Global dynamics of large solution for the compressible Navier–Stokes–Korteweg equations

Abstract

In this paper, we study the Navier–Stokes–Korteweg equations governed by the evolution of compressible fluids with capillarity effects. We first investigate the global well-posedness of solution in the critical Besov space for large initial data. Contrary to pure parabolic methods in Charve et al. (Indiana Univ Math J 70:1903–1944, 2021), we also take the strong dispersion due to large capillarity coefficient $\kappa$ into considerations. By establishing a dissipative–dispersive estimate, we are able to obtain uniform estimates and incompressible limits in terms of $\kappa$ simultaneously. Secondly, we establish the large time behaviors of the solution. We would make full use of both parabolic mechanics and dispersive structure which implicates our decay results without limitations for upper bound of derivatives while requiring no smallness for initial assumption.

Appendix

In this appendix, we would recall some classical theory concerns Fourier localization technique. The Fourier transform of a function $f\in {\mathcal {S}}$ (the Schwarz class) is denoted by

$$\begin{aligned} {\widehat{f}}(\xi )={\mathcal {F}}[f](\xi ):=\int _{\mathbb {R}^{d}}f(x)e^{-i\xi \cdot x}dx. \end{aligned}$$

For $1\le p\le \infty$, we denote by $L^{p}=L^{p}(\mathbb {R}^{d})$ the usual Lebesgue space on $\mathbb {R}^{d}$ with the norm $\Vert \cdot \Vert _{L^{p}}$.

For convenience of reader, we would like to recall the Littlewood–Paley decomposition, Besov spaces and related analysis tools. The reader is referred to Chap. 2 and Chap. 3 of [1] for more details. Let $\chi$ be a smooth function valued in [0, 1], such that $\chi$ is supported in the ball ${\textbf{B}}(0,\frac{4}{3})=\{\xi \in \mathbb {R}^{d}:|\xi |\le \frac{4}{3}\}$. Set $\varphi (\xi )=\chi (\xi /2)-\chi (\xi )$. Then $\varphi$ is supported in the shell ${\textbf{C}}(0,\frac{3}{4},\frac{8}{3})=\{\xi \in \mathbb {R}^{d}:\frac{3}{4}\le |\xi |\le \frac{8}{3}\}$ so that

$$\begin{aligned} \sum _{q\in \mathbb {Z}}\varphi (2^{-q}\xi )=1, \quad \forall \xi \in \mathbb {R}^{d}\backslash \{{0}\}. \end{aligned}$$

For any tempered distribution $f\in {\mathcal {S}}'$, one can define the homogeneous dyadic blocks and homogeneous low-frequency cut- off operators:

$$\begin{aligned}{} & {} \dot{\Delta }_{q}f:=\varphi (2^{-q}D)f={\mathcal {F}}^{-1}(\varphi (2^{-q}\xi ){\mathcal {F}}f), \quad q\in \mathbb {Z}; \\{} & {} {\dot{S}}_{q}f:=\chi (2^{-q}D)f={\mathcal {F}}^{-1}(\chi (2^{-q}\xi ){\mathcal {F}}f), \quad q\in \mathbb {Z}. \end{aligned}$$

Furthermore, we have the formal homogeneous decomposition as follows

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

Also, throughout the paper, $f^{h}$ and $f^{\ell }$ represent the high frequency part and low frequency part of f respectively where

$$\begin{aligned} f^{\ell }\triangleq {\dot{S}}_{q_{0}}f;\quad f^{h}\triangleq (1-{\dot{S}}_{q_{0}})f \end{aligned}$$

with some given constant $q_{0}$.

Denote by ${\mathcal {S}}'_{0}:=\mathcal {S'}/{\mathcal {P}}$ the tempered distributions modulo polynomials ${\mathcal {P}}$. As we known, the homogeneous Besov spaces can be characterised in terms of the above spectral cut-off blocks.

1.1 Homogeneous Besov space

Definition 5.1

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

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

where

$$\begin{aligned} \Vert f\Vert _{{\dot{B}}^s_{p,r}}:=\Big (\sum _{q\in \mathbb {Z}}(2^{qs}\Vert \dot{\Delta }_qf\Vert _{L^{p}})^{r}\Big )^{1/r} \end{aligned}$$

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

We often use the following classical properties of Besov spaces (see [1]):

$\bullet$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

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

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

$\bullet$Interpolation: The following inequality is satisfied for $1\le p,r_{1},r_{2}, r\le \infty , \sigma _{1}\ne \sigma _{2}$ and $\theta \in (0,1)$:

$$\begin{aligned} \Vert f\Vert _{{\dot{B}}_{p,r}^{\theta \sigma _{1}+(1-\theta )\sigma _{2}}}\lesssim \Vert f\Vert _{{\dot{B}}_{p,r_{1}}^{\sigma _{1}}}^{\theta } \Vert f\Vert _{{\dot{B}}_{p,r_2}^{\sigma _{2}}}^{1-\theta } \end{aligned}$$

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

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

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

The embedding properties will be used several times throughout the paper.

Proposition 5.1

• 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 $\sigma \in \mathbb {R}$, $1\le p_{1}\le p_{2}\le \infty$ and $1\le r_{1}\le r_{2}\le \infty ,$ 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, $p<\infty$).

In addition, we also recall the classical Bernstein inequality:

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

(5.1)

that holds for all function f such that $\textrm{Supp}\,{\mathcal {F}}f\subset \left\{ \xi \in \mathbb {R}^{d}: |\xi |\le R\lambda \right\}$ 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 $\textrm{Supp}\,{\mathcal {F}}f\subset \{\xi \in \mathbb {R}^{d}: R_{1}\lambda \le |\xi |\le R_{2}\lambda \}$ 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 [1]):

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

(5.2)

An obvious consequence of (5.1) and (5.2) is that $\Vert D^{k}f\Vert _{{\dot{B}}^{s}_{p, r}}\thickapprox \Vert f\Vert _{{\dot{B}}^{s+k}_{p, r}}$ for all $k\in \mathbb {N}$.

Moreover, a class of mixed space-time Besov spaces are also used when studying the evolution PDEs, which were firstly proposed by J.-Y. Chemin and N. Lerner in [9].

Definition 5.2

For $T>0,s\in \mathbb {R}, 1\le r,\theta \le \infty$, the homogeneous Chemin-Lerner spaces $\widetilde{L}^\theta _{T}({\dot{B}}^s_{p,r})$ are defined by

$$\begin{aligned} \widetilde{L}^\theta _{T}({\dot{B}}^s_{p,r}):=\Big \{f\in L^\theta (0,T;{\mathcal {S}}'_{0}):\Vert f\Vert _{\widetilde{L}^\theta _{T}({\dot{B}}^s_{p,r})}<\infty \Big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{\widetilde{L}^\theta _{T}({\dot{B}}^s_{p,r})}:=\Bigg (\sum _{q\in \mathbb {Z}}(2^{qs}\Vert \dot{\Delta }_qf\Vert _{L^\theta _{T}(L^{p})})^{r}\Bigg )^{1/r} \end{aligned}$$

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

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

Remark 5.1

It holds that

$$\begin{aligned} \left\| f\right\| _{{\widetilde{L}}^{\theta }_{T}({\dot{B}}^{s}_{p,r})}\le \left\| f\right\| _{L^{\theta }_{T}({\dot{B}}^{s}_{p,r})}\,\,\, \text{ if } \,\, \, r\ge \theta ;\ \ \ \ \left\| f\right\| _{{\widetilde{L}}^{\theta }_{T}({\dot{B}}^{s}_{p,r})}\ge \left\| f\right\| _{L^{\theta }_{T}({\dot{B}}^{s}_{p,r})}\,\,\, \text{ if }\,\,\, r\le \theta . \end{aligned}$$

1.2 Product estimates and composition estimates

The product estimates in Besov spaces play a fundamental role in bounding bilinear terms in (2.1) (see [1]).

Proposition 5.2

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

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

If $s_{1},s_{2}\le \frac{d}{p}$, $s_{1}+s_{2}>d\max \{0,\frac{2}{p}-1\}$, then

$$\begin{aligned} \Vert ab\Vert _{{\dot{B}}^{s_{1}+s_{2}-\frac{d}{p}}_{p,1}}\lesssim \Vert a\Vert _{{\dot{B}}^{s_{1}}_{p,1}}\Vert b\Vert _{{\dot{B}}^{s_{2}}_{p,1}}. \end{aligned}$$

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

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

System (2.1) also involves compositions of functions that are handled according to the following estimates.

Proposition 5.3

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 $F(f)\in {\dot{B}}^{s}_{p,r}\cap L^{\infty }$ for $f\in {\dot{B}}^{s}_{p,r}\cap L^{\infty }$, and

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

with $C>0$ depending only on $\Vert f\Vert _{L^{\infty }}$, $F'$ (and higher derivatives), s, p and d.

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

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

where $C>0$ is some constant depends on $\Vert f\Vert _{\dot{B}^{\frac{d}{p}}_{p,1}}$, F, s, p and d.

1.3 Proof of Proposition 3.2

Taking advantages of interpolation, it is enough to prove the case $p=\infty$. Inspired by Young inequality and heat kernel estimates under Fourier localization, we immediately have

$$\begin{aligned} \Vert e^{i\sqrt{\kappa } Ht}e^{\Delta t}f_{j}\Vert _{L^\infty }\lesssim e^{-2^{2j}t}\Vert e^{i\sqrt{\kappa } Ht}\psi _{j}\Vert _{L^\infty }\Vert f_{j}\Vert _{L^1}. \end{aligned}$$

(5.3)

where $\widehat{\psi }(\xi )=\varphi (\xi )$. Then (3.33) is given if the following estimate holds true:

$$\begin{aligned} \Vert e^{i\sqrt{\kappa } Ht}\psi _{j}\Vert _{L^\infty }\lesssim (\sqrt{\kappa } t)^{-\frac{d}{2}}. \end{aligned}$$

(5.4)

The (5.4) is proved by stationary phase method give in [17] and we shall focus on case $d\ge 2$ while $d=1$ is more direct by van der Corput’s lemma, see [38]. Actually, denote by $J_{m} (r)$ the Bessel function, one could write

$$\begin{aligned} e^{i\sqrt{\kappa } Ht}\psi _{j}=2^{jd}\int ^{\infty }_{0}e^{it\widetilde{H}(2^{j}r)}\varphi (r)r^{d-1}(r2^{j}|x|)^{-\frac{n-2}{2}}J_{\frac{n-2}{2}}(r2^{j}|x|)dr \end{aligned}$$

(5.5)

where $\widetilde{H}(r)=r\sqrt{1+\kappa r^2}$. Then it is not difficult to see

$$\begin{aligned}{} & {} \widetilde{H}'(r)\thicksim 1;\,\,\,\widetilde{H}''(r)\thicksim \kappa r,\,\,\,\widetilde{H}^{(m)}(r)\le (\kappa r)^{1-m},\,\,\,r\le \kappa ^{-\frac{1}{2}};\\{} & {} \widetilde{H}'(r)\thicksim \sqrt{\kappa }r;\,\,\,\widetilde{H}''(r)\thicksim \sqrt{\kappa },\,\,\,\widetilde{H}^{(m)}(r)\le \sqrt{\kappa }r^{2-m},\,\,\,r\ge \kappa ^{-\frac{1}{2}}. \end{aligned}$$

Therefore, we would consider case (i). $2^{j}\le \kappa ^{-\frac{1}{2}}$ and (ii). $2^{j}\le \kappa ^{-\frac{1}{2}}$ respectively. For case (i), the corresponding behaviors are closely linked with wave operator and we start with $|x|\le 2$. In fact, denote $D_{r}=\frac{1}{it\widetilde{H}'(2^j r)2^j}\frac{d}{dr}$, integral by parts in terms of $D_{r}$ immediately yields

$$\begin{aligned} e^{i\sqrt{\kappa } Ht}\psi _{j}= & {} 2^{jd}\int ^{\infty }_{0}D^{k}_{r}\big (e^{it\widetilde{H}(2^{j}r)}\big )\varphi (r)r^{d-1}(r2^{j}|x|)^{-\frac{n-2}{2}}J_{\frac{n-2}{2}}(r2^{j}|x|)dr\nonumber \\= & {} \frac{2^{jd}}{(it2^j)^k}\sum ^{k}_{m=0}\sum ^{m}_{l_{m}}\int ^{\infty }_{0}e^{it\widetilde{H}(2^{j}r)}\prod _{l_{m}}\partial ^{l_{m}}_{r}\left( \frac{1}{\widetilde{H}'(2^j r)}\right) \nonumber \\&\cdot&\partial ^{k-m}_{r}\left( \varphi (r)r^{d-1}(r2^{j}|x|)^{-\frac{n-2}{2}}J_{\frac{n-2}{2}}(r2^{j}|x|)\right) dr. \end{aligned}$$

(5.6)

where $m=l_{1}+l_{2}+...+l_{m}$. Keep in mind that for any $m\ge 0$

$$\begin{aligned} \frac{d^{m}}{d^{m}_{r}}\left( \frac{1}{\widetilde{H}'(2^j r)}\right) \le c,\,\,\,\textrm{for}\,\,\,2^{j}\le \kappa ^{-\frac{1}{2}}, \end{aligned}$$

the vanishing property of the Bessel function at the origin indicates

$$\begin{aligned} |e^{i\sqrt{\kappa } Ht}\psi _{j}|\le & {} Ct^{-k}2^{j(d-k)}. \end{aligned}$$

(5.7)

Hence, taking $k=\frac{d}{2}$ yields (5.4). For case $|x|\ge 2$, we rewrite (5.5) into

$$\begin{aligned} e^{i\sqrt{\kappa } Ht}\psi _{j}=2^{jd}\int ^{\infty }_{0}e^{it(\widetilde{H}(2^{j}r)-r|x|)}\varphi (r)r^{d-1}h(r|x|)dr \end{aligned}$$

(5.8)

where

$$\begin{aligned} {\mathcal {R}}(e^{ir}h(r))=cr^{-\frac{n-2}{2}}J_{\frac{n-2}{2}}(r). \end{aligned}$$

At this moment, we start with |x| fulfills $\frac{1}{2}\inf \limits _{r}t2^{j}\widetilde{H}'(2^{j}r)\le |x|\le 2\sup \limits _{r} t2^{j}\widetilde{H}'(2^{j}r)$. Denote ${\bar{H}}(r)=\widetilde{H}(2^{j}r)-r|x|$, it is clear that ${\bar{H}}''(r)=2^{2j}\widetilde{H}''(2^{j}r)\ge \kappa 2^{3j}$, therefore, by van der Corput’s lemma and pointwise estimates for h (see in [17]), there holds

$$\begin{aligned} |e^{i\sqrt{\kappa } Ht}\psi _{j}|\le & {} C(|t|\kappa 2^{3j})^{-\frac{1}{2}}\int ^{\infty }_{0}\big |\frac{d}{dr}(\varphi (r)r^{d-1}h(r|x|))\big |dr\le t^{-\frac{d}{2}}\kappa ^{-\frac{1}{2}}2^{\frac{d-2}{2}j}, \end{aligned}$$

which implies (5.4) by the fact $2^{j}\le \kappa ^{-\frac{1}{2}}$. For $|x|\le \frac{1}{2}\inf \limits _{r}t2^{j}\widetilde{H}'(2^{j}r)$ and $2\sup \limits _{r} t2^{j}\widetilde{H}'(2^{j}r)\le |x|$, there holds

$$\begin{aligned} \big |{\bar{H}}'(r)\big |\ge c2^{j},\,\,\,c>0, \end{aligned}$$

we could repeat same calculations as (5.6)–(5.7) and conclude with (5.4).

The case $2^{j}\ge \kappa ^{-\frac{1}{2}}$ is treated as low frequencies above, in which part could be regarded as a Schr$\ddot{o}$dinger  semi-group with parameter $\sqrt{\kappa }$. We omit the detailed proof and conclude with (5.4).