Large-Time Behavior of Solutions in the Critical Spaces for the Non-isentropic Compressible Navier–Stokes Equations with Capillarity

Abstract

In this paper, we investigate the large-time behavior for the non-isentropic compressible Navier–Stokes equations with capillarity in the whole space ${\mathbb {R}}^{d}$ ($d\ge 3$). Under an additional smallness assumption of the low frequencies of initial data, the time-decay estimates of $L^{q}$–$L^{r}$ type for global strong solutions near constant equilibrium (away from vacuum) can be deduced by establishing the time-weighted energy inequality. On the other hand, a pure energy approach (without the spectral analysis) different from the time-weighted energy method is performed, which allows us not only to get the time-decay rates but also to remove the smallness condition of low frequencies of initial data. The treatment of new nonlinear terms arising from capillary mainly depends on non classical Besov product estimates and the refined use of Sobolev embeddings and interpolations.

Appendix A

For convenience of reader, we briefly recall Littlewood-Paley decomposition, Besov spaces and some analysis tools. For more details, the interested reader may refer to Chaps. 2 and 3 of [51].

1.1 A.1 Littlewood–Paley Decomposition and Besov Spaces

Let us begin with the homogeneous Littlewood–Paley decomposition. To do this, one may fix a smooth radial non increasing function $\chi$ with $\mathrm {Supp}\,\chi \subset B\left( 0,\frac{4}{3}\right)$ and $\chi \equiv 1$ on $B\left( 0,\frac{3}{4}\right)$. Set $\varphi (\xi ) =\chi (\xi /2)-\chi (\xi )$. It is not difficult to check that

$$\begin{aligned} \sum _{j\in {\mathbb {Z}}}\varphi ( 2^{-j}\cdot ) =1\ \ \text {in}\ \ {\mathbb {R}}^{d}\setminus \{ 0\} \ \ \text {and}\ \ \mathrm {Supp}\,\varphi \subset \{ \xi \in {\mathbb {R}}^{d}:3/4\le |\xi |\le 8/3\} . \end{aligned}$$

Denoting $h\triangleq {\mathcal {F}}^{-1}\varphi$, we then define the dyadic blocks ${\dot{\Delta }}_{j}$ ($j\in {\mathbb {Z}}$) as follows

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

Formally, one has the following homogeneous decomposition for any tempered distribution $f\in {\mathcal {S}}^{\prime }({\mathbb {R}}^{d})$

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

(A1)

The equality holds true in the set ${\mathcal {S}}'$ of tempered distributions whenever f belongs to

$$\begin{aligned} {\mathcal {S}}'_{0}=\big \{f\in {\mathcal {S}}'|\lim _{j\rightarrow -\infty }\Vert {\dot{S}}_{j}u\Vert _{L^{\infty }}=0\big \}, \end{aligned}$$

where ${\dot{S}}_{j}f$ stands for the low frequency cut-off defined by ${\dot{S}}_{j}f\triangleq \chi (2^{-j}D)f$.

Let us now define the homogeneous Besov spaces ${\dot{B}}^{s}_{p,r}$ in terms of Littlewood-Paley decomposition.

Definition 1

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

$$\begin{aligned} {\dot{B}}^{s}_{p,r}\triangleq \{f\in {\mathcal {S}}'_{0}:\Vert f\Vert _{{\dot{B}}^{s}_{p,r}}<+\infty \}, \end{aligned}$$

where

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

(A2)

When investigating the evolution PDEs, it is somehow natural to take a class of mixed space-time Besov spaces, which was first introduced by Chemin and Lerner [52] (see also [53] for the particular case of Sobolev spaces).

Definition 2

For $T>0, s\in {\mathbb {R}}$ and $1\le r,\rho \le \infty$, the homogeneous Chemin–Lerner space ${\widetilde{L}}^{\rho }_{T}({\dot{B}}^{s}_{p,r})$ is defined by

$$\begin{aligned} {\widetilde{L}}^{\rho }_{T}({\dot{B}}^{s}_{p,r})\triangleq \{f\in L^{\rho }(0,T;{\mathcal {S}}'_{0}):\Vert f\Vert _{{\widetilde{L}}^{\rho }_{T}({\dot{B}}^{s}_{p,r})}<+\infty \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{{\widetilde{L}}^{\rho }_{T}({\dot{B}}^{s}_{p,r})}\triangleq \Vert (2^{js}\Vert {\dot{\Delta }}_{j}f\Vert _{L^{\rho }_{T}(L^{p})})\Vert _{\ell ^{r}({\mathbb {Z}})}. \end{aligned}$$

(A3)

The index T will be omitted if $T=+\infty$ and we denote by the following functional space:

A direct application of Minkowski’s inequality implies that

$$\begin{aligned} \Vert f\Vert _{{\widetilde{L}}^{\rho }_{T}(B^{s}_{p,r})}\le \Vert f\Vert _{L^{\rho }_{T}(B^{s}_{p,r})} \ \ \text{ if } \ \ r\ge \rho ;\ \ \Vert f\Vert _{{\widetilde{L}}^{\rho }_{T}(B^{s}_{p,r})}\ge \Vert f\Vert _{L^{\rho }_{T}(B^{s}_{p,r})} \ \ \text{ if } \ \ r\le \rho . \end{aligned}$$

Restricting the above norms (A2) and (A3) to the low or high frequencies parts of distributions will be fundamental in this paper. For example, let us fix some integer $j_{0}$ (the value of which will come from the proof of the main results) and put²

$$\begin{aligned}&\Vert f\Vert _{{\dot{B}}_{p,1}^{s}}^{\ell } \triangleq \sum _{j\le j_{0}}2^{js}\Vert {\dot{\Delta }}_{j}f\Vert _{L^{p}} \ \ \text{ and } \ \ \Vert f\Vert _{{\dot{B}}_{p,1}^{s}}^{h}\triangleq \sum _{j\ge j_{0}-1}2^{js}\Vert {\dot{\Delta }}_{j}f\Vert _{L^{p}}, \\&\Vert f\Vert _{{\tilde{L}}_{T}^{\infty } ({\dot{B}}_{p,1}^{s})}^{\ell } \triangleq \sum _{j\le j_{0}}2^{js}\Vert {\dot{\Delta }}_{j}f\Vert _{L_{T}^{\infty } (L^{p})} \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _{{\tilde{L}}_{T}^{\infty } ({\dot{B}}_{p,1}^{s})}^{h}\triangleq \sum _{j\ge j_{0}-1}2^{js}\Vert {\dot{\Delta }}_{j}f\Vert _{L_{T}^{\infty } (L^{p})}. \end{aligned}$$

1.2 A.2 Some Analysis Tools

Now, let us review some nonlinear tools in the $L^{p}$ framework, which will play a key role in our analysis.

We shall use repeatedly the following classical properties (see [51]):

Proposition 2

(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 $s\in {\mathbb {R}}$, $1\le p_{1}\le p_{2}\le \infty$ and $1\le r_{1}\le r_{2}\le \infty ,$ then ${\dot{B}}^{s}_{p_1,r_1}\hookrightarrow {\dot{B}}^{s-d(\frac{1}{p_{1}}-\frac{1}{p_{2}})}_{p_{2},r_{2}}$.

• 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, $p<\infty$).

Proposition 3

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

• Complex interpolation: 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 $f\in {\dot{B}}_{p,r}^{\rho s_{1}+(1-\rho )s_{2}}$ for all $\rho \in (0,1)$ and

  $$\begin{aligned} \Vert f\Vert _{{\dot{B}}_{p,r}^{\rho s_{1}+(1-\rho )s_{2}}}\le \Vert f\Vert _{{\dot{B}}_{p,r_{1}}^{s_{1}}}^{\rho } \Vert f\Vert _{{\dot{B}}_{p,r_{2}}^{s_{2}}}^{1-\rho } \ \ \hbox {with} \ \ \frac{1}{r}=\frac{\rho }{r_{2}}+\frac{1-\rho }{r_{2}}. \end{aligned}$$

• Real interpolation: If $f\in {\dot{B}}^{s_{1}}_{p,\infty }\cap {\dot{B}}^{s_{2}}_{p,\infty }$ and $s_{1}<s_{2}$, then $f\in {\dot{B}}_{p,1}^{\rho s_{1}+(1-\rho )s_{2}}$ for all $\rho \in (0,1)$ and

  $$\begin{aligned} \Vert f\Vert _{{\dot{B}}_{p,1}^{\rho s_{1}+(1-\rho )s_{2}}}\le \frac{C}{\rho (1-\rho )(s_{2}-s_{1})}\Vert f\Vert _{{\dot{B}}_{p,\infty }^{s_{1}}}^{\rho } \Vert f\Vert _{{\dot{B}}_{p,\infty }^{s_{2}}}^{1-\rho }. \end{aligned}$$

Proposition 4

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

  $$\begin{aligned} C^{-1}\lambda ^{s-\frac{d}{p}}\Vert f\Vert _{{\dot{B}}_{p,r}^{s}} \le \Vert f(\lambda \cdot )\Vert _{{\dot{B}}_{p,r}^{s}}\le C\lambda ^{s-\frac{d}{p}}\Vert f\Vert _{{\dot{B}}_{p,r}^{s}}. \end{aligned}$$

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

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

  $$\begin{aligned} F(D):{\dot{B}}_{p,r}^{s}\rightarrow {\dot{B}}_{p,r}^{s-m}. \end{aligned}$$

  In particular, the gradient operator maps ${\dot{B}}^{s}_{p,r}$ to ${\dot{B}}^{s-1}_{p,r}$.

In addition, let us recall the following classical Bernstein inequality:

$$\begin{aligned} \Vert D^{k}f\Vert _{L^{b}} \le C^{1+k} \lambda ^{k+d(\frac{1}{a}-\frac{1}{b})}\Vert f\Vert _{L^{a}} \end{aligned}$$

(A4)

that holds for all function f such that $\mathrm {Supp}\,{\mathcal {F}}u\subset \{\xi \in {\mathbb {R}}^{d}: |\xi |\le R\lambda \}$ for some $R>0$ and $\lambda >0$, if $k\in {\mathbb {N}}$ and $1\le a\le b\le \infty$.

More generally, if we assume f to satisfy $\mathrm {Supp}\,{\mathcal {F}}f\subset \{\xi \in {\mathbb {R}}^{d}: R_{1}\lambda \le |\xi |\le R_{2}\lambda \}$ for some $0<R_{1}<R_{2}$ and $\lambda >0$, then for any smooth homogeneous of degree m function A on ${\mathbb {R}}^{d}\setminus \{0\}$ and $1\le a\le \infty ,$ we have (see e.g. Lemma 2.2 in [51]):

$$\begin{aligned} \Vert A(D)f\Vert _{L^{a}}\lesssim \lambda ^{m}\Vert f\Vert _{L^{a}}. \end{aligned}$$

(A5)

An obvious consequence of (A4) and (A5) is that $\Vert D^{k}f\Vert _{{\dot{B}}^{s}_{p, r}}\thickapprox \Vert f\Vert _{{\dot{B}}^{s+k}_{p, r}}$ for all $k\in {\mathbb {N}}.$

The classical product estimates are used repeatedly in bounding bilinear terms.

Proposition 5

([51, 54]). Let $1\le p,r\le \infty$. Then we have the following:

• (a) If $s>0$, then

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

• (b) If $s_{1}, s_{2}\le \frac{d}{p}$ and $s_{1}+s_{2}>d\max (0,\frac{2}{p}-1)$, then

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

• (c) If $s_{1}\le \frac{d}{p}, s_{2}<\frac{d}{p}$ and $s_{1}+s_{2}\ge d\max (0,\frac{2}{p}-1)$, then

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

To match different Lebesgue norms at low frequencies and high frequencies, we need to present some non classical product estimates in the $L^{p}$ framework. Precisely,

Proposition 6

([41, 44, 45]). Let the real numbers $s_{1},$ $s_{2},$ $p_1$ and $p_2$ be such that

$$\begin{aligned} s_{1}+s_{2}>0,\ \ s_{1}\le \frac{d}{p_{1}}, \ \ s_{2}\le \frac{d}{p_{2}}, \ \ s_{1}\ge s_{2} \ \ \hbox {and} \ \ \frac{1}{p_{1}}+\frac{1}{p_{2}}\le 1. \end{aligned}$$

Then we have

$$\begin{aligned} \Vert fg\Vert _{{\dot{B}}^{s_{2}}_{q,1}}\lesssim \Vert f\Vert _{{\dot{B}}^{s_{1}}_{p_{1},1}}\Vert g\Vert _{{\dot{B}}^{s_{2}}_{p_{2},1}} \ \ \hbox {with} \ \ \frac{1}{q}=\frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{s_{1}}{d}. \end{aligned}$$

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

$$\begin{aligned} \frac{d}{p_{1}}+\frac{d}{p_{2}}-d\le s \le \min \left( \frac{d}{p_{1}},\frac{d}{p_{2}}\right) \ \ \hbox {and} \ \ \frac{1}{q}=\frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{s}{d}, \end{aligned}$$

we have

$$\begin{aligned} \Vert fg\Vert _{{\dot{B}}^{-s}_{q,\infty }}\lesssim \Vert f\Vert _{{\dot{B}}^{s}_{p_{1},1}}\Vert g\Vert _{{\dot{B}}^{-s}_{p_{2},\infty }}. \end{aligned}$$

Together with the regularity requirement in Theorem 2, Proposition 6 can lead to the following inequalities.

Corollary 1

([41, 55]). Let $1-\frac{d}{2}<s_{1}\le s_{0}$ and p satisfy (8). It holds that

$$\begin{aligned} \Vert fg\Vert _{{\dot{B}}^{-s_{1}}_{2,\infty }}\lesssim & {} \Vert f\Vert _{{\dot{B}}^{\frac{d}{p}}_{p,1}}\Vert g\Vert _{{\dot{B}}^{-s_{1}}_{2,1}}, \end{aligned}$$

(A6)

$$\begin{aligned} \Vert fg\Vert _{{\dot{B}}^{\frac{d}{p}-\frac{d}{2}-s_{1}}_{2,\infty }}\lesssim & {} \Vert f\Vert _{{\dot{B}}^{\frac{d}{p}-\frac{d}{2}-s_{1}}_{p,1}}\Vert g\Vert _{{\dot{B}}^{\frac{d}{p}}_{2,1}}, \end{aligned}$$

(A7)

$$\begin{aligned} \Vert fg\Vert ^{\ell }_{{\dot{B}}^{-s_{0}}_{2,\infty }}\lesssim & {} \Vert f\Vert _{{\dot{B}}^{\frac{d}{p}-1}_{p,1}}\Vert g\Vert _{{\dot{B}}^{1-\frac{d}{p}}_{p,1}}. \end{aligned}$$

(A8)

Indeed, note that if $s_{1}=s_{0}$, then we get from (A7) that

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

(A9)

In addition, the third estimate in Proposition 5 can be also extended to the non classical form (see [48]).

Proposition 7

Let the real number $s_{1}$, $s_{2}$, $p_{1}$ and $p_{2}$ be such that

$$\begin{aligned} s_{1}+s_{2}\ge 0, \ s_{1}\le \frac{d}{p_{1}}, \ s_{2}<\min \left( \frac{d}{p_{1}},\ \frac{d}{p_{2}}\right) \ \ \hbox {and} \ \ \frac{1}{p_{1}}+\frac{1}{p_{2}}\le 1. \end{aligned}$$

Then it holds that

$$\begin{aligned} \Vert fg\Vert _{{\dot{B}}^{s_{1}+s_{2}-\frac{d}{p_{1}}}_{p_{2},\infty }}\lesssim \Vert f\Vert _{{\dot{B}}^{s_{1}}_{p_{1},1}} \Vert g\Vert _{{\dot{B}}^{s_{2}}_{p_{2},\infty }}. \end{aligned}$$

As a direct consequence, the following product estimates in the pure energy argument with interpolation will be employed.

Corollary 2

Let $1-\frac{d}{2}<s_{1}\le s_{0}$ and p satisfy (8). It holds that

$$\begin{aligned} \Vert fg\Vert _{{\dot{B}}^{-s_{1}}_{2,\infty }}\lesssim & {} \Vert f\Vert _{{\dot{B}}^{\frac{d}{p}}_{p,1}} \Vert g\Vert _{{\dot{B}}^{-s_{1}}_{2,\infty }}, \end{aligned}$$

(A10)

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

(A11)

System (7) also involves compositions of functions and they are bounded in terms of the following conclusion.

Proposition 8

([41, 44, 51, 55]). Let $F:\mathbb {R}\rightarrow {\mathbb {R}}$ be smooth with $F(0)=0$. For all $1\le p,r\le \infty$ and $s>0$ we have $F(u)\in {\dot{B}}^{s}_{p,r}\cap L^{\infty }$ for $u\in {\dot{B}}^{s}_{p,r}\cap L^{\infty }$, and

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

with C depending only on $\Vert u\Vert _{L^{\infty }}$, $F'$ (and higher derivatives), s, p and d.

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

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

Finally, we end this section with the endpoint maximal regularity property of the heat equation which considers complex diffusion coefficient. The proof is similar to the case of real coefficient. The interested reader is referred to [37] for more details.

Proposition 9

Let $T>0$, $s\in {\mathbb {R}}$ and $1\le \rho _{2},p,r\le \infty$. Let u satisfy

$$\begin{aligned} \left\{ \begin{array}{lll} \partial _{t}u-\nu \Delta u=f,\\ u|_{t=0}=u_{0}, \end{array} \right. \end{aligned}$$

where $\nu \in {\mathbb {C}}$ is a complex number with $Re\, \nu >0$. Then, there exists a constant C depending only on d and such that for all $\rho _{1}\in [\rho _{2},\infty ]$, one have

$$\begin{aligned} (Re\,\nu )^{\frac{1}{\rho _1}}\Vert u\Vert _{{{\tilde{L}}}_{T}^{\rho _1}\big (\dot{B}^{s+\frac{2}{\rho _1}}_{p,r}\big )}\le C \left( \Vert u_{0}\Vert _{{\dot{B}}^{s}_{p,r}}+(Re\,\nu )^{\frac{1}{\rho _{2}}-1}\Vert f\Vert _{{{\tilde{L}}}^{\rho _{2}}_{T}({\dot{B}}^{s-2+\frac{2}{\rho _{2}}}_{p,r})}\right) . \end{aligned}$$