$\Vert (\varrho -{\varrho }_{\infty })(t){\Vert }_{L^2}\le C(1+t{)}^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{2})-\frac{1}{2}}and\Vert u(t){\Vert }_{L^2}\le C(1+t{)}^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{2})}.$$

$$\left\{\begin{array}[]{l}\partial_{t}a+\mathrm{div}u=f,\\[2.84526pt] \partial_{t}u-\mathcal{A}u+\nabla a+\nabla(-\Delta)^{-1}a=g,\end{array}\right.$$

$$\displaystyle f\triangleq-\mbox{\rm div}\;\!\big{(}au\big{)},$$

$$\displaystyle g\triangleq-u\cdot\nabla u-I(a)\mathcal{A}u-k(a)\nabla a+\frac{1}{1+a}\mathrm{div}\big{(}2\widetilde{\mu}(a)\,D(u)+\widetilde{\lambda}(a)\mathrm{div}u\,{\rm Id}\,\big{)}$$

$$\displaystyle\mathcal{A}\triangleq\mu_{\infty}\Delta+\big{(}\lambda_{\infty}+\mu_{\infty}\big{)}\nabla\mbox{\rm div}\;\!\ \ \hbox{such that}\ \ 2\mu_{\infty}+\lambda_{\infty}=1\ \ \hbox{and}\ \ \mu_{\infty}>0$$

$$({\mu }_{\infty }\triangleq \mu (1)and{\lambda }_{\infty }\triangleq \lambda (1)),I(a)\triangleq \frac{a}{1+a},k(a)\triangleq \frac{P^{\prime }(1+a)}{1+a}-1,$$

$$\tilde{\mu }(a)\triangleq \mu (1+a)-\mu (1),\tilde{\lambda }(a)\triangleq \lambda (1+a)-\lambda (1).$$

$$\Vert (\varrho -1)(t){\Vert }_{{\dot{B}}_{p,1}^s}$$

$$\Vert u(t){\Vert }_{{\dot{B}}_{p,1}^s}$$

$$\Vert {\Lambda }^l(\varrho -1)(t){\Vert }_{L^r}\lesssim (1+t{)}^{-\frac{d}{2}(\frac{1}{2}-\frac{1}{r})-\frac{s_1+l+1}{2}}$$

$$\Vert {\Lambda }^{m}u(t){\Vert }_{L^r}\lesssim (1+t{)}^{-\frac{d}{2}(\frac{1}{2}-\frac{1}{r})-\frac{s_1+m}{2}}$$

$$\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\cdot)=1\ \ \hbox{in}\ \ \mathbb{R}^{d}\setminus\{0\}\ \ \hbox{and}\ \ \mathrm{Supp}\,\varphi\subset\big{\{}\xi\in\mathbb{R}^{d}:3/4\leq|\xi|\leq 8/3\big{\}}.$$

$$\dot{\Delta}_{j}f\triangleq\varphi(2^{-j}D)f=\mathcal{F}^{-1}\big{(}\varphi(2^{-j}\cdot)\mathcal{F}f\big{)}=2^{jd}h(2^{j}\cdot)\star f\ \ \hbox{with}\ \ h\triangleq\mathcal{F}^{-1}\varphi.$$

$$f=\sum _{j\in \mathbb{Z}}{\dot{\Delta }}_{j}f.$$

$$\underset{j\to -\infty }{\lim }\Vert {\dot{S}}_{j}f{\Vert }_{L^{\infty }}=0,$$

$$\dot{B}^{\sigma}_{p,r}\triangleq\big{\{}f\in S^{\prime}_{0}:\|f\|_{\dot{B}^{\sigma}_{p,r}}<+\infty\big{\}},$$

$$\Vert f{\Vert }_{{\dot{B}}_{p,r}^{\sigma }}\triangleq \Vert (2^{j\sigma }\Vert {\dot{\Delta }}_{j}f{\Vert }_{L^p}){\Vert }_{{\ell }^r(\mathbb{Z})}.$$

$$\widetilde{L}^{\theta}_{T}(\dot{B}^{\sigma}_{p,r})\triangleq\big{\{}f\in L^{\theta}(0,T;S^{\prime}_{0}):\|f\|_{\widetilde{L}^{\theta}_{T}(\dot{B}^{\sigma}_{p,r})}<+\infty\big{\}},$$

$$\Vert f{\Vert }_{{\tilde{L}}_T^{\theta }({\dot{B}}_{p,r}^{\sigma })}\triangleq \Vert (2^{j\sigma }\Vert {\dot{\Delta }}_{j}f{\Vert }_{L_T^{\theta }(L^p)}){\Vert }_{{\ell }^r(\mathbb{Z})}.$$

$$\widetilde{\mathcal{C}}_{b}(\mathbb{R_{+}};\dot{B}_{p,r}^{\sigma})\triangleq\big{\{}f\in\mathcal{C}(\mathbb{R_{+}};\dot{B}_{p,r}^{\sigma})\ \hbox{s.t}\ \big{\|}f\big{\|}_{\widetilde{L}^{\infty}(\dot{B}_{p,r}^{\sigma})}<+\infty\big{\}}.$$

$$\Vert f{\Vert }_{{\tilde{L}}_T^{\theta }({\dot{B}}_{p,r}^{\sigma })}\le \Vert f{\Vert }_{L_T^{\theta }({\dot{B}}_{p,r}^{\sigma })}\text{mbox}ifr\ge \theta ;\Vert f{\Vert }_{{\tilde{L}}_T^{\theta }({\dot{B}}_{p,r}^{\sigma })}\ge \Vert f{\Vert }_{L_T^{\theta }({\dot{B}}_{p,r}^{\sigma })}\text{mbox}ifr\le \theta .$$

$$\Vert f{\Vert }_{{\dot{B}}_{p,1}^{\sigma }}^{\ell }\triangleq \sum _{j\le j_0}{2}^{j\sigma }\Vert {\dot{\Delta }}_{j}f{\Vert }_{L^p}\text{mbox}and\Vert f{\Vert }_{{\dot{B}}_{p,1}^{\sigma }}^h\triangleq \sum _{j\ge j_0-1}{2}^{j\sigma }\Vert {\dot{\Delta }}_{j}f{\Vert }_{L^p},$$

Full text

1. Introduction

The barotropic compressible Navier-Stokes-Poisson equations can be written as

[TABLE]

which can be used to simulate the transport of charged particles in semiconductor devices under the influence of electric fields (see for example [22] for more explanations). Here, $u=u(t,x)\in\mathbb{R}^{d}$ (with $(t,x)\in[0,+\infty)\times\mathbb{R}^{d}$ ) and $\varrho=\varrho(t,x)$ stand for the velocity field of charged particles and the density, respectively. The function $\psi=\psi(t,x)$ denotes the electrostatic potential force. The barotropic assumption means that the pressure $P$ is given suitably smooth function of $\varrho$. The notation $D(u)\triangleq\frac{1}{2}(\nabla u+{}^{T}\!\nabla u)$ stands for the deformation tensor, and $\nabla$ and div  are the gradient and divergence operators with respect to the space variable. The Lamé coefficients $\lambda$ and $\mu$ (the bulk and shear viscosities) are density-dependent functions, which are supposed to be smooth functions of density and to satisfy

[TABLE]

The initial condition of (1.1) is prescribed by

[TABLE]

We focus on solutions that are close to some constant equilibrium $(\varrho_{\infty},0)$ with $\varrho_{\infty}>0$, at infinity.

As for the many systems arising from mathematical physics, it is well known that scaling invariance plays a fundamental role. The main aim of this paper is to investigate the asymptotic behavior of (strong) global solutions to Cauchy problem (1.1)-(1.3) in the critical Besov spaces, that is in functional spaces endowed with norms that is invariant for all $l>0$ by the following transformation

[TABLE]

Indeed, that definition of criticality corresponds to the scaling invariance of system (1.1) if neglecting the lower order pressure term and electronic field term. To the best of our knowledge, the idea is now classical and comes from the research of incompressible Navier-Stokes system, see [4, 13, 19] and references therein. It is worth mentioning that if there isn’t the Poisson potential, then system (1.1) reduces to the usual compressible Navier-Stokes system for baratropic fluids. There are some known results for compressible Navier-Stokes equations in the critical Besov spaces, see [5, 8, 10, 11, 12, 15, 23, 30, 31] and references therein. As regards the global existence for (1.1) in the critical Besov spaces, Hao and Li [17] established the global existence of small strong solution to (1.1) in the $L^{2}$ critical hybrid Besov space (in dimension $d\geq 3$). Subsequently, Zheng [32] extended the result of [17] to the $L^{p}$ framework. However, they only considered the non oscillation case $2\leq p<d$ with $d\geq 3$. In [9], Chikami and Ogawa performed Lagrangian approach and then used Banach fixed point theorem to establish the local existence and uniqueness of solutions, which allows one to handle dimension $d\geq 2$ in the general $L^{p}$ ($1<p<2d$) critical Besov spaces. Recently, Chikami and Danchin [6] constructed the unique global strong solution near constant equilibrium in the critical $L^{p}$ framework and in any dimension $d\geq 2$, where the cases of $2\leq p\leq d$ and $p>d$ are both involved. For simplicity, those physical coefficients $\mu$ and $\lambda$ are assumed to be constant. In fact, their results still hold true in case that $\mu$ and $\lambda$ depend smoothly on the density. For convenience, we state it as follows (the reader is also referred to [6]).

Theorem 1.1**.**

Let $d\geq 2$ and $p$ fulfill

[TABLE]

Assume that P^{\prime}\big{(}\varrho_{\infty}\big{)}>0 and that (1.2) is satisfied. There exists a small positive constant c=c\big{(}p,d,\mu,\lambda,P,\varrho_{\infty}\big{)} and a universal integer $j_{0}\in\mathbb{N}$ on $\mu$ and $\lambda$ such that if $a_{0}\triangleq(\varrho_{0}-\varrho_{\infty})\in\dot{B}_{p,1}^{\frac{d}{p}}$, if $u_{0}\in\dot{B}_{p,1}^{\frac{d}{p}-1}$ and if in addition $(a_{0}^{\ell},\nabla u_{0}^{\ell})\in\dot{B}_{2,1}^{\frac{d}{2}-2}$ with

[TABLE]

then the Cauchy problem (1.1)-(1.3) admits a unique global-in-time solution $(\varrho,u)$ with $\varrho=\varrho_{\infty}+a$ and $(a,u)$ in the space $X_{p}$ defined by:

[TABLE]

Furthermore, we get for some constant $C=C(p,d,\mu,\lambda,P,\varrho_{\infty})$,

[TABLE]

for any $t>0$, where

[TABLE]

Next, a natural question is to explore the large time asymptotic description of solutions in Theorem 1.1 in the general $L^{p}$ critical Besov spaces. Here, let us recall the spectral analysis briefly, which has been studied by Li, Matsumura & Zhang [20]. By the detailed analysis of the Fourier transform of the Green function for the linearized system of (1.1), it follows from [20] that if the initial perturbation $\left(\varrho_{0}-\varrho_{\infty},u_{0}\right)\in L^{q}(\mathbb{R}^{3})\cap H^{s}(\mathbb{R}^{3})$ with $q\in[1,2]$ and $s\geq 3$, the optimal time decay rates of $L^{q}$-$L^{2}$ type is

[TABLE]

It is observed that the electric field has significant effects on the large time behavior of the density and velocity field, which is a different ingrdient in comparison with the situation of compressible Navier-Stokes equations. Indeed, we see that the $L^{2}$ norm of the velocity $u$ grows in time at the rate $\langle t\rangle^{\frac{1}{2}}$ possibly if taking $q=2$ in (1.5), which seems to be a contradiction with the dissipativity of (1.1). Note that the energy structure of (1.1), it is natural to assume that $\nabla\psi_{0}\in L^{2}$. With the aid of the Poisson equation, we find that the condition $\nabla\psi_{0}\in L^{2}$ is equivalent to that $\Lambda^{-1}(\varrho_{0}-\varrho_{\infty})\in L^{2}$ with $\Lambda^{-1}f\triangleq\mathcal{F}^{-1}(|\xi|^{-1}\mathcal{F}f)$. Inspired by this, Wang [27] posed a stronger assumption, say $(\Lambda^{-1}(\varrho_{0}-\varrho_{\infty}),u_{0})\in L^{q}(\mathbb{R}^{3})$ with $q\in[1,2]$, which leads to the substituting decay in comparison with (1.5):

[TABLE]

In this sense, the electric field does not slow down but rather enhances the dissipation of density such that it enjoys additional half time-decay rate than velocity in time, with the relatively stronger assumption than [20]. So far there are lots of works dedicated to the convergence rates of solutions with high Sobolev regularity to (1.1)-(1.3), see [20, 21, 27, 28, 29] and references therein.

Let us also take a look at some important progress concerning (1.1)-(1.3) in the critical framework. Bie, Wang & Yao [2] developed the method of [12] so as to get the large-time asymptotic behavior of the constructed solutions in [32]. However, owing to the global-in-time results, they only consider the non oscillation case $2\leq p<d$ and $d\geq 3$. Recently, Chikami and Danchin [6] proposed the description of the time-decay which allows one to handle dimension $d\geq 2$ in the $L^{2}$ critical Besov space. In [26], the author & Xu developed a new regularity assumption of low frequencies, where the regularity $s_{1}$ belongs to $(1-\frac{d}{2},s_{0}]$ with $s_{0}\triangleq\frac{2d}{p}-\frac{d}{2}$, and established the sharp time-weighted inequality, which led to the optimal time-decay rates of strong solutions. These recent works (see for example [2, 6, 20, 21, 28, 29, 26] and references therein) mainly relies on the refined time-weighted energy approach in the Fourier semi-group framework, so the smallness assumption of low frequencies of initial data plays a key role. In this paper, we develop a pure energy method of [30] without the spectral analysis under the same regularity assumption as in [26], which enables one to remove the smallness of low frequencies of initial data, and then establish the optimal time-decay rates of solutions to (1.1)-(1.3) in the general $L^{p}$ critical Besov spaces. It is convenient to rewrite (1.1) as the nonlinear perturbation form of $(\varrho_{\infty},0)$, looking at the nonlinearities as source terms. To make a clearer introduction to our result, we assume that $\varrho_{\infty}=1$ and $P^{\prime}(1)=1$. Consequently, in terms of the new variables $(a,u)$, system (1.1) becomes

[TABLE]

where

[TABLE]

with

[TABLE]

Denote $\Lambda^{s}f\triangleq\mathcal{F}^{-1}(|\xi|^{s}\mathcal{F}f)$ for $s\in\mathbb{R}$. Now, we state main results as follows.

Theorem 1.2**.**

Let those assumptions of Theorem 1.1 hold and $(\varrho,u)$ be the corresponding global solution to (1.1). If in addition $a^{\ell}_{0}\in\dot{B}_{2,\infty}^{-s_{1}-1}$ and $u^{\ell}_{0}\in\dot{B}_{2,\infty}^{-s_{1}}$ ($1-\frac{d}{2}<s_{1}\leq s_{0}\ \ \hbox{with}\ \ s_{0}\triangleq\frac{2d}{p}-\frac{d}{2}$) such that $\|a_{0}\|^{\ell}_{\dot{B}_{2,\infty}^{-s_{1}-1}}$ and $\|u_{0}\|^{\ell}_{\dot{B}_{2,\infty}^{-s_{1}}}$ are bounded, then we have

[TABLE]

for all $t\geq 0$, where \widetilde{s}_{1}\triangleq s_{1}+d\big{(}\frac{1}{2}-\frac{1}{p}\big{)}.

Moreover, one has the density decay estimates of $\dot{B}^{-s_{1}-1}_{2,\infty}$-$L^{r}$ type and the velocity decay estimates of $\dot{B}^{-s_{1}}_{2,\infty}$-$L^{r}$ type.

Corollary 1.1**.**

Let those assumptions of Theorem 1.2 be satisfied. Then the corresponding solution $\varrho$ fulfills

[TABLE]

for $p\leq r\leq\infty$ and $l\in\mathbb{R}$ satisfying -\widetilde{s}_{1}-1<l+d\big{(}\frac{1}{p}-\frac{1}{r}\big{)}\leq\frac{d}{p}-2, and $u$ fulfills

[TABLE]

for $p\leq r\leq\infty$ and $m\in\mathbb{R}$ satisfying -\widetilde{s}_{1}<m+d\big{(}\frac{1}{p}-\frac{1}{r}\big{)}\leq\frac{d}{p}-1.

Remark 1.1**.**

In comparison with the recent work [2, 26, 6], the innovative ingredient is that the smallness of low frequencies is no longer needed in Theorem 1.2 and Corollary 1.1. Moreover, noting that condition (1.4) allows us to consider the case $p>d$ for which the velocity regularity exponent $\frac{d}{p}-1$ may be negative in physical dimensions $d=2,3$. The result thus applies to large highly oscillating initial velocity. Owing to the dissipative effect coming from the Poisson potential, we see that the $\dot{B}^{s}_{p,1}$ norm of density is faster at the rate $(1+t)^{-\frac{1}{2}}$ than that of velocity, which is an essential and different ingredient in comparison with compressible Navier-Stokes equations (see for example [12, 30, 31]).

Remark 1.2**.**

In our work [26], there is a little loss on time-decay rates due to using different Sobolev embeddings at low frequencies and high frequencies. For example, in the case of $s_{1}=s_{0}$, it was shown that the density (the velocity) itself decayed to equilibrium in $L^{p}$ norm with the rate of (1+t)^{-d\big{(}\frac{1}{p}-\frac{1}{4}\big{)}-\frac{1}{2}} ((1+t)^{-d\big{(}\frac{1}{p}-\frac{1}{4}\big{)}}) for $t\rightarrow\infty$. The present results avoid this minor flaw and indicate the optimal decay of the density (the velocity) as fast as $(1+t)^{-\frac{d}{2p}-\frac{1}{2}}$ ($(1+t)^{-\frac{d}{2p}}$), which are satisfactory.

Let us end this section by sketching the strategy for the proof of Theorem 1.2. Throughout the process, the main task of this paper is to establish a Lyapunov-type inequality in time for energy norms (see (5.5)) by means of the pure energy method of [31] (independent of spectral analysis). Indeed, in the Sobolev framework of high Sobolev regularity, the idea was initiated by Strain & Guo [24] for several Boltzmann type equations and then developed by Guo & Wang[14] for compressible Navier-Stokes equations and Wang [27] for compressible Navier-Stokes-Poisson equations. In the critical regularity framework however, there is a loss of one derivative of density (see the term $u\cdot\nabla a$) in the transport equation. Clearly, their method fails to take effect in critical spaces.

Due to the coupled Poisson potential, there is a nonlocal term $\nabla(-\Delta)^{-1}a$ (that is equivalent to $\widetilde{a}=\Lambda^{-1}a$) available in the velocity equation. By applying the hyperbolic energy approach, one can get the parabolic decay for the low frequencies of $(\widetilde{a},u)$. In the high-frequency regime, the nonlocal term $\nabla(-\Delta)^{-1}a$ is no longer effective and the large behaves of solutions the same as that of compressible Navier-Stoke system. Consequently, one can get the dissipative mechanism of (1.1)-(1.2). On the other hand, owing to the general regularity that $s_{1}$ belongs to the whole range $(1-\frac{d}{2},s_{0}]$ with $s_{0}=\frac{2d}{p}-\frac{d}{2}$, the low-frequency analysis is more complicated. Precisely, in light of low and high frequency decomposition, one splits the nonlinear term \big{(}\Lambda^{-1}f,g\big{)} into \big{(}\Lambda^{-1}f^{\ell},g^{\ell}\big{)} and \big{(}\Lambda^{-1}f^{h},g^{h}\big{)} (see section 4). To bound the nonlinear term \big{(}\Lambda^{-1}f^{\ell},g^{\ell}\big{)}, we develop some non classical Besov product estimates (4.4)-(4.5) to get desired result. For the term \big{(}\Lambda^{-1}f^{h},g^{h}\big{)}, we proceed differently the analysis depending on whether $2\leq p\leq d$ (non oscillation) and $p>d$ (oscillation). The former case depends on Besov product estimates (see (4.6)), while the later case (that is relevant in physical dimension $d=2,3$) lies in non-classical product estimates in Proposition 2.5. Combining these estimates leads to the evolution of Besov norm of solutions. Finally, nonlinear product estimates and real interpolations allow us to obtain the Lyapunov-type inequality (5.5) for energy norms.

The rest of the paper unfolds as follows: In section 2, we briefly recall Littlewood-Paley decomposition, Besov spaces and useful analysis tools. In section 3, we establish the low-frequency and high-frequency estimates of solutions. Section 4 is devoted to bounding the evolution of negative Besov norms, which plays the key role in deriving the Lyapunov-type inequality for energy norms. In the last section (Section 5), we show the proofs of Theorem 1.2 and Corollary 1.1.

2. Preliminary

Throughout the paper, $C>0$ stands for a generic “constant”. For brevity, we write $f\lesssim g$ instead of $f\leq Cg$. The notation $f\approx g$ means that $f\lesssim g$ and $g\lesssim f$. For any Banach space $X$ and $f,g\in X$, we agree that $\|(f,g)\|_{X}\triangleq\|f\|_{X}+\|g\|_{X}$. For all $T>0$ and $\theta\in[1,+\infty]$, we denote by $L_{T}^{\theta}(X)\triangleq L^{\theta}([0,T];X)$ the set of measurable functions $f:[0,T]\rightarrow X$ such that $t\mapsto\|f(t)\|_{X}$ is in $L^{\theta}(0,T)$.

2.1. Littlewood-Paley decomposition and Besov spaces

Let us recall Littlewood-Paley decomposition and Besov spaces for convenience. The reader is referred to Chap. 2 and Chap. 3 of [1] for more details. Choose a smooth radial non increasing function $\chi$ with \mathrm{Supp}\,\chi\subset B\big{(}0,\frac{4}{3}\big{)} and $\chi\equiv 1$ on B\big{(}0,\frac{3}{4}\big{)}. Set $\varphi(\xi)=\chi(\xi/2)-\chi(\xi)$. It is not difficult to check that

[TABLE]

The homogeneous dyadic blocks $\dot{\Delta}_{j}$ ($j\in\mathbb{Z}$) are defined by

[TABLE]

Consequently, one has the unit decomposition for any tempered distribution $f\in S^{\prime}(\mathbb{R}^{d})$

[TABLE]

As it holds only modulo polynomials, it is convenient to consider the subspace of those tempered distributions $f$ such that

[TABLE]

where $\dot{S}_{j}f$ stands for the low frequency cut-off defined by $\dot{S}_{j}f\triangleq\chi(2^{-j}D)f$. Indeed, if (2.2) is fulfilled, then (2.1) holds in $S^{\prime}(\mathbb{R}^{d})$. For convenience, we denote by $S^{\prime}_{0}(\mathbb{R}^{d})$ the subspace of tempered distributions satisfying (2.2).

In terms with Littlewood-Paley decomposition, Besov spaces are defined as follows.

Definition 2.1**.**

For $\sigma\in\mathbb{R}$ and $1\leq p,r\leq\infty,$ the homogeneous Besov spaces $\dot{B}^{\sigma}_{p,r}$ is defined by

[TABLE]

where

[TABLE]

On the other hand, a class of mixed space-time Besov spaces are also used, which was initiated by J.-Y. Chemin and N. Lerner [7] (see also [3] for the particular case of Sobolev spaces).

Definition 2.2**.**

For $T>0,\sigma\in\mathbb{R},1\leq r,\theta\leq\infty$, the homogeneous Chemin-Lerner space $\widetilde{L}^{\theta}_{T}(\dot{B}^{\sigma}_{p,r})$ is defined by

[TABLE]

where

[TABLE]

For notational simplicity, index $T$ is omitted if $T=+\infty$. We agree with the notation

[TABLE]

The Chemin-Lerner space $\widetilde{L}^{\theta}_{T}(\dot{B}^{\sigma}_{p,r})$ may be linked with the standard spaces $L_{T}^{\theta}(\dot{B}_{p,r}^{\sigma})$ by means of Minkowski’s inequality.

Remark 2.1**.**

It holds that

[TABLE]

Restricting the above norms (2.3) and (2.4) to the low or high frequencies parts of distributions will be fundamental in our method. For instance, let us fix some integer $j_{0}$ (the value of which will follow from the proof of the high-frequency estimates) and put111Note that for technical reasons, we need a small overlap between low and high frequencies.

[TABLE]

[TABLE]

2.2. Analysis tools in Besov spaces

Let us recall the classical properties (see [1]):

Proposition 2.1**.**

• •

Scaling invariance:* For any $\sigma\in\mathbb{R}$ and $(p,r)\in[1,\infty]^{2}$, there exists a constant $C=C(\sigma,p,r,d)$ such that for all $\lambda>0$ and $f\in\dot{B}_{p,r}^{\sigma}$, we have*

[TABLE]

• •

Completeness:* $\dot{B}^{\sigma}_{p,r}$ is a Banach space whenever $\sigma<\frac{d}{p}$ or $\sigma\leq\frac{d}{p}$ and $r=1$.*

• •

Action of Fourier multipliers:* If $F$ is a smooth homogeneous of degree $m$ function on $\mathbb{R}^{d}\backslash\{0\}$ then*

[TABLE]

Proposition 2.2**.**

Let $1\leq p,r_{1},r_{2},r\leq\infty$.

• •

Complex interpolation:* If $f\in\dot{B}_{p,r_{1}}^{\sigma_{1}}\cap\dot{B}_{p,r_{2}}^{\sigma_{2}}$ and $\sigma_{1}\neq\sigma_{2}$, then $f\in\dot{B}_{p,r}^{\theta\sigma_{1}+(1-\theta)\sigma_{2}}$ for all $\theta\in(0,1)$ and*

[TABLE]

with $\frac{1}{r}=\frac{\theta}{r_{1}}+\frac{1-\theta}{r_{2}}$.

• •

Real interpolation:* If $f\in\dot{B}_{p,r_{1}}^{\sigma_{1}}\cap\dot{B}_{p,r_{2}}^{\sigma_{2}}$ and $\sigma_{1}<\sigma_{2}$, then $f\in\dot{B}_{p,r}^{\theta\sigma_{1}+(1-\theta)\sigma_{2}}$ for all $\theta\in(0,1)$ and*

[TABLE]

The following embedding properties will be used frequently throughout this paper.

Proposition 2.3**.**

(Embedding for Besov spaces on $\mathbb{R}^{d}$)

• •

For any $p\in[1,\infty]$ we have the continuous embedding $\dot{B}^{0}_{p,1}\hookrightarrow L^{p}\hookrightarrow\dot{B}^{0}_{p,\infty}.$

• •

If $\sigma\in\mathbb{R}$, $1\leq p_{1}\leq p_{2}\leq\infty$ and $1\leq r_{1}\leq r_{2}\leq\infty,$ then $\dot{B}^{\sigma}_{p_{1},r_{1}}\hookrightarrow\dot{B}^{\sigma-d\,(\frac{1}{p_{1}}-\frac{1}{p_{2}})}_{p_{2},r_{2}}$.

• •

The space $\dot{B}^{\frac{d}{p}}_{p,1}$ is continuously embedded in the set of bounded continuous functions (going to zero at infinity if, additionally, $p<\infty$).

The product estimate in Besov spaces plays a fundamental role in bounding bilinear terms of (1.6) (see for example [1, 12, 25, 26, 30]).

Proposition 2.4**.**

Let $\sigma>0$ and $1\leq p,r\leq\infty$. Then $\dot{B}^{\sigma}_{p,r}\cap L^{\infty}$ is an algebra and

[TABLE]

Let the real numbers $\sigma_{1},$ $\sigma_{2},$ $p_{1}$ and $p_{2}$ fulfill

[TABLE]

Then we have

[TABLE]

Additionally, for exponents $\sigma>0$ and $1\leq p_{1},p_{2},q\leq\infty$ satisfying

[TABLE]

we have

[TABLE]

Proposition 2.4 is not enough to bound the low frequency part of some nonlinear terms in the proof of Theorem 1.2, so we need to the following non-classical product estimate (see [25, 26, 12, 30]).

Proposition 2.5**.**

Let $j_{0}\in\mathbb{Z},$ and denote $z^{\ell}\triangleq\dot{S}_{j_{0}}z$, $z^{h}\triangleq z-z^{\ell}$ and, for any $\sigma\in\mathbb{R}$,

[TABLE]

There exists a universal integer $N_{0}$ suc that for any $2\leq p\leq 4$ and $\sigma>0$, we have

[TABLE]

with $s_{0}\triangleq\frac{2d}{p}-\frac{d}{2}$ and $\frac{1}{p^{*}}\triangleq\frac{1}{2}-\frac{1}{p},$ and $C$ depending only on $j_{0}$, $d$ and $\sigma$.

System (1.6) also involves compositions of functions (through $I(a)$, $k(a)$, $\widetilde{\lambda}(a)$ and $\widetilde{\mu}(a)$) that are handled according to the following conclusion.

Proposition 2.6**.**

Let $F:\mathbb{R}\rightarrow\mathbb{R}$ be smooth with $F(0)=0$. For all $1\leq p,r\leq\infty$ and $\sigma>0$ we have $F(f)\in\dot{B}^{\sigma}_{p,r}\cap L^{\infty}$ for $f\in\dot{B}^{\sigma}_{p,r}\cap L^{\infty},$ and

[TABLE]

with $C$ depending only on $\|f\|_{L^{\infty}}$, $F^{\prime}$ (and higher derivatives), $\sigma$, $p$ and $d$.

In the case \sigma>-\min\big{(}\frac{d}{p},\frac{d}{p^{\prime}}\big{)} then $f\in\dot{B}^{\sigma}_{p,r}\cap\dot{B}^{\frac{d}{p}}_{p,1}$ implies that $F(f)\in\dot{B}^{\sigma}_{p,r}\cap\dot{B}^{\frac{d}{p}}_{p,1}$, and we have

[TABLE]

3. Low-frequency and high-frequency analysis

Let us establish a Lyapunov-type inequality for energy norms by means of a pure energy approach. For clarity, the proof is divided into two steps. In this section, we focus on the low-frequency and high-frequency estimates.

3.1. Low-frequency estimates

Let \Lambda^{s}z\triangleq\mathcal{F}^{-1}\big{(}|\xi|^{s}\mathcal{F}z\big{)} ($s\in\mathbb{R}$). Denote by $\Omega=\Lambda^{-1}\mathrm{curl}\,u$ the incompressible part of $u$ and by $\omega=\Lambda^{-1}\mathrm{div}\,u$ the compressible part of $u$. Then system (1.6) becomes

[TABLE]

with $h=\Lambda^{-1}\mathrm{div}\,g$ and $m=\Lambda^{-1}\mathrm{curl}\,g$.

Observe that the incompressible component $\Omega$ satisfies

[TABLE]

It is not difficult to infer that

[TABLE]

for $1\leq p\leq\infty$ and $t\geq 0$, where $z^{\ell}\triangleq\dot{S}_{j_{0}}z$. As pointed out in Introduction, at low frequencies, it is natural to consider the following system

[TABLE]

with $\widetilde{a}=\Lambda^{-1}a$.

Lemma 3.1**.**

let $j_{0}$ be some integer. Then it holds that for all $t\geq 0$

[TABLE]

Proof.

Let $z_{j}\triangleq\dot{\Delta}_{j}z$. We may apply the operator $\dot{\Delta}_{j}\dot{S}_{j_{0}}$ to (3.2). Taking advantage of the standard energy method, we get the following four equalities:

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Combining (3.4)-(3.7), one can conclude that

[TABLE]

with \mathcal{L}_{j}^{2}\triangleq 2\big{(}\|\widetilde{a}^{\ell}_{j}\|^{2}_{L^{2}}+\|\omega^{\ell}_{j}\|^{2}_{L^{2}}+\|\Lambda\widetilde{a}^{\ell}_{j}\|^{2}_{L^{2}}\big{)}+\|\Lambda^{2}\widetilde{a}^{\ell}_{j}\|^{2}_{L^{2}}-2(\omega^{\ell}_{j}|\Lambda^{2}\widetilde{a}^{\ell}_{j}). Thanks to the low-frequency cut-off, we get from Young inequality that $\mathcal{L}_{j}^{2}\thickapprox\|(\widetilde{a}^{\ell}_{j},\Lambda\widetilde{a}^{\ell}_{j},\Lambda^{2}\widetilde{a}^{\ell}_{j},\omega^{\ell}_{j})\|^{2}_{L^{2}}\thickapprox\|(\Lambda\widetilde{a}^{\ell}_{j},\widetilde{a}^{\ell}_{j},\omega^{\ell})\|^{2}_{L^{2}}\thickapprox\|(\widetilde{a}^{\ell}_{j},\omega^{\ell})\|^{2}_{L^{2}}$. Consequently, it follows that

[TABLE]

which leads to

[TABLE]

Therefore, multiplying both sides by 2^{j\big{(}\frac{d}{2}-1\big{)}} and summing up on $j\in\mathbb{Z}$ yield

[TABLE]

Hence, (3.3) is followed by (3.1) and (3.8) directly. ∎

3.2. High-frequency estimates

In the high-frequency regime, the nonlocal term $\nabla(-\Delta)^{-1}a$ no longer effective and the Green function behaves the same as that of compressible Navier–Stokesequations. Consequently, we apply the $L^{p}$ energy method and then establish the high-frequency estimates (see [31] for more details).

Lemma 3.2**.**

let $j_{0}$ be chosen suitably large. It holds that for all $t\geq 0$

[TABLE]

where

[TABLE]

4. The evolution of negative Besov norm

This section is devoted to the evolution of Besov norms at low frequencies, which plays the key role in deriving the Lyapunov-type inequality for energy norms.

Lemma 4.1**.**

Let $1-\frac{d}{2}<s_{1}\leq s_{0}$ and $p$ satisfy (1.4). It hold that:

[TABLE]

where

[TABLE]

Proof.

First of all, let us keep in mind that the global solution $(a,u)$ given by Theorem 1.1 satisfies

[TABLE]

It follows from (3.4) and (3.7) that

[TABLE]

By performing a routine procedure, we can conclude that

[TABLE]

Next, let us handle the nonlinear norm $\|(\Lambda^{-1}f,g)\|^{\ell}_{\dot{B}^{-s_{1}}_{2,\infty}}$. To do this, it is suitable to decompose $\Lambda^{-1}f$ and $g$ according to low-frequency and high-frequency parts as follows:

[TABLE]

with

[TABLE]

and

[TABLE]

with

[TABLE]

where

[TABLE]

and

[TABLE]

As shown by [31], we can get the following two estimates

[TABLE]

In order to finish the proof of lemma 4.1, it suffices to bound those “new” nonlinear terms, which are different terms in comparison with compressible Navier-Stokes equations. For those terms of $\Lambda^{-1}f$ and $g$ with $a^{\ell}$ or $u^{\ell}$, we shall use the following two inequalities

[TABLE]

for $1-\frac{d}{2}<s_{1}\leq s_{0}$. Indeed, the proofs of (4.4)-(4.5) may be found in [31]. For the term with $\Lambda^{-1}\mathrm{div}(au^{\ell})$, we note that, owing to (4.4),

[TABLE]

where $\Lambda^{-1}\mathrm{div}$ is an homogeneous Fourier multiplier of degree [math]. Let us next look at the term involving $k(a)\nabla a^{\ell}$. Thanks to (4.4) and Proposition 2.6, we arrive at

[TABLE]

Similarly, we have

[TABLE]

To deal with the term with $g_{4}(a,u^{\ell})$, it follows from the fact $s_{1}\leq s_{1}+\frac{d}{2}-\frac{d}{p}$, (4.5) and Proposition 2.6 that

[TABLE]

For those “new” terms in $\Lambda^{-1}f^{h}$ and $g^{h}$, precisely,

[TABLE]

we shall proceed these calculations differently depending on whether $2\leq p\leq d$ or $p>d$. For the case $2\leq p\leq d$, we shall take advantage of the following inequality

[TABLE]

which has been shown by [26]. With the aid of (4.6), we get

[TABLE]

To bound the term corresponding to $k(a)\nabla a^{h}$, we can use (4.6), (4.2) and Proposition 2.6 with $\frac{d}{p}-1>-\frac{d}{p}$ $(p<d<2d)$, and get

[TABLE]

Similarly, we arrive at

[TABLE]

For the term containing $g_{4}(a,u^{h})$, we write that, due to (4.6) and Proposition 2.6,

[TABLE]

In what follows, we consider the oscillation case $p>d$. Using the fact that $s_{1}\leq s_{0}$ and applying (2.5) with $\sigma=1-\frac{d}{p}>0$ yield

[TABLE]

Using the embeddings $\dot{B}^{\frac{d}{p}}_{2,1}\hookrightarrow L^{p^{*}}$, $\dot{B}^{\frac{d}{2}-1}_{2,1}\hookrightarrow\dot{B}^{\frac{d}{p}-1}_{p,1}$ and the relations $\frac{d}{2}-1\leq\frac{d}{p}$ and $\frac{d}{p}-1<1-\frac{d}{p}<\frac{d}{p}$ for $p>d$ satisfying (1.4), we arrive at

[TABLE]

Taking $F=a$ and $G=u$ in (4.8) yields

[TABLE]

We observe that, using the composition inequality in Lebesgue spaces, the embeddings $\dot{B}^{\frac{d}{p}}_{2,1}\hookrightarrow L^{p^{*}}$ and $\dot{B}^{s_{0}}_{p,1}\hookrightarrow L^{p^{*}}$ and the relations $\frac{d}{p}>\frac{d}{2}-2$ and $s_{0}\leq\frac{d}{p}$, we obtain

[TABLE]

Hence, taking advantage of (4.7), Proposition 2.6 and applying the fact $\frac{d}{p}-2<\frac{d}{p}-1<0<1-\frac{d}{p}<\frac{d}{p}$ and the embedding $\dot{B}^{\frac{d}{2}-2}_{2,1}\hookrightarrow\dot{B}^{\frac{d}{p}-2}_{p,1}$ yield

[TABLE]

For the term with $g_{3}(a,u^{h})$, we mimic the procedure leading to (4.9) and get

[TABLE]

Let us finally bound the term with $g_{4}(a,u^{h})$. Applying (2.6) with $\sigma=1-\frac{d}{p}$ yields for any smooth function $K$ vanishing at [math],

[TABLE]

As $p^{*}\geq p$, it follows from Bernstein inequality that $\|\dot{\Delta}_{j}\nabla u^{h}\|_{L^{p^{*}}}\lesssim\|\dot{\Delta}_{j}\nabla u^{h}\|_{L^{p}}$ for $j_{0}\leq j<j_{0}+N_{0}$. Hence, with the aid of Proposition 2.6 and the relations $s_{1}\leq s_{0}$ and $1-\frac{d}{p}<\frac{d}{p}$, we get

[TABLE]

By inserting above all estimates into (4.3) gives (4.1). ∎

According to the definition of $\mathcal{E}_{p}(t)$ in Theorem 1.1, we deduce that

[TABLE]

and

[TABLE]

since $\mathcal{E}_{p,0}\ll 1$. Finally, combining (4.10) and (4.11), one can make use of nonlinear generalisations of the Gronwall’s inequality (see for example, Page 360 of [18]) and obtain

[TABLE]

where $C_{0}>0$ depends on the norms $\|a_{0}\|^{\ell}_{\dot{B}^{-s_{1}-1}_{2,\infty}}$ and $\|u_{0}\|^{\ell}_{\dot{B}^{-s_{1}}_{2,\infty}}$.

5. Proofs of main results

In this section, we present the proofs of Theorem 1.2 and Corollary 1.1.

5.1. Proof of Theorem 1.2

We get from Lemmas 3.1 and 3.2 that

[TABLE]

where

[TABLE]

Note that the fact $p\geq 2$, the last term above can be bounded easily by $\mathcal{E}_{p}(t)(\|u^{\ell}\|_{\dot{B}_{2,1}^{\frac{d}{2}+1}}+\|u\|_{\dot{B}_{p,1}^{\frac{d}{p}+1}}^{h})$. Next, it follows from Proposition 2.4 and Bernstein inequality that

[TABLE]

On the other hand, we observe that

[TABLE]

Furthermore, we arrive at

[TABLE]

So we are left with the proof of

[TABLE]

In order to prove (5.2), we need the following two inequalities (see [6, 11]):

[TABLE]

[TABLE]

where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ and $d^{*}=\frac{2d}{d-2}$.

Note that $\Lambda^{-1}\mathrm{div}$ is an homogeneous Fourier multiplier of degree [math], we write that

[TABLE]

Taking advantage of Bony’s para-product decomposition gives $au=T_{a}u+R(a,u)+T_{u}a.$ According to (5.3) and (5.4) with $\sigma=\frac{d}{p}$, we have

[TABLE]

With the aid of the interpolations, embeddings and Young inequality, we observe that the right-side norm of the above three inequalities can be bounded by

[TABLE]

Hence, we conclude that

[TABLE]

It follows from [31] that

[TABLE]

For the term with $k(a)\nabla a$, we use the decomposition

[TABLE]

With the aid of (5.3) and (5.4) with $\sigma=\frac{d}{p}$ as well as Proposition 2.6, we get

[TABLE]

and, owing to (5.3) with $\sigma=\frac{d}{p}-1$ and Proposition 2.6, (4.2)

[TABLE]

Hence, we have

[TABLE]

The term $g_{3}(a,u)$ of $g$ is of the type $H(a)\nabla^{2}u$ with $H(a)=0$. Consequently, we write that

[TABLE]

We conclude that exactly as the previous term $k(a)\nabla a$ that

[TABLE]

The term $g_{4}(a,u)$ of $g$ is of the type $\nabla K(a)\otimes\nabla u$ with $K(0)=0$. Let us use the decomposition

[TABLE]

By virtue of (5.3) and (5.4) with $\sigma=\frac{d}{p}$ and Proposition 2.6, we have

[TABLE]

It follows from (5.3) with $\sigma=\frac{d}{p}-1$ and Proposition 2.6 that

[TABLE]

Then we discover that

[TABLE]

Hence, the the inequality (5.2) is proved.

Inserting the above estimates into (5.1) and performing the fact that $\mathcal{E}_{p}(t)\lesssim\mathcal{E}_{p,0}\ll 1$ for all $t\geq 0$ guaranteed by Theorem 1.1, we end up with

[TABLE]

In what follows, we observe that interpolation plays the key role to obtain the time-decay estimates. Owing to the fact $-s_{1}<\frac{d}{2}-1\leq\frac{d}{p}<\frac{d}{2}+1$, we get from Proposition 2.2 (the real interpolation) that

[TABLE]

With the aid of (4.12), we arrive at

[TABLE]

On the other hand, it follows from the fact $\|(\nabla a,u)\|^{h}_{\dot{B}^{\frac{d}{p}-1}_{p,1}}\leq\mathcal{E}_{p}(t)\lesssim\mathcal{E}_{p,0}\ll 1$ for all $t\geq 0$ that

[TABLE]

Consequently, there exists a constant $\widetilde{c}_{0}>0$ such that the following Lyapunov-type inequality holds

[TABLE]

Solving (5.5) directly yields

[TABLE]

It follows from Proposition 2.3 and (5.6) that

[TABLE]

and thus that

[TABLE]

On the other hand, let $\widetilde{s}_{1}\triangleq s_{1}+d(\frac{1}{2}-\frac{1}{p})$. We get from Propositions 2.2 and 2.3 that

[TABLE]

if s\in\big{(}-\widetilde{s}_{1}-1,\frac{d}{p}-2\big{)}\ \ \hbox{and}\ \ \theta_{1}=\frac{\frac{d}{p}-2-s}{\frac{d}{2}-1+s_{1}}\in(0,1), and that

[TABLE]

if s\in\big{(}-\widetilde{s}_{1},\frac{d}{p}-1\big{)}\ \ \hbox{and}\ \ \theta_{2}=\frac{\frac{d}{p}-1-s}{\frac{d}{2}-1+s_{1}}\in(0,1). Noticing the fact that

[TABLE]

for all $t\geq 0$, with the aid of (5.6), (5.9), (5.10), we have

[TABLE]

and

[TABLE]

for all $t\geq 0$, which lead to

[TABLE]

for $s\in(-\widetilde{s}_{1}-1,\frac{d}{p}-2)$, and

[TABLE]

for $s\in(-\widetilde{s}_{1},\frac{d}{p}-1)$. Hence, combining with (5.7), (5.8), (5.11), (5.12) yields Theorem 1.2.

5.2. Proof of Corollary 1.1

For brevity, it suffices to establish those time-optimal decay rates of $a$. Thanks to $p\leq r\leq\infty$, it follows from the embeddings $\dot{B}^{0}_{r,1}\hookrightarrow L^{r}$ and \dot{B}_{p,1}^{l+d\big{(}\frac{1}{p}-\frac{1}{r}\big{)}}\hookrightarrow\dot{B}_{r,1}^{l} $(l\in\mathbb{R})$, (5.7) and (5.11) that

[TABLE]

for $p\leq r\leq\infty$ and $l\in\mathbb{R}$ satisfying -\widetilde{s}_{1}-1<l+d\big{(}\frac{1}{p}-\frac{1}{r}\big{)}\leq\frac{d}{p}-2. This completes the proof of Corollary 1.1.

Acknowledgments

Last but not least, he is very grateful to Professor J. Xu for the suggestion on this question.