Compressible Navier–Stokes equations without heat conduction in $L^p$-framework

Abstract

In this paper, we mainly consider global well-posedness and long time behavior of compressible Navier–Stokes equations without heat conduction in $L^p$-framework. This is a generalization of Peng and Zhai (SIMA 55(2):1439–1463, 2023), where they obtained the corresponding result in $L^2$-framework. Based on the key observation that we can release the regularity of non-dissipative entropy S in high frequency in Peng and Zhai (2023), we ultimately achieve the desired $L^p$ estimate in the high frequency via complicated calculations on the nonlinear terms. In addition, we get the $L^p$-decay rate of the solution.

Appendix

We will state several important lemmas and propositions on the homogeneous Besov space $\dot{B}^{s}_{p,r}$. First, let $\mathcal {S}(\mathbb {R}^{n})$ be the Schwartz class of rapidly decreasing function. Given $f\in \mathcal {S}(\mathbb {R}^{n})$, its Fourier transform $\mathcal {F}f=\widehat{f}$ is defined by

$$\begin{aligned} \widehat{f}(\xi )=\int \limits _{\mathbb {R}^{n}}e^{-ix\cdot \xi }f(x)\textrm{d}x. \end{aligned}$$

Let $(\chi , \varphi )$ be a couple of smooth functions valued in [0, 1] such that $\chi$ is supported in the ball $\{\xi \in \mathbb {R}^{n}: \ |\xi |\le \frac{4}{3}\}$, $\varphi$ is supported in the shell $\{\xi \in \mathbb {R}^{n}: \ \frac{3}{4}\le |\xi |\le \frac{8}{3}\}$, $\varphi (\xi ):=\chi (\xi /2)-\chi (\xi )$ and

$$\begin{aligned} \chi (\xi )+\sum _{j\ge 0}\varphi (2^{-j}\xi )=1~ \textrm{for}\ \forall \ \xi \in \mathbb { R}^{n}, ~~~\sum _{j\in \mathbb {Z}}\varphi (2^{-j}\xi )=1 ~\textrm{for} \ \forall \ \xi \in \mathbb {R}^{n}\setminus \{0\}. \end{aligned}$$

For $f\in \mathcal {S}'$, the homogeneous frequency localization operators $\dot{\Delta }_j$ and $\dot{S}_j$ are defined by

$$\begin{aligned} \dot{\Delta }_{j}f\triangleq \varphi (2^{-j}D)f=\mathcal {F}^{-1}(\varphi (2^{-j}\xi )\mathcal {F}f)\quad \textrm{and}\quad \dot{S}_{j}f\triangleq \chi (2^{-j}D)f=\mathcal {F}^{-1}(\chi (2^{-j}\xi )\mathcal {F}f). \end{aligned}$$

We denote the space $\mathcal {S}'_{h}(\mathbb {R}^n)$ by the dual space of $\mathcal {S}'(\mathbb {R}^n)=\{f\in \mathcal {S}(\mathbb {R}^n):\,D^\alpha \hat{f}(0)=0\}$, which can also be identified by the quotient space of $\mathcal {S}'(\mathbb {R}^n)/{\mathbb {P}}$ with the polynomial space ${\mathbb {P}}$. The formal equality

$$\begin{aligned} f=\sum _{j\in \mathbb {Z}}\dot{\Delta }_jf \end{aligned}$$

holds true for $f\in \mathcal {S}'_{h}(\mathbb {R}^n)$ and is called the homogeneous Littlewood-Paley decomposition, and then we have the fact that

$$\begin{aligned} \dot{S}_jf=\sum _{q\le j-1}\dot{\Delta }_qf. \end{aligned}$$

One easily verifies that with our choice of $\varphi$,

$$\begin{aligned} \dot{\Delta }_j\dot{\Delta }_qf\equiv 0\quad \text {if}\quad |j-q|\ge 2\quad \text {and} \quad \dot{\Delta }_j(\dot{S}_{q-1}f\dot{\Delta }_q f)\equiv 0\quad \hbox {if} \quad |j-q|\ge 5. \end{aligned}$$

Definition 3.1

(Homogeneous Besov space) For $s\in \mathbb {R}$ and $1\le p,r\le \infty$, the homogeneous Besov space $\dot{B}^{s}_{p,r}$ is defined by

$$\begin{aligned} \dot{B}^s_{p,r}\triangleq \left\{ f\in \mathcal {S}_h':||f||_{\dot{B}^s_{p,r}}<+\infty \right\} , \end{aligned}$$

(3.44)

where

$$\begin{aligned} ||f||_{\dot{B}^s_{p,r}}\triangleq ||2^{js}||\dot{\Delta }_jf||_{L^p}||_{ l ^r(\mathbb {Z})}. \end{aligned}$$

(3.45)

Definition 3.2

(Chemin-Lerner spaces) Let $T>0$, $s\in \mathbb {R}$, $1<r,p,q\le \infty$. The space $\widetilde{L}^q_{T}(\dot{B}^s_{p,r})$ is defined by

$$\begin{aligned} \widetilde{L}^q_{T}(\dot{B}^s_{p,r})\triangleq \left\{ f\in L^q(0,T;\mathcal {S}'_h):||f||_{\widetilde{L}^q_{T}(\dot{B}^s_{p,r})}<+\infty \right\} , \end{aligned}$$

(3.46)

where

$$\begin{aligned} ||f||_{\widetilde{L}^q_{T}(\dot{B}^s_{p,r})}\triangleq ||2^{js}||\dot{\Delta }_jf||_{L^q(0,T;L^p)}||_{ l ^r(\mathbb {Z})}. \end{aligned}$$

(3.47)

Remark 3.1

It holds that

$$\begin{aligned} ||f||_{\widetilde{L}^q_{T}(\dot{B}^s_{p,r})}\le ||f||_{L^q_{T}(\dot{B}^s_{p,r})}\quad \textrm{if}\quad r\ge q;\quad ||f||_{\widetilde{L}^q_{T}(\dot{B}^s_{p,r})}\ge ||f||_{L^q_{T}(\dot{B}^s_{p,r})}\quad \textrm{if}\quad r\le q. \end{aligned}$$

Restricting the above norms (4.2), (4.3) to the low or high frequencies parts of distributions will be crucial in our approach. For example, let us fix some integer $j_0$ and set

$$\begin{aligned} ||f||^l_{\dot{B}^{s}_{p,1}}\triangleq & {} \sum _{j\le j_0}2^{js}||\dot{\Delta }_jf||_{L^p},\quad ||f||^h_{\dot{B}^{s}_{p,1}}\triangleq \sum _{j\ge j_0-1}2^{js}||\dot{\Delta }_jf||_{L^p};\\ ||f||^l_{\widetilde{L}^\infty _{T}(\dot{B}^s_{p,1})}\triangleq & {} \sum _{j\le j_0}2^{js}||\dot{\Delta }_jf||_{L^\infty _T(L^p)},\quad ||f||^h_{\widetilde{L}^\infty _{T}(\dot{B}^s_{p,1})}\triangleq \sum _{j\ge j_0-1}2^{js}||\dot{\Delta }_jf||_{L^\infty _T(L^p)}. \end{aligned}$$

Lemma 3.1

(Bernstein inequalities) Let $\mathscr {B}$ be a ball and $\mathscr {C}$ be a ring of $\mathbb {R}^n$. For $\lambda >0$, integer $k\ge 0$, $1\le p\le q\le \infty$ and a smooth homogeneous function $\sigma$ in $\mathbb {R}^n\backslash \{0\}$ of degree m, then there holds

$$\begin{aligned} ||\nabla ^kf||_{L^q}\le & {} C^{k+1}\lambda ^{k+n(\frac{1}{p}-\frac{1}{q})}||f||_{L^p},\quad \textrm{whenever}\, \textrm{supp}\widehat{f}\subset \lambda \mathscr {B},\\ C^{-k-1}\lambda ^k||f||_{L^q}\le & {} ||\nabla ^kf||_{L^p}\le C^{k+1}\lambda ^k||f||_{L^p},\quad \textrm{whenever}\, \textrm{supp}\widehat{f}\subset \lambda \mathscr {C},\\ ||\sigma (\nabla )f||_{L^q}\le & {} C_{\sigma ,m}\lambda ^{m+n(\frac{1}{p}-\frac{1}{q})}||f||_{L^p},\quad \textrm{whenever}\, \textrm{supp}\widehat{f}\subset \lambda \mathscr {C}. \end{aligned}$$

Proposition 3.2

[1](Embedding for Besov space on $\mathbb {R}^n$)

• 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 $s\in \mathbb {R}$, $1\le p_1\le p_2\le \infty$, and $1\le r_1\le r_2\le \infty$ then $\dot{B}^{s}_{p_1,r_1}\hookrightarrow \dot{B}^{s-n(\frac{1}{p_1}-\frac{1}{p_2})}_{p_2,r_2}$.

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

Proposition 3.3

[16] If supp$\mathcal {F}f\subset \left\{ \xi \in \mathbb {R}^n:R_1\lambda \le |\xi |\le R_2\lambda \right\}$, then there exists C depending only on d, $R_1$, $R_2$ so that for all $1<p<\infty$,

$$\begin{aligned} C\lambda ^2\left( \frac{p-1}{p}\right) \int \limits _{\mathbb {R}^n}|f|^pdx\le (p-1)\int \limits _{\mathbb {R}^n}|\nabla f|^2|f|^{p-2}\textrm{d}x=-\int \limits _{\mathbb {R}^n}\Delta f|f|^{p-2}f\textrm{d}x. \end{aligned}$$

(3.48)

Proposition 3.4

[1](Interpolation inequality) Let $1\le p,r,r_1,r_2\le \infty$, if $f\in \dot{B}^{s_1}_{p,r_1}\cap \dot{B}^{s_2}_{p,r_2}$ and $s_1\ne s_2$, then $f\in \dot{B}^{\theta s_1+(1-\theta )s_2}_{p,r}$ for all $\theta \in (0,1)$ and

$$\begin{aligned} ||f||_{\dot{B}^{\theta s_1+(1-\theta )s_2}_{p,r}}\le ||f||^\theta _{\dot{B}^{s_1}_{p,r_1}}||f||^{1-\theta }_{\dot{B}^{s_2}_{p,r_2}} \end{aligned}$$

(3.49)

with $\frac{1}{r}=\frac{\theta }{r_1}+\frac{1-\theta }{r_2}$.

Proposition 3.5

[1, 13] Let $s>0$, $1\le p$, $r\le \infty$, then $\dot{B}^{s}_{p,r}\cap L^\infty$ is an algerbra and

$$\begin{aligned} ||fg||_{\dot{B}^{s}_{p,r}} \lesssim ||f||_{L^\infty }||g||_{\dot{B}^{s}_{p,r}} + ||g||_{L^\infty }||f||_{\dot{B}^{s}_{p,r}}. \end{aligned}$$

(3.50)

Let $s_1+s_2>0$, $s_1 \le \frac{n}{p_1}$, $s_2 \le \frac{n}{p_2}$, $s_1\ge s_2$, $\frac{1}{p_1}+\frac{1}{p_2}\le 1$. Then, it holds that

$$\begin{aligned} ||fg||_{\dot{B}^{s_2}_{q,1}} \lesssim ||f||_{\dot{B}^{s_1}_{p_1,1}}||g||_{\dot{B}^{s_2}_{p_2,1}}, \end{aligned}$$

(3.51)

where $\frac{1}{q} = \frac{1}{p_1} + \frac{1}{p_2} - \frac{s_1}{n}$.

Proposition 3.6

[41] Let the real numbers $s_1,\ s_2,\ p_1$ and $p_2$ be such that

$$\begin{aligned} s_1+s_2\ge 0,\ s_1\le \frac{n}{p_1},\ s_2<\min \left( \frac{n}{p_1},\frac{n}{p_2}\right) \ and\ \frac{1}{p_1}+\frac{1}{p_2}\le 1. \end{aligned}$$

Then, it holds that

$$\begin{aligned} \Vert fg\Vert _{\dot{B}_{p_2,\infty }^{s_1+s_2-\frac{n}{p_1}}}\lesssim \Vert f\Vert _{\dot{B}_{p_1,1}^{s_1}}\Vert g\Vert _{\dot{B}_{p_2,\infty }^{s_2}}. \end{aligned}$$

(3.52)

Corollary 3.1

Let the real numbers $1-\frac{n}{2}<\sigma _1\le \frac{2n}{p}-\frac{n}{2}$ and p satisfy (1.7). The following two inequalities hold true:

$$\begin{aligned} \Vert fg\Vert _{\dot{B}_{2,\infty }^{-\sigma _1}}\lesssim \Vert f\Vert _{\dot{B}_{p,1}^{\frac{n}{p}}}\Vert g\Vert _{\dot{B}_{2,\infty }^{-\sigma _1}}, \end{aligned}$$

(3.53)

as well as

$$\begin{aligned} \Vert fg\Vert _{\dot{B}_{2,\infty }^{\frac{n}{p}-\frac{n}{2}-\sigma _1}}\lesssim \Vert f\Vert _{\dot{B}_{p,1}^{\frac{n}{p}-1}}\Vert g\Vert _{\dot{B}_{2,\infty }^{\frac{n}{p}-\frac{n}{2}-\sigma _1+1}}. \end{aligned}$$

(3.54)

Proposition 3.7

[42] Let $j_0\in \mathbb {Z}$, and denote $z^l\triangleq \dot{S}_{j_0}z,\ z^h\triangleq z-z^l$ and, for any $s\in \mathbb {R}$,

$$\begin{aligned} \Vert z\Vert _{\dot{B}_{2,\infty }^{s}}^l\triangleq \sup _{j\le j_0}2^{js}\Vert \dot{\Delta }_jz\Vert _{L^2}. \end{aligned}$$

There exists a universal integer $N_0$ such that for any $2\le p\le 4$ and $s>0$, we have

$$\begin{aligned}{} & {} \Vert fg^h\Vert _{\dot{B}_{2,\infty }^{-\sigma _0}}^l\le C(\Vert f\Vert _{\dot{B}_{p,1}^{s}}+\Vert \dot{S}_{k_0+N_0}f\Vert _{L^{p^*}})\Vert g^h\Vert _{\dot{B}_{p,\infty }^{-s}}, \end{aligned}$$

(3.55)

$$\begin{aligned}{} & {} \Vert f^h g\Vert _{\dot{B}_{2,\infty }^{-\sigma _0}}^l\le C(\Vert f^h\Vert _{\dot{B}_{p,1}^{s}}+\Vert \dot{S}_{k_0+N_0}f^h\Vert _{L^{p^*}})\Vert g\Vert _{\dot{B}_{p,\infty }^{-s}}, \end{aligned}$$

(3.56)

where $\sigma _0\triangleq \frac{2n}{p}-\frac{n}{2}$, $\frac{1}{p^*}+\frac{1}{p}=\frac{1}{2}$, C depends only on $k_0,\ n,\ s$.

Additionally, for exponents $s>0$, $1 \le p_1,p_2,q \le \infty$ satisfying

$$\begin{aligned} \frac{n}{p_1}+\frac{n}{p_2}-n\le s\le \min \left( \frac{n}{p_1},\frac{n}{p_2}\right) \ \ and\ \ \frac{1}{q} = \frac{1}{p_1}+\frac{1}{p_2}-\frac{s}{n}. \end{aligned}$$

Then, it holds that

$$\begin{aligned} ||fg||_{\dot{B}^{-s}_{q,\infty }} \lesssim ||f||_{\dot{B}^{s}_{p_1,1}}||g||_{\dot{B}^{-s}_{p_2, \infty }}. \end{aligned}$$

(3.57)

Proposition 3.8

[11] Let $(s,r)\in \mathbb {R}\times [1,\infty ]$ and $1\le p,p_1,p_2\le \infty$ with $1/p=1/p_1+1/p_2$: For the paraproduct, the following continuity properties hold true

$$\begin{aligned} ||\dot{T}_ab||_{\dot{B}^{s}_{p,r}} \lesssim ||a||_{_{L^{p_1}}}||b||_{\dot{B}^{s}_{p_2,r}} \end{aligned}$$

(3.58)

and

$$\begin{aligned} ||\dot{T}_ab||_{\dot{B}^{s+t}_{p,r}} \lesssim ||a||_{\dot{B}^t_{p_1,\infty }}||b||_{\dot{B}^{s}_{p_2,r}},\ \ if\ t<0. \end{aligned}$$

(3.59)

If $s_1+s_2>0$ and $1/r=1/r_1+1/r_2\le 1$, then

$$\begin{aligned} ||\dot{R}(a,b)||_{\dot{B}^{s_1+s_2}_{p,r}} \lesssim ||a||_{\dot{B}^{s_1}_{p_1,r_1}}||b||_{\dot{B}^{s_2}_{p_2,r_2}}. \end{aligned}$$

(3.60)

If $s_1+s_2=0$ and $1/r_1+1/r_2\ge 1$, then

$$\begin{aligned} ||\dot{R}(a,b)||_{\dot{B}^{0}_{p,\infty }} \lesssim ||a||_{\dot{B}^{s_1}_{p_1,r_1}}||b||_{\dot{B}^{s_2}_{p_2,r_2}}. \end{aligned}$$

(3.61)

From Proposition 3.8, we easily deduce the following proposition:

Proposition 3.9

[42] The Bony decomposition satisfies that

$$\begin{aligned} ||\dot{T}_ab||_{\dot{B}^{s-1+ \frac{n}{2} - \frac{\textrm{d}}{p}}_{2,1}}\lesssim & {} ||a||_{\dot{B}^{\frac{n}{p}-1}_{p,1}}||b||_{\dot{B}^{s}_{p,1}},\ \ if\ n \ge 2\ and\ 1 \le p \le \min \left( 4,\frac{2n}{n-2}\right) , \end{aligned}$$

(3.62)

$$\begin{aligned} ||\dot{R}(a,b)||_{\dot{B}^{s-1+ \frac{n}{2} - \frac{n}{p}}_{2,1}}\lesssim & {} ||a||_{_{\dot{B}^{\frac{n}{p}-1}_{p,1}}}||b||_{\dot{B}^{s}_{p,1}},\ \ if\ s>1-\min \left( \frac{n}{p},\frac{n}{p'}\right) \ and\ 1\le p\le 4, \end{aligned}$$

(3.63)

where $\frac{1}{p'}+\frac{1}{p}=\frac{1}{2}$.

Lemma 3.2

[13] Let $n\ge 2$, $1\le p, q\le \infty$, $v\in \dot{B}^{s}_{q,1}(\mathbb {R}^n)$ and $\nabla u \in \dot{B}^{\frac{n}{p}}_{p,1}$.

Asumme that

$$\begin{aligned} -n \min \left( \frac{1}{p}, 1-\frac{1}{q}\right) <s\le 1+n\min \left( \frac{1}{p}, \frac{1}{q}\right) . \end{aligned}$$

Then, it holds the commutator estimate

$$\begin{aligned} ||[\dot{\Delta }_j, u\cdot \nabla ]v||_{L^p} \lesssim d_j2^{-js}||\nabla u||_{\dot{B}^{\frac{n}{p}}_{p,1}}||v||_{\dot{B}^{s}_{q,1}}. \end{aligned}$$

(3.64)

In the limit case $s=-n\min \left( \frac{1}{p}, 1-\frac{1}{q}\right)$, we have

$$\begin{aligned} \sup 2^{js}||[\dot{\Delta }_j, u\cdot \nabla ]||_{L^p}\lesssim ||\nabla u||_{\dot{B}^{\frac{n}{p}}_{p,1}}||v||_{\dot{B}^{s}_{q,\infty }}. \end{aligned}$$

(3.65)

Proposition 3.10

[42] Let $F: \mathbb {R} \mapsto \mathbb {R}$ be a smooth function with $F(0)=0$, $1\le p, r\le \infty$ and $s>0$. Then, $F: \dot{B}^{s}_{p,r}(\mathbb {R}^n)\cap (L^\infty (\mathbb {R}^n))$ and

$$\begin{aligned} ||F(u)||_{\dot{B}^{s}_{p,r}}\le C||u||_{\dot{B}^{s}_{p,r}} \end{aligned}$$

(3.66)

with C a constant depending only on $|||u||_{L^\infty }$, s, p, n and derivatives of F.

If $s>-\min (\frac{n}{p}, \frac{n}{p'})$, then $F: \dot{B}^{s}_{p,r}(\mathbb {R}^n)\cap \dot{B}^{\frac{n}{p}}_{p,1}(\mathbb {R}^n) \mapsto \dot{B}^{s}_{p,r}(\mathbb {R}^n)\cap \dot{B}^{\frac{n}{p}}_{p,1}(\mathbb {R}^n)$, and

$$\begin{aligned} ||F(u)||_{\dot{B}^{s}_{p,r}}\le C\left( 1+||u||_{\dot{B}^{\frac{n}{p}}_{p,1}}\right) ||u||_{\dot{B}^{s}_{p,r}}, \end{aligned}$$

(3.67)

where $\frac{1}{p}+\frac{1}{p'}=\frac{1}{2}$.