Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System

Abstract

The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on ${\mathbb{R}}^2$. The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) for compressible Navier–Stokes and compressible Navier–Stokes–Korteweg systems. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space–time $L^2$ of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by $L^2$ on space, decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz’s energy estimate, and the Fefferman–Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.

1. Introduction

We study the asymptotic behavior of solutions to the following compressible Navier–Stokes–Korteweg system in ${\mathbb{R}}^2$, called CNSK:

$$\left\{\begin{array}{c}{\partial }_t\rho +\mathrm{div}M=0,\hfill \\ {\partial }_{t}M+\mathrm{div}\left(\frac{M\otimes M}{\rho }\right)+\nabla P\left(\rho \right)=\mathrm{div}\left(\mathcal{S}\left(\frac{M}{\rho }\right)+\mathcal{K}\left(\rho \right)\right),\hfill \\ \rho (x,0)={\rho }_0,M(x,0)=M_0.\hfill \end{array}\right.$$

(1)

Here, $\rho =\rho (x,t)$ and $M=(M_1(x,t),M_2(x,t))$ are unknown density and momentum, respectively, at time $t\in {\mathbb{R}}^{+}$ and position $x\in {\mathbb{R}}^2$; ${\rho }_0={\rho }_0\left(x\right)$ and $M_0=M_0\left(x\right)$ are given initial data; $\mathcal{S}$ and $\mathcal{K}$ denote the viscous stress tensor and Korteweg stress tensor, respectively, given by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left(\mathcal{S}\right)}_{i,j}\left(\frac{M}{\rho }\right)=\left({\mu }^{\prime }\mathrm{div}\frac{M}{\rho }\right){\delta }_{i,j}+2\mu d_{ij}\left(\frac{M}{\rho }\right),\hfill \\ {\left(\mathcal{K}\right)}_{i,j}\left(\rho \right)=\frac{\kappa }{2}(\Delta {\rho }^2-{|\nabla \rho |}^2){\delta }_{i,j}-\kappa \frac{\partial \rho }{\partial x_i}\frac{\partial \rho }{\partial x_j},\hfill \end{array}\right.\end{array}$$

(2)

where $d_{ij}\left(\frac{M}{\rho }\right)=\frac{1}{2}\left(\frac{\partial }{\partial x_i}{\left(\frac{M}{\rho }\right)}_j+\frac{\partial }{\partial x_j}{\left(\frac{M}{\rho }\right)}_i\right)$; $\mu$ and ${\mu }^{\prime }$ are the viscosity coefficients, supposed to be constants satisfying

$$\mu >0,\mu +{\mu }^{\prime }\ge 0.$$

$\kappa$ is the capillary constant satisfying $\kappa \ge 0$. If $\kappa =0$ in the Korteweg tensor, the usual compressible Navier–Stokes equation (CNS) appears: $P=P\left(\rho \right)$ is pressure assumed to be a smooth function of $\rho$ satisfying $P^{\prime }\left({\rho }_{\ast }\right)>0$, where ${\rho }_{\ast }$ is a given positive constant and ${}^{\top }({\rho }_{\ast },0)$ is a given constant state for (1). We consider solutions to (1) around the constant state.

(1) is the system of equations of motion of liquid–vapor type two-phase flow with phase transition in a compressible fluid, similarly as in [1]. To describe the phase transition, this model uses the diffusive interface. Hence, the phase boundary is regarded as a narrow transition layer and change of the density prescribes fluid state. Due to the diffusive interface, it is enough to consider one set of equations and a single spatial domain and difficulty of topological change of interface do not occur. If we assume that $\kappa =0$, the CNS that describes the motion of one-phase compressible fluid is obtained. Hence, (1) is obtained from adding higher-order derivative terms for $\rho$, including $\nabla \Delta \rho$ and $\nabla \rho \otimes \nabla \rho$ to CNS.

For the derivation of (1), Van der Waals [2] suggested that a phase-transition boundary be regarded as a thin transition zone, i.e., a diffusive interface caused by a steep gradient of density. On the basis of his idea, Korteweg [3] modified the stress tensor of the Navier–Stokes equation to that including term $\nabla \rho \otimes \nabla \rho$. Dunn and Serrin [4] generalized Korteweg’s work and strictly provided System (1) with (2). In their recent works, Heida and Málek [5] derived (1) by the entropy production method.

We focus on the diffusion wave that stems from hyperbolic and parabolic aspects of the system. The diffusion wave is given by convolution between heat kernel and the fundamental solution to the wave equation. The importance of the diffusion wave for problems in one-dimensional cases was first recognized by Liu [6] for the study of stability of shock waves for viscous conservation laws. The multidimensional diffusion wave with a time-decay estimate of solutions was studied for CNS by Hoff and Zumbrun [7,8], and Kobayashi and Shibata [9]; for the viscoelastic equation on ${\mathbb{R}}^n$$(n\ge 2)$, by Shibata [10]. Let $u={}^{\top }(\rho -{\rho }_{\ast },M)$ be a solution to CNS and set $E:=\parallel u_0{\parallel }_{H^{s+\ell }\cap L^1}$, where $u_0={}^{\top }({\rho }_0-{\rho }_{\ast },M_0)={}^{\top }(\rho \left(0\right)-{\rho }_{\ast },M\left(0\right))$, s is an integer part of $n/2$ and ℓ is integer satisfying $\ell \ge 3$. Then, the authors in [7,8,9] showed that the linear parts decay faster than nonlinear parts do in the Duhamel formula, and the asymptotic behavior in $L^p\left({\mathbb{R}}^n\right)$$(p>2,n\ge 2)$ of solutions was presented as

$$\begin{array}{c}\hfill u\left(t\right)\sim \underset{\mathrm{solutions\; to\; linearized\; system}}{\underbrace{\left(\begin{array}{c}0\\ {\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}\end{array}\right)+\left(\begin{array}{c}\rho \left(t\right)-{\rho }_{\ast }\\ M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}\end{array}\right)}}+\underset{\mathrm{nonlinear\; parts}}{\underbrace{\cdots }}\mathrm{in}{L}^p\left({\mathbb{R}}^n\right)\end{array}$$

(3)

as t goes to infinity. Here, notation $u\left(t\right)\sim f\left(t\right)$ in $L^p\left({\mathbb{R}}^n\right)$ is defined as

$$\underset{t\to \infty }{lim\; sup}{\parallel u\left(t\right)-f\left(t\right)\parallel }_{L^p\left({\mathbb{R}}^n\right)}\le C$$

for a positive number C independent of t, similar notation is used hereafter. ${\mathcal{K}}_{\nu }={\mathcal{K}}_{\nu }(t,x)$ is the standard heat kernel and $M_{0,in}$ is a divergence-free part of $M_0$, given by

$${\mathcal{K}}_{\nu }={\mathcal{F}}^{-1}\left(e^{-{\nu \left|\xi \right|}^{2}t}\right),M_{0,in}={\mathcal{F}}^{-1}\left\{\left(I_n-\frac{\xi {}^{\top }\xi }{{\left|\xi \right|}^2}\right)M_0\right\}.$$

More precisely, it holds that

$${\parallel u\left(t\right)\parallel }_{L^p\left({\mathbb{R}}^n\right)}\le C{(1+t)}^{-\frac{n}{2}(1-\frac{1}{p})},{\parallel u\left(t\right)-G\left(t\right)\ast u_0\parallel }_{L^p\left({\mathbb{R}}^n\right)}\le C{(1+t)}^{-\frac{n}{2}(1-\frac{1}{p})-\frac{1}{2}}$$

and

$$\begin{array}{c}\hfill \parallel \left(\begin{array}{c}\rho \left(t\right)-{\rho }_{\ast }\\ M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}\end{array}\right){\parallel }_{L^p\left({\mathbb{R}}^n\right)}\le C{(1+t)}^{-\frac{n}{2}(1-\frac{1}{p})+\left(\frac{n-1}{4}\right)(\frac{2}{p}-1)}\end{array}$$

for $t>0$, where $G=G\left(t\right)$ is the Green function of linearized CNS and $\left(\frac{n-1}{4}\right)(\frac{2}{p}-1)\le 0$ when $2\le p\le \infty$. ${\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$ and $M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$ are the Stokes flow and potential flow parts of M, respectively, in the Helmholtz decomposition. $\rho \left(t\right)-{\rho }_{\ast }$ and $M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$ are given by the Green matrix of the linearized system, which consists of the convolution with the Green functions of the diffusion and the wave equations and are called the diffusion-wave part. In addition, when $p=2$, the behaviors of both of $\rho \left(t\right)-{\rho }_{\ast }$ and $M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$ coincide with those of ${\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$ as the parabolic-type decay rate. Kobayashi and Tsuda studied the diffusion-wave property for (1) in [11].

In this paper, we consider the linearized system for (1). Under some initial conditions given by the Hardy space ${\mathcal{H}}^1$(defined below), we show some space–time $L^2$ estimates for the density and the Stokes flow part of the momentum. The potential flow part of the momentum is also shown to grow at the rate of logarithmic order in spatial-time $L^2$ norm. The precise initial condition given by the Hardy space is shown below. Here, we assume a stronger initial condition by ${\mathcal{H}}^1$ for density than that by $L^1$, in contrast to [11]; thus, our results may show a gain of regularity by the Hardy space in the decay estimates. Such a gain is also obtained for heat equations (see Appendix A). Nonlinear terms are expected to decay faster than the linear terms in the Duhamel formula do, as in [7,11]. As a consequence, the leading terms of the asymptotic expansion of solution u for (1) are given by

$$\begin{array}{c}\hfill u\left(t\right)\sim \underset{\mathrm{solutions\; to\; linearized\; system}}{\underbrace{\left(\begin{array}{c}0\\ M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}\end{array}\right)+\left(\begin{array}{c}\rho \left(t\right)-{\rho }_{\ast }\\ {\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}\end{array}\right)}}+\underset{\mathrm{nonlinear\; parts}}{\underbrace{\cdots }}\mathrm{in}{L}^2(0,\infty ;{\mathbb{R}}^2).\end{array}$$

(4)

Precisely, the following estimates hold true for solutions to the linearized CNSK:

$${\displaystyle {\int }_0^t{\parallel M\left(\tau \right)-{\mathcal{K}}_{\nu }\left(\tau \right)\ast M_{0,in}\parallel }_{L^2\left({\mathbb{R}}^2\right)}^{2}d\tau \sim logtast\to \infty ,}$$

$$\parallel \rho -{\rho }_{\ast }{\parallel }_{L^2(0,\infty ;L^2\left({\mathbb{R}}^2\right))}+{\parallel {\mathcal{K}}_{\nu }\ast M_{0,in}\parallel }_{L^2(0,\infty ;L^2\left({\mathbb{R}}^2\right))}<\infty .$$

The above behaviors of the diffusion-wave parts $\rho \left(t\right)-{\rho }_{\ast }$ and $M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$ are clearly different from (3). Measuring by $L^2$ on space, $M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$ decays slower than the Stokes flow part of M does. By the dependence on $\kappa$ of constants, the above estimate (4) also holds true for CNS (Theorems 2 and 3). We also obtain a decay rate of $L^2$ norm of density (Theorem 4). Furthermore, if $M_0\in {\mathcal{H}}^1$, space–time $L^2$ boundedness is obtained for $M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$, $\rho \left(t\right)-{\rho }_{\ast }$ and ${\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$.

The proofs of the main results are based on Morawetz-type energy estimates for a linearized system. The diffusion-wave part of density $\rho \left(t\right)-{\rho }_{\ast }$ is bounded in space–time $L^2$. We rewrite (1) to some linear doubly dispersion equation for $\rho$ and apply a modified version of Morawetz’s energy estimate. A preliminary function is introduced in the Morawetz estimate (see (12) below), which is defined by use of a doubly Laplace-type equation. The existence of solution to the linear doubly Laplace-type equation is shown by use of the linear theory on ${\mathcal{H}}^1$, which may be of its own interest. Through the preliminary function, we perform Morawetz-type energy estimates utilizing the Fefferman–Stein inequality on the duality between ${\mathcal{H}}^1$ and the space of functions of bounded mean oscillation. Another diffusion-wave part $M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$ is shown to grow at the rate of order $logt$ as t goes to infinity. Here, we use fundamental solutions for the linearized system given in [11]. Since a high-frequency part of the solutions exponentially decays, a low-frequency part only has to be estimated here. By direct computation with the explicit form of the Green matrix, we obtain the growth order for $M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$. For the Stokes flow part ${\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$, space–time $L^2$ boundedness is derived in Theorem A3 bellow. These estimates are combined for a diffusion wave, and the Stokes flow parts yields asymptotic expansion (4).

This paper is organized as follows. In Section 2 some notations and lemmas are given. In Section 3, the main results are presented. In Section 4, the proofs of the estimates for the diffusion wave parts are demonstrated.

2. Preliminaries

In this section, we introduce notations such as function spaces that are used in this paper. We also present lemmas needed in the proof of the main result.

The norm on X is denoted by ${\parallel ·\parallel }_X$ for a given Banach space X.

Let $1\leqq p\leqq \infty .$ $L^p$ is the usual Lebesgue space of $p$th powered integrable and essentially bounded functions on ${\mathbb{R}}^2$ for a finite p and $p=\infty$, respectively. Let k be a non-negative integer. $W^{k,p}$ and $H^k$ are the usual Sobolev spaces of order k, based on $L^2$ and $L^p$, respectively. As usual, $H^0$ is defined by $H^0:=L^2$.

We also use notation $L^p$ to denote the function space of all vector fields $w={}^{\top }(w_1,w_2)$ on ${\mathbb{R}}^2$ satisfying $w_j\in L^p$$(j=1,2)$, and ${\parallel ·\parallel }_{L^p}$ is norm ${\parallel ·\parallel }_{{\left(L^p\right)}^2}$ for brevity if no confusion occurs. Similarly, a function space X is the linear space of all vector fields $w={}^{\top }(w_1,w_2)$ on ${\mathbb{R}}^2$ satisfying $w_j\in X$$(j=1,2)$, and ${\parallel ·\parallel }_X$ is norm ${\parallel ·\parallel }_{X^2}$ if no confusion occurs.

Let $u={}^{\top }(\varphi ,m)$ with $\varphi \in H^k$ and $m={}^{\top }(m_1,m_2)\in H^j$. Then, norm ${\parallel u\parallel }_{H^k\times H^j}$ is defined as that of u on $H^k\times H^j$

$${\parallel u\parallel }_{H^k\times H^j}:={\left({\parallel \varphi \parallel }_{H^k}^2+{\parallel w\parallel }_{H^j}^2\right)}^{\frac{1}{2}}.$$

In particular, if $j=k$, we put

$$H^k:=H^k\times {\left(H^k\right)}^2{,\parallel u\parallel }_{H^k}:={\parallel u\parallel }_{H^k\times {\left(H^k\right)}^2}(u={}^{\top }(\varphi ,m)).$$

Let X and Y be given Banach spaces. For $u={}^{\top }(\varphi ,m)\in X\times Y$ with $m={}^{\top }(m_1,m_2)$, we similarly set

$${\parallel u\parallel }_{X\times Y}:={\left({\parallel \varphi \parallel }_X^2+{\parallel m\parallel }_Y^2\right)}^{\frac{1}{2}}.$$

More generally, in the case that $Y=X^2$, let

$$X:=X\times X^2{,\parallel u\parallel }_X:={\parallel u\parallel }_{X\times X^2}.$$

Symbols $\widehat{f}$ and $\mathcal{F}\left[f\right]$ stand for the Fourier transform of f with respect to space variable x

$$\begin{array}{c}\hfill \widehat{f}\left(\xi \right)=\mathcal{F}\left[f\right]\left(\xi \right):={\int }_{{\mathbb{R}}^2}f\left(x\right)e^{-ix·\xi }dx,\xi \in {\mathbb{R}}^2.\end{array}$$

Furthermore, the inverse Fourier transform of f is defined by

$$\begin{array}{c}\hfill {\mathcal{F}}^{-1}\left[f\right]\left(x\right):={\left(2\pi \right)}^{-2}{\int }_{{\mathbb{R}}^2}f\left(\xi \right)e^{i\xi ·x}d\xi ,x\in {\mathbb{R}}^2.\end{array}$$

For a non-negative number s, $\left[s\right]$ is the Gaussian symbol that denotes the integer part of s. Symbol $“\ast ”$ denotes the convolution on space variable x.

Now, we prepare Hardy space ${\mathcal{H}}^1$ and BMO space.

Definition 1.

Hardy space ${\mathcal{H}}^1={\mathcal{H}}^1\left({\mathbb{R}}^2\right)$ consists of integrable functions on ${\mathbb{R}}^2$, $f\in L^1\left({\mathbb{R}}^2\right)$ such that

$$\begin{array}{c}\hfill {\parallel f\parallel }_{{\mathcal{H}}^1\left({\mathbb{R}}^2\right)}={\int }_{{\mathbb{R}}^2}\underset{r>0}{sup}|{\varphi }_r\ast f\left(x\right)|dx\end{array}$$

is finite, where ${\varphi }_r\left(x\right)=r^{-n}\varphi \left(r^{-1}x\right)$ for $r>0$, and ϕ is a smooth function on ${\mathbb{R}}^2$ with compact support in an unit ball with center of the origin, $B_1\left(0\right)=\{x\in {\mathbb{R}}^2;\left|x\right|<1\}$. The definition does not depend on the choice of a function ϕ.

Definition 2.

Let f be locally integrable in ${\mathbb{R}}^2$, $f\in L_{loc}^1\left({\mathbb{R}}^2\right)$. We say that f is of bounded mean oscillation, abbreviated as $BMO$, if

$$\begin{array}{c}\hfill {\parallel f\parallel }_{\mathrm{BMO}}=\underset{B\subset {\mathbb{R}}^2}{sup}\frac{1}{\left|B\right|}{\int }_B|f-{\left(f\right)}_B|dx<\infty ,\end{array}$$

where the supremum ranges over all finite balls B⊂${\mathbb{R}}^2$, $\left|B\right|$ is the 2-dimensional Lebesgue measure of B, and ${\left(f\right)}_B$ denotes the integral mean of f over B, namely ${\left(f\right)}_B=\frac{1}{\left|B\right|}{\int }_{B}f\left(x\right)dx$.

The class of functions of $BMO$, modulo constants, is a Banach space with norm ${\parallel ·\parallel }_{BMO}$ defined above.

We crucially use the decisive Fefferman–Stein inequality, which means the duality between ${\mathcal{H}}^1\left({\mathbb{R}}^2\right)$ and $BMO\left({\mathbb{R}}^2\right)$, i.e., ${\left({\mathcal{H}}^1\left({\mathbb{R}}^2\right)\right)}^{\ast }=BMO\left({\mathbb{R}}^2\right)$. For the proof, see [12].

Lemma 1.

(Fefferman-Stein inequality) There is a positive constant C, such that, if $f\in {\mathcal{H}}^1\left({\mathbb{R}}^2\right)$ and $g\in BMO\left({\mathbb{R}}^2\right)$, then

$$\begin{array}{c}\hfill \left|{\int }_{{\mathbb{R}}^2}fgdx\right|\le {C\parallel f\parallel }_{{\mathcal{H}}^1\left({\mathbb{R}}^2\right)}{\parallel g\parallel }_{BMO\left({\mathbb{R}}^2\right)}.\end{array}$$

We also recall the well-known Poincaré inequality.

Lemma 2.

It holds that

$${\parallel f\parallel }_{BMO\left({\mathbb{R}}^2\right)}\le {\parallel \nabla f\parallel }_{L^2\left({\mathbb{R}}^2\right)}$$

for $f\in H^1\left({\mathbb{R}}^2\right)$.

We denote by $C_{0,\sigma }^{\infty }$ the set of all vector-valued functions $\varphi ={}^{\top }({\varphi }_1,{\varphi }_2)$ whose each ${\varphi }_j$$(j=1,2)$ is $C^{\infty }$ function having compact support, and satisfying that $\mathrm{div}\varphi =0$. For $1\le q<\infty$, $L_{\sigma }^q$ is the closure of $C_{0,\sigma }^{\infty }$ with respect to the $L^q$ norm.

A spatial weighted function space $W_w^{1,2}\left({\mathbb{R}}^2\right)$ is defined by

$$\begin{array}{c}\hfill W_w^{1,2}\left({\mathbb{R}}^2\right)=\left\{u:\frac{u}{w\left(x\right)}\in L^2\left({\mathbb{R}}^2\right),\nabla u\in L^2\left({\mathbb{R}}^2\right)\right\},\end{array}$$

where $w\left(x\right)$ is a spatial weight defined by $w\left(x\right)=(1+|x\left|\right)log(2+|x\left|\right)$.

The following Hölder type inequality was proved by Amrouche and Nguyen [13].

Lemma 3.

([13] Corollary 2.10) Let $f\in L_{\sigma }^1\left({\mathbb{R}}^2\right)$. Then it holds true that ${\int }_{{\mathbb{R}}^2}f\left(x\right)dx=0$ and that, for such f and any $g\in W_w^{1,2}\left({\mathbb{R}}^2\right)\cap L^{\infty }\left({\mathbb{R}}^2\right)$,

$$\begin{array}{c}\hfill \left|{\int }_{{\mathbb{R}}^2}fgdx\right|\le {C\parallel f\parallel }_{L^1}{\parallel \nabla g\parallel }_{L^2}.\end{array}$$

Since $W^{1,2}\left({\mathbb{R}}^2\right)\subset W_w^{1,2}\left({\mathbb{R}}^2\right)$, Lemma 3 also yields the following.

Corollary 1.

Let $f\in L_{\sigma }^1\left({\mathbb{R}}^2\right)$. Then, there holds that ${\int }_{{\mathbb{R}}^2}f\left(x\right)dx=0$ and, for such f and any $g\in W^{1,2}\left({\mathbb{R}}^2\right)\cap L^{\infty }\left({\mathbb{R}}^2\right)$,

$$\begin{array}{c}\hfill \left|{\int }_{{\mathbb{R}}^2}fgdx\right|\le {C\parallel f\parallel }_{L^1}{\parallel \nabla g\parallel }_{L^2}.\end{array}$$

3. Main Results

In this section, we consider the linearized system corresponding to (1) and present some decay estimates for its solution. akey estimate to show (4) is space–time $L^2$ boundedness of the density for the linearized system. First, (1) is reformulated and linearized as follows. Hereafter, we assume that ${\rho }_{\ast }=1$ without loss of generality. We also set

$$\varphi =\rho -1,m=\frac{M}{\gamma },\gamma =\sqrt{P^{\prime }\left(1\right)}.$$

Substituting $\varphi$ and m into (1), we have system of equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_t\varphi +\gamma \mathrm{div}m=0,\hfill \\ {\partial }_{t}m-\nu \Delta m-\tilde{\nu }\nabla \mathrm{div}m+\gamma \nabla \varphi -{\kappa }_0\nabla \Delta \varphi =f\left(u\right),\hfill \\ {\varphi |}_{t=0}={\varphi }_0{,m|}_{t=0}=m_0,\hfill \end{array}\right.\end{array}$$

(5)

where we use notation

$$u={}^{\top }(\varphi ,m),{\varphi }_0={\rho }_0-1,m_0=\frac{M_0}{\gamma },\nu =\mu ,\tilde{\nu }=\mu +{\mu }^{\prime },{\kappa }_0=\frac{\kappa }{\gamma }$$

and put

$$\begin{array}{ccc}\hfill f\left(u\right)& =& -\{\gamma \mathrm{div}(m\otimes m)+\gamma \mathrm{div}(P_{\left(1\right)}\left(\varphi \right)\varphi m\otimes m)+\frac{1}{\gamma }\nabla \left(P_{\left(2\right)}\left(\varphi \right){\varphi }^2\right)\hfill \\ & & -\nu \Delta \left(P_{\left(1\right)}\left(\varphi \right)\varphi m\right)-\tilde{\nu }\nabla \mathrm{div}\left(P_{\left(1\right)}\left(\varphi \right)\varphi m\right)-\mathrm{div}\mathsf{\Phi }\left(\varphi \right)\},\hfill \\ \hfill P_{\left(1\right)}\left(\varphi \right)& =& {\int }_0^1{f}^{\prime }(1+\tau \varphi )d\tau ,f\left(\tau \right)=\frac{1}{\tau },\tau \in \mathbb{R},\hfill \\ \hfill P_{\left(2\right)}\left(\varphi \right)& =& {\int }_0^1(1-\tau )P″\left(1+\tau \varphi \right)d\tau ,\hfill \\ \hfill \mathsf{\Phi }\left(\varphi \right)& =& {\kappa }_0\left\{\varphi \Delta \varphi I_2+(\nabla \varphi )·(\nabla \varphi )I_2-\frac{{|\nabla \varphi |}^2}{2}{I}_2-\nabla \varphi \otimes \nabla \varphi \right\}.\hfill \end{array}$$

Therefore, (1) is linearized as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_t\varphi +\gamma \mathrm{div}m=0,\hfill \\ {\partial }_{t}m-\nu \Delta m-\tilde{\nu }\nabla \mathrm{div}m+\gamma \nabla \varphi -{\kappa }_0\nabla \Delta \varphi =0,\hfill \\ {\varphi |}_{t=0}={\varphi }_0{,m|}_{t=0}=m_0.\hfill \end{array}\right.\end{array}$$

(6)

By (6), $\varphi$ satisfies the following doubly dissipative equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{tt}\varphi -(\nu +\tilde{\nu })\Delta {\varphi }_t-{\gamma }^2\Delta \varphi +\gamma {\kappa }_0{\Delta }^2\varphi =0,\hfill \\ \varphi (x,0)={\varphi }_0,{\partial }_t\varphi (x,0)=-\gamma \mathrm{div}{m}_0.\hfill \end{array}\right.\end{array}$$

Due to the positivity of $\nu$ and $\tilde{\nu }$, we may suppose that $\nu +\tilde{\nu }=1$ and $\gamma =1$ without loss of generality. Then, $\varphi$ satisfies

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{tt}\varphi -\Delta {\varphi }_t-\Delta \varphi +{\kappa }_0{\Delta }^2\varphi =0,\hfill \\ \varphi (x,0)={\varphi }_0,{\partial }_t\varphi (x,0)=-\mathrm{div}{m}_0.\hfill \end{array}\right.\end{array}$$

Now, we state the existence of solutions to (7) in the energy class, defined in the following.

Definition 3.

A function ϕ defined on $(0,\infty )\times {\mathbb{R}}^2$ is called to be a solution to (7) if ϕ belongs to $C([0,\infty );H^2)$ with ${\partial }_t\varphi \in C([0,\infty );L^2)$ and satisfies (7) in the distribution sense, i.e., satisfies the following conditions:

(i) For each $T>0$, $\phi \in H^2$ and a.e. $0\le t<T,$

$${({\partial }_{tt}\varphi ,\phi )}_{L^2}+{(\nabla {\varphi }_t,\nabla \phi )}_{L^2}+{(\nabla \varphi ,\nabla \phi )}_{L^2}+{\kappa }_0{(\Delta \varphi ,\Delta \phi )}_{L^2}=0;$$

(ii) ${\varphi }_0(x,0)={\varphi }_0$ and ${\partial }_t\varphi (x,0)=-\mathrm{div}{m}_0$.

Theorem 1.

For each $({\varphi }_0,m_0)\in H^2\times H^1$, there exists a unique solution $\varphi \in C([0,\infty );H^2)$ with ${\partial }_t\varphi \in C([0,\infty );L^2)$ to (7) such that

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{2}(\parallel {\varphi }_t\left(t_2\right){\parallel }_{L^2}^2+\parallel \nabla \varphi \left(t_2\right){\parallel }_{L^2}^2+{\kappa }_0\parallel \Delta \varphi \left(t_2\right){\parallel }_{L^2}^2)+{\int }_{t_1}^{t_2}{\parallel \nabla {\varphi }_{\tau }\left(\tau \right)\parallel }_{L^2}^{2}d\tau }\hfill \\ & & =\frac{1}{2}(\parallel {\varphi }_t\left(t_1\right){\parallel }_{L^2}^2+\parallel \nabla \varphi \left(t_1\right){\parallel }_{L^2}^2+{\kappa }_0\parallel \Delta \varphi \left(t_1\right){\parallel }_{L^2}^2)\hfill \end{array}$$

(7)

holds for any $0\le t_1<t_2<+\infty$.

Theorem 1 is valid by the standard Galerkin method based on energy inequality (7) in a similar manner to the proof of Theorem 3.1 in Huafei and Yadong [14]. In ([14] Theorem 3.1), they added $-\Delta u_{tt}$ in a linear system and considered nonlinearity. We apply a similar manner as [14] to the proof without $-\Delta u_{tt}$ and nonlinearity; thus, we do not obtain in Theorem 1 that ${\partial }_t\varphi \in L^{\infty }(0,\infty ;H^1)$ in contrast to ([14] Theorem 3.1); we omitted the details.

In $L^2(0,\infty ;L^2)$ for a solution $\varphi$ to (7).

Theorem 2.

Suppose that ${\varphi }_0\in H^2\cap {\mathcal{H}}^1$, $m_0\in H^1$ and $m_0+\nabla {\varphi }_0\in {\mathcal{H}}^1$. Set

$$J_0=({\kappa }_0+{\kappa }_0^2)\{\parallel {}^{\top }({\varphi }_0,m_0){\parallel }_{H^1}^2+\parallel \Delta {\varphi }_0{\parallel }_{L^2}^2\}+(1+{\kappa }_0)\parallel m_0+\nabla {\varphi }_0{\parallel }_{{\mathcal{H}}^1}^2+{\parallel {\varphi }_0\parallel }_{{\mathcal{H}}^1\cap L^2}^2.$$

Let ϕ be a solution to (7). Then, it holds true that

$${\displaystyle {\int }_0^t{\parallel \varphi \left(\tau \right)\parallel }_{L^2}^{2}d\tau \le C{J}_0}$$

for any $t>0$, where C is a positive constant independent of t and ${\kappa }_0$.

In the case that ${\kappa }_0=0$, we also have the time–space $L^2$ estimate for linearized CNS.

Theorem 3.

Let $({\varphi }_0,m_0)\in H^1$, ${\varphi }_0\in {\mathcal{H}}^1$ and $m_0+\nabla {\varphi }_0\in {\mathcal{H}}^1$. Set

$$J_1=\parallel m_0+\nabla {\varphi }_0{\parallel }_{{\mathcal{H}}^1}^2+{\parallel {\varphi }_0\parallel }_{{\mathcal{H}}^1\cap L^2}^2.$$

Let ϕ be a solution to (A2) in Appendix B. Then, there holds that

$${\displaystyle {\int }_0^t{\parallel \varphi \left(\tau \right)\parallel }_{L^2}^{2}d\tau \le C{J}_1}$$

for any $t>0$, where C is a positive constant independent of t.

Next, we have a time-decay estimate of the solution in the energy class to (7). By Theorem 1 and the Sobolev inequality $\varphi \in C([0,\infty );L^2)$. We have the following:

Theorem 4.

Under the assumption of Theorem 2, it holds that

$${(1+t)\parallel \varphi \left(t\right)\parallel }_{L^2}^2\le C{J}_0$$

for any $t>0$, where C is a positive constant independent of t.

We now recall the existence of solutions to linear system (6) in the energy class in order to consider another diffusion-wave part $m-{\mathcal{K}}_{\nu }\ast m_{0,in}$. System (6) is rewritten as

$${\partial }_{t}u+Au=0,$$

(8)

where

$$\begin{array}{c}\hfill u={}^{\top }(\varphi ,m),A=\left(\begin{array}{cc}0& \gamma \mathrm{div}\\ \gamma \nabla -{\kappa }_0\nabla \Delta & -\nu \Delta -\tilde{\nu }\nabla \mathrm{div}\end{array}\right)\end{array}$$

(9)

Let us introduce a semigroup $S\left(t\right)=e^{-tA}$ generated by A;

$$S\left(t\right)=e^{-tA}={\mathcal{F}}^{-1}{e}^{-t{\widehat{A}}_{\xi }}\mathcal{F},$$

where

$$\begin{array}{c}\hfill {\widehat{A}}_{\xi }=\left(\begin{array}{cc}0& i\gamma {}^{\top }\xi \\ i\gamma \xi +i{\kappa }_0{\left|\xi \right|}^2\xi & {\nu \left|\xi \right|}^2{I}_n+\tilde{\nu }\xi {}^{\top }\xi \end{array}\right)(\xi \in {\mathbb{R}}^2).\end{array}$$

Theorem 5.

([15] Proposition 3.3) Let s be a non-negative integer satisfying $s\ge 2$. Then, $S\left(t\right)=e^{-tA}$ is a contraction semigroup for (8) on $H^s\times H^{s-1}$. In addition, for each $u_0={}^{\top }({\varphi }_0,m_0)\in H^s\times H^{s-1}$ and all $T>0$, $S\left(t\right)$ satisfies

$$S(·)u_0\in C([0,T];H^s\times H^{s-1}),S\left(0\right)u_0=u_0$$

and there holds the estimate

$$\begin{array}{c}\hfill \parallel S\left(t\right)u_0{\parallel }_{H^s\times H^{s-1}}\le {\parallel u_0\parallel }_{H^s\times H^{s-1}}\end{array}$$

(10)

for $u_0={}^{\top }({\varphi }_0,m_0)\in H^s\times H^{s-1}$ and $t\ge 0$.

Remark 1.

Proposition 3.3 in [15] is stated on the three-dimensional case. However, the proof is based on the standard energy estimate for the resolvent problem in the Fourier space, and it can also be applied to our two-dimensional case.

Lastly, another diffusion-wave part $m-{\mathcal{K}}_{\nu }\ast m_{0,in}$ is shown to grow in $L^2(0,\infty ;L^2)$ at the rate of logarithmic order.

Theorem 6.

Let $u_0={}^{\top }({\varphi }_0,m_0)\in H^2\times H^1$ and u be a solution of (6), $u\left(t\right)=S\left(t\right)u_0$, as in Theorem 5. Suppose that $m_0\in L^2\cap L^1$, $\left|x\right|m_0\in L^1$ and ${\widehat{m}}_0\left(0\right)\ne 0$. Then, it holds true that

$${\int }_0^t{\parallel m\left(\tau \right)-{\mathcal{K}}_{\nu }\left(\tau \right)\ast m_{0,in}\parallel }_{L^2}^{2}d\tau \sim logtast\to \infty ,$$

precisely,

$${\displaystyle \underset{t\to \infty }{lim\; sup}|{\int }_0^t{\parallel m\left(\tau \right)-{\mathcal{K}}_{\nu }\left(\tau \right)\ast m_{0,in}\parallel }_{L^2}^{2}d\tau -logt|\le C,}$$

where $C=C\left(u_0\right)$ is a positive constant independent of t.

Remark 2.

In addition of the initial condition in Theorem 2, we assume that $m_0\in {\mathcal{H}}^1$; then, it holds that

$$\begin{array}{c}\hfill {\displaystyle {\int }_0^t{\parallel m\left(\tau \right)-{\mathcal{K}}_{\nu }\left(\tau \right)\ast m_{0,in}\parallel }_{L^2}^{2}d\tau \le C,t>0,}\end{array}$$

(11)

where $C=C\left(u_0\right)$ is a positive constant independent of t. This shows a gain of regularity by the membership in the Hardy space of data, similarly as in the decay estimates for density in Theorems 2 and 3. a similar phenomenon was already observed in [16,17] for dissipative wave equations. The proof is given by direct computations based on the explicit form of fundamental solution (40) below and a similar argument as in Kobayashi and Misawa [16,17]. We omitted the details here.

We state the space–time $L^2$ boundedness for Stokes flow part ${\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$. Indeed, from (11) together with Theorem A3 in Appendix C and ([18] Chapter 3, Section 3, Theorem 3), we find that if $m_0\in {\mathcal{H}}^1$ is added in the assumption of Theorem 2, space–time $L^2$ boundedness holds true for $M\left(t\right)-{\mathcal{K}}_{\nu }\ast M_{0,in}\left(t\right)$, $\rho \left(t\right)-{\rho }_{\ast }$ and ${\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}$.

4. Proof of Main Results

4.1. Proof of Theorems 2 and 4

In this subsection, we prove Theorems 2 and 4. The proof is performed by modifying Morawetz’s energy estimate. For a solution $\varphi$ to (7), we define function w by

$$\begin{array}{c}\hfill {\displaystyle w={\int }_0^t\varphi \left(\tau \right)d\tau -\mathrm{div}\mathsf{\Phi },}\end{array}$$

(12)

where $\mathsf{\Phi }$ is a solution to the doubly Laplace equation

$$\begin{array}{c}\hfill (-\Delta +{\kappa }_0{\Delta }^2)\mathsf{\Phi }=m_0+\nabla {\varphi }_0.\end{array}$$

(13)

For the existence of a solution to (13), we have

Theorem 7.

Suppose that ${\varphi }_0\in H^2\cap {\mathcal{H}}^1\left({\mathbb{R}}^2\right)$, $m_0\in H^1\left({\mathbb{R}}^2\right)$ and $m_0+\nabla {\varphi }_0\in {\mathcal{H}}^1\left({\mathbb{R}}^2\right)$. Then, there exists a solution Φ of

$$\begin{array}{c}\hfill (-\Delta )\mathsf{\Phi }={(I-{\kappa }_0\Delta )}^{-1}(m_0+\nabla {\varphi }_0)in {\mathbb{R}}^2\end{array}$$

(14)

such that

$$\begin{array}{ccc}& & {\parallel \nabla \mathsf{\Phi }\parallel }_{L^2\left({\mathbb{R}}^2\right)}+{\parallel \Delta \mathsf{\Phi }\parallel }_{L^2\left({\mathbb{R}}^2\right)}\le C{\parallel m_0+\nabla {\varphi }_0\parallel }_{{\mathcal{H}}^1\left({\mathbb{R}}^2\right)},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & {\parallel \mathrm{div}\Delta \mathsf{\Phi }\parallel }_{L^2\left({\mathbb{R}}^2\right)}\le C(\parallel \Delta {\varphi }_0{\parallel }_{L^2\left({\mathbb{R}}^2\right)}+\parallel {}^{\top }({\varphi }_0,m_0){\parallel }_{H^1\left({\mathbb{R}}^2\right)}).\hfill \end{array}$$

(16)

The proof of Theorem 7 is in Appendix D.

By the definition of w, we derive

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{w}_{tt}-\Delta w_t-\Delta w+{\kappa }_0{\Delta }^{2}w=0,\hfill \\ w\left(0\right)=-\mathrm{div}\mathsf{\Phi },w_t\left(0\right)={\varphi }_0.\hfill \end{array}\right.\end{array}$$

(17)

To estimate

$${\displaystyle {\int }_0^t{\parallel \varphi \left(\tau \right)\parallel }_{L^2}^{2}d\tau ={\int }_0^t{\parallel w_{\tau }\left(\tau \right)\parallel }_{L^2}^{2}d\tau ,}$$

we take the $L^2$ inner product of ${\left(17\right)}_1$ with w; thus, we have

$$\begin{array}{c}\hfill \frac{d}{dt}(w_t,w)+\frac{1}{2}\frac{d}{dt}{\parallel \nabla w\parallel }_{L^2}^2+{\parallel \nabla w\parallel }_{L^2}^2+{\kappa }_0{\parallel \Delta w\parallel }_{L^2}^2={\parallel w_t\parallel }_{L^2}^2,\end{array}$$

which is integrated on time interval $(0,t)$, yielding

$$\begin{array}{c}{\int }_0^t{\parallel w_{\tau }\left(\tau \right)\parallel }_{L^2}^{2}d\tau \hfill \\ ={\int }_0^t{\parallel \nabla w\left(\tau \right)\parallel }_{L^2}^{2}d\tau +{\int }_0^t{\kappa }_0{\parallel \Delta w\left(\tau \right)\parallel }_{L^2}^{2}d\tau \hfill \\ +\frac{{\parallel \nabla w\left(t\right)\parallel }_{L^2}^2}{2}+(w_t,w)-\frac{{\parallel \nabla w\left(0\right)\parallel }_{L^2}^2}{2}-(w_t\left(0\right),w\left(0\right)),\hfill \end{array}$$

(18)

where $w\left(0\right)=-\mathrm{div}\mathsf{\Phi }$. Now, we estimate the terms in the right-hand side of (18). Terms ${\parallel \nabla w\left(0\right)\parallel }_{L^2}^2$ and $(w_t\left(0\right),w\left(0\right))$ are directly estimated by (15). A test function $w_t$ in (17), being integrated on time interval $(0,t)$ and using (15) and (16), yields estimate

$$\begin{array}{c}\parallel w_t{\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla w\left(t\right)\parallel }_{L^2}^2+{\kappa }_0{\parallel \Delta w\left(t\right)\parallel }_{L^2}^2+{\int }_0^t{\parallel \nabla w_{\tau }\left(\tau \right)\parallel }_{L^2}^{2}d\tau \\ \le C(\parallel {\varphi }_0{\parallel }_{L^2}^2+\parallel m_0+\nabla {\varphi }_0{\parallel }_{{\mathcal{H}}^1}^2)+C{\kappa }_0(\parallel \Delta {\varphi }_0{\parallel }_{L^2}^2+\parallel {}^{\top }({\varphi }_0,m_0){\parallel }_{H^1}^2).\end{array}$$

(19)

On the other hand, from taking the $L^2$ inner product of $-{\kappa }_0\Delta w$ with (17) and integrating on $(0,t)$, we obtain that

$$\begin{array}{c}\frac{1}{2}{\kappa }_0{\parallel \Delta w\left(t\right)\parallel }_{L^2}^2+{\kappa }_0{\int }_0^t{\parallel \Delta w\left(\tau \right)\parallel }_{L^2}^{2}d\tau +{\kappa }_0(\nabla w_t,\nabla w)+{\kappa }_0^2{\int }_0^t{\parallel \nabla \Delta w\left(\tau \right)\parallel }_{L^2}^{2}d\tau \\ =\frac{1}{2}{\kappa }_0{\parallel \Delta w\left(0\right)\parallel }_{L^2}^2+{\kappa }_0{\int }_0^t{\parallel \nabla w_{\tau }\left(\tau \right)\parallel }_{L^2}^{2}d\tau +{\kappa }_0(\nabla w_t\left(0\right),\nabla w\left(0\right))\end{array}$$

and thus,

$$\begin{array}{c}\frac{1}{2}{\kappa }_0{\parallel \Delta w\left(t\right)\parallel }_{L^2}^2+{\kappa }_0{\int }_0^t{\parallel \Delta w\left(\tau \right)\parallel }_{L^2}^{2}d\tau +{\kappa }_0^2{\int }_0^t{\parallel \nabla \Delta w\left(\tau \right)\parallel }_{L^2}^{2}d\tau \\ \le C{\kappa }_0\{\parallel \nabla {\varphi }_0{\parallel }_{L^2}^2+{\parallel \nabla \mathrm{div}\mathsf{\Phi }\parallel }_{L^2}^2+{\parallel \Delta \mathrm{div}\mathsf{\Phi }\parallel }_{L^2}^2+{\int }_0^t\parallel \nabla w_{\tau }\left(\tau \right){\parallel }_{L^2}^{2}d\tau \}\hfill \\ +C{\kappa }_0{(\parallel \nabla w\left(t\right)\parallel }_{L^2}^2+\parallel \nabla w_t\left(t\right){\parallel }_{L^2}^2).\hfill \end{array}$$

(20)

Terms having $\mathsf{\Phi }$ in (20) are estimated by (15) and (16). The fourth and fifth terms on the right-hand side of (20) are also estimated by (19). For term $\parallel \nabla w_t{\left(t\right)\parallel }_{L^2}^2={\parallel \nabla \varphi \left(t\right)\parallel }_{L^2}^2$, we apply the standard energy estimate obtained from (7) with a test function ${\varphi }_t$

$$\begin{array}{c}\frac{1}{2}(\parallel {\varphi }_t{\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla \varphi \left(t\right)\parallel }_{L^2}^2+{\kappa }_0{\parallel \Delta \varphi \left(t\right)\parallel }_{L^2}^2)+{\int }_0^t{\parallel \nabla {\varphi }_{\tau }\left(\tau \right)\parallel }_{L^2}^{2}d\tau \\ =\frac{1}{2}(\parallel {\varphi }_t{\left(0\right)\parallel }_{L^2}^2+{\parallel \nabla \varphi \left(0\right)\parallel }_{L^2}^2+{\kappa }_0{\parallel \Delta \varphi \left(0\right)\parallel }_{L^2}^2)\\ \le C(\parallel \mathrm{div}{m}_0{\parallel }_{L^2}^2+\parallel \nabla {\varphi }_0{\parallel }_{L^2}^2)+C{\kappa }_0{\parallel \Delta {\varphi }_0\parallel }_{L^2}^2\end{array}$$

and thus,

$$\begin{array}{ccc}\hfill {\displaystyle {\kappa }_0{\int }_0^t{\parallel \Delta w\left(\tau \right)\parallel }_{L^2}^{2}d\tau }& \le & C({\kappa }_0+{\kappa }_0^2)\{\parallel {}^{\top }({\varphi }_0,m_0){\parallel }_{H^1}^2+\parallel \Delta {\varphi }_0{\parallel }_{L^2}^2\}\hfill \\ & & +C{\kappa }_0{\parallel m_0+\nabla {\varphi }_0\parallel }_{{\mathcal{H}}^1}^2.\hfill \end{array}$$

(21)

For the estimation of the $L^2-$norm on a space of w and the one on the time–space of its spatial gradient, let

$$\begin{array}{c}\hfill v={\int }_0^{t}w\left(s\right)ds\end{array}$$

(22)

and proceed to the energy estimates of v. From the direct calculation, v satisfies

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{v}_{tt}-\Delta v-\Delta v_t+{\kappa }_0{\Delta }^{2}v={\varphi }_0+\Delta div\mathsf{\Phi },\hfill \\ v\left(0\right)=0,v_t\left(0\right)=-div\mathsf{\Phi }.\hfill \end{array}\right.\end{array}$$

(23)

A test function $v_t$ in (23) gives

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}\parallel v_t{\left(t\right)\parallel }_{L^2}^2+\frac{1}{2}\frac{d}{dt}{\parallel \nabla v\left(t\right)\parallel }_{L^2}^2+\parallel \nabla v_t{\left(t\right)\parallel }_{L^2}^2+\frac{1}{2}{\kappa }_0\frac{d}{dt}{\parallel \Delta v\left(t\right)\parallel }_{L^2}^2\hfill \\ & & =({\varphi }_0+\Delta \mathrm{div}\mathsf{\Phi },v_t\left(t\right))\hfill \\ & & =\frac{d}{dt}({\varphi }_0+\Delta \mathrm{div}\mathsf{\Phi },v\left(t\right)),\hfill \end{array}$$

(24)

being integrated on $(0,t)$ and yielding

$$\begin{array}{ccc}& & \frac{1}{2}\parallel v_t{\left(t\right)\parallel }_{L^2}^2+\frac{1}{2}{\parallel \nabla v\left(t\right)\parallel }^2+{\int }_0^t\parallel \nabla v_s{\left(s\right)\parallel }_{L^2}^{2}ds+\frac{1}{2}{\kappa }_0{\parallel \Delta v\left(s\right)\parallel }_{L^2}^2\hfill \\ & & =({\varphi }_0+\Delta \mathrm{div}\mathsf{\Phi },v\left(t\right))+\frac{1}{2}{\parallel w\left(0\right)\parallel }_{L^2}^2.\hfill \end{array}$$

(25)

Now, the first term of the right-hand side of (25) is

$$\begin{array}{c}\hfill \left({\varphi }_0+\Delta div\mathsf{\Phi },v\left(t\right)\right)=\left({\varphi }_0,v\left(t\right)\right)+\left(\Delta \mathrm{div}\mathsf{\Phi },v\left(t\right)\right)\end{array}$$

(26)

of which the first term is evaluated by the use of Lemmas 1 and 2 and Young’s inequality as

$$\begin{array}{ccc}\hfill \left|\left({\varphi }_0,v\left(t\right)\right)\right|\le C\parallel {\varphi }_0{\parallel }_{{\mathcal{H}}^1}{\parallel v\left(t\right)\parallel }_{\mathrm{BMO}}& \le & C\parallel {\varphi }_0{\parallel }_{{\mathcal{H}}^1}{\parallel \nabla v\left(t\right)\parallel }_{L^2}\hfill \\ & \le & \frac{1}{8}{\parallel \nabla v\left(t\right)\parallel }_{L^2}^2+C{\parallel {\varphi }_0\parallel }_{{\mathcal{H}}^1}^2\hfill \end{array}$$

and, the second term is controlled by (15) as

$$\begin{array}{ccc}\hfill \left|\left(\Delta \mathrm{div}\mathsf{\Phi },v\left(t\right)\right)\right|& =& \left|\left(\Delta \mathsf{\Phi },\nabla v\left(t\right)\right)\right|\hfill \\ & \le & \parallel m_0+\nabla {\varphi }_0{\parallel }_{{\mathcal{H}}^1}{\parallel \nabla v\left(t\right)\parallel }_{L^2}\hfill \\ & \le & \frac{1}{8}{\parallel \nabla v\left(t\right)\parallel }_{L^2}^2+C{\parallel m_0+\nabla {\varphi }_0\parallel }_{{\mathcal{H}}^1}^2.\hfill \end{array}$$

Since $w\left(0\right)=-\mathrm{div}\mathsf{\Phi }$, the right-hand side of (25) is bounded by

$$\begin{array}{ccc}& & ({\varphi }_0+\Delta \mathrm{div}\mathsf{\Phi },v\left(t\right))+\frac{1}{2}{\parallel w\left(0\right)\parallel }_{L^2}^2\hfill \\ & & \le \frac{1}{4}{\parallel \nabla v\left(t\right)\parallel }_{L^2}^2+C\parallel {\varphi }_0{\parallel }_{{\mathcal{H}}^1}^2+C{\parallel m_0+\nabla {\varphi }_0\parallel }_{{\mathcal{H}}^1}^2.\hfill \end{array}$$

Thus, it follows that

$$\begin{array}{ccc}& & \frac{1}{2}\parallel v_t{\left(t\right)\parallel }^2+\frac{1}{4}{\parallel \nabla v\left(t\right)\parallel }^2+{\int }_0^t\parallel \nabla v_s{\left(s\right)\parallel }^{2}ds+\frac{1}{2}{\kappa }_0{\parallel \Delta v\left(s\right)\parallel }_{L^2}^2\hfill \\ & & \le C\parallel {\varphi }_0{\parallel }_{{\mathcal{H}}^1}^2+C{\parallel m_0+\nabla {\varphi }_0\parallel }_{{\mathcal{H}}^1}^2.\hfill \end{array}$$

(27)

Gathering (18), (19), (21), and (27), we obtain Theorem 2.

On the basis of Theorems 2 and 4 is proved as follows. We set a total energy of w as

$$E\left(w\left(t\right)\right)=\parallel w_t{\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla w\left(t\right)\parallel }_{L^2}^2+{\kappa }_0{\parallel \Delta w\left(t\right)\parallel }_{L^2}^2.$$

By the proof of Theorem 2 and integration by parts, we find that

$$\begin{array}{ccc}\hfill C{J}_0& \ge & {\int }_0^{t}E\left(w\left(s\right)\right)ds\hfill \\ & =& tE\left(w\left(t\right)\right)-{\int }_0^{t}s\frac{d}{ds}E\left(w\left(s\right)\right)ds.\hfill \end{array}$$

Since, by integration by parts again,

$$\begin{array}{ccc}\hfill \frac{d}{dt}E\left(w\left(t\right)\right)& =& \frac{d}{dt}\left\{{\parallel \nabla w\left(t\right)\parallel }_{L^2}^2+\parallel w_t{\left(t\right)\parallel }_{L^2}^2+{\kappa }_0{\parallel \Delta w\left(t\right)\parallel }_{L^2}^2\right\}\hfill \\ & =& 2(\nabla w_t\left(t\right),\nabla w\left(t\right))+2(w_{tt}\left(t\right),w_t\left(t\right))+2({\kappa }_0\Delta w,\Delta w_t)\hfill \\ & =& -2(w_t\left(t\right),\Delta w\left(t\right))+2(w_{tt}\left(t\right),w_t\left(t\right))+2({\kappa }_0{\Delta }^{2}w,w_t)\hfill \\ & =& 2(w_{tt}\left(t\right)-\Delta w\left(t\right)+{\kappa }_0{\Delta }^{2}w\left(t\right),w_t\left(t\right))\hfill \\ & =& 2(\Delta w_t\left(t\right),w_t\left(t\right))\hfill \\ & =& -2\parallel \nabla w_t{\left(t\right)\parallel }_{L^2}^2\hfill \\ & =& -{2\parallel \nabla \varphi \left(t\right)\parallel }_{L^2}^2,\hfill \end{array}$$

we have

$$\begin{array}{ccc}\hfill C{J}_0& \ge & tE\left(w\left(t\right)\right)+2{\int }_0^{t}s{\parallel \nabla \varphi \left(s\right)\parallel }^{2}ds.\hfill \end{array}$$

(28)

This, together with (19), gives the assertion of Theorem 4. The proof is completed. □

4.2. Proof of Theorem 6

In this section, we show the validity of Theorem 6. By taking the Fourier transform of (6) with respect to space variable x, we have the following ordinary differential equation with a parameter $\xi$.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_t\widehat{\varphi }(t,\xi )+i\gamma \xi ·\widehat{m}(t,\xi )=0,\hfill \\ {\partial }_t\widehat{m}(t,\xi )+{\nu \left|\xi \right|}^2\widehat{m}(t,\xi )+\tilde{\nu }\xi (\xi ·\widehat{m}(t,\xi ))+i\gamma \xi \widehat{\varphi }(t,\xi )+i\xi {\kappa }_0{\left|\xi \right|}^2\widehat{\varphi }(t,\xi )=0,\hfill \\ \widehat{\varphi }(0,\xi )={\widehat{\varphi }}_0,\widehat{m}(0,\xi )={\widehat{m}}_0.\hfill \end{array}\right.\end{array}$$

(29)

Therefore, the solutions of (6) are given by the following formulas by [11]. Let ${\displaystyle K=\frac{2\sqrt{{\kappa }_0\gamma }}{\nu +\tilde{\nu }}}$ and ${\displaystyle B=\frac{2\gamma }{\nu +\tilde{\nu }}}$. For $\left|\xi \right|\ne 0,B/\sqrt{1-K^2}$ when $0<K<1$ and $\left|\xi \right|\ne 0$ when $K\ge 1$, the Fourier transforms of $\varphi$ and m are given explicitly by formulas

$$\begin{array}{ccc}\hfill \widehat{\varphi }& =& {\displaystyle \frac{{\lambda }_{+}\left(\xi \right)e^{{\lambda }_{-}\left(\xi \right)t}-{\lambda }_{-}\left(\xi \right)e^{{\lambda }_{+}\left(\xi \right)t}}{{\lambda }_{+}\left(\xi \right)-{\lambda }_{-}\left(\xi \right)}{\widehat{\varphi }}_0-i\gamma \frac{e^{{\lambda }_{+}\left(\xi \right)t}-e^{{\lambda }_{-}\left(\xi \right)t}}{{\lambda }_{+}\left(\xi \right)-{\lambda }_{-}\left(\xi \right)}\xi ·{\widehat{m}}_0,}\hfill \\ \hfill \widehat{m}& =& e^{-{\nu \left|\xi \right|}^{2}t}{\widehat{m}}_0-i\xi (\gamma +{\kappa }_0{\left|\xi \right|}^2)\left({\displaystyle \frac{e^{{\lambda }_{+}\left(\xi \right)t}-e^{{\lambda }_{-}\left(\xi \right)t}}{{\lambda }_{+}\left(\xi \right)-{\lambda }_{-}\left(\xi \right)}}\right){\widehat{\varphi }}_0\hfill \\ & & +\left({\displaystyle \frac{{\lambda }_{+}\left(\xi \right)e^{{\lambda }_{+}\left(\xi \right)t}-{\lambda }_{-}\left(\xi \right)e^{{\lambda }_{-}\left(\xi \right)t}}{{\lambda }_{+}\left(\xi \right)-{\lambda }_{-}\left(\xi \right)}-e^{-{\nu \left|\xi \right|}^{2}t}}\right)\frac{\xi (\xi ·{\widehat{m}}_0)}{{\left|\xi \right|}^2},\hfill \end{array}$$

(30)

where ${\lambda }_{\pm }$ are given by

$$\begin{array}{c}\hfill {\lambda }_{\pm }\left(\xi \right)=-{A\left(\right|\xi |}^2\pm \sqrt{{\left|\xi \right|}^4-B^2{\left|\xi \right|}^2-K^2{\left|\xi \right|}^4})\end{array}$$

(31)

with a positive constant ${\displaystyle A=\frac{\nu +\tilde{\nu }}{2}}$ and stand for roots of the characteristic equation of (29). If $0<K<1$ and ${\displaystyle min\left\{\frac{1}{2},\frac{B}{2\sqrt{1-K^2}}\right\}\le \left|\xi \right|\le 2\frac{B}{\sqrt{1-K^2}}}$, $\widehat{\varphi }$ and $\widehat{m}$ are represented as

$$\begin{array}{ccc}\hfill \widehat{\varphi }& =& {\displaystyle \frac{1}{2\pi i}{\oint }_{\Gamma }\frac{{(z+A|\xi |}^2)e^{zt}}{z^2+(\nu +\tilde{\nu }){\left|\xi \right|}^{2}z+{\kappa }_0{\gamma \left|\xi \right|}^4+{\gamma }^2{\left|\xi \right|}^2}dz{\widehat{\varphi }}_0}\hfill \end{array}$$

$$\begin{array}{ccc}& & {\displaystyle -\frac{\gamma }{2\pi }{\oint }_{\Gamma }\frac{e^{zt}}{z^2+(\nu +\tilde{\nu }){\left|\xi \right|}^{2}z+{\kappa }_0{\gamma \left|\xi \right|}^4+{\gamma }^2{\left|\xi \right|}^2}dz\xi ·{\widehat{m}}_0,}\hfill \\ \hfill \widehat{m}& =& {\displaystyle e^{-{\nu \left|\xi \right|}^{2}t}{\widehat{m}}_0-\frac{\gamma \xi }{2\pi }{\oint }_{\Gamma }\frac{e^{zt}}{z^2+(\nu +\tilde{\nu }){\left|\xi \right|}^{2}z+{\kappa }_0{\gamma \left|\xi \right|}^4+{\gamma }^2{\left|\xi \right|}^2}dz{\widehat{\varphi }}_0}\hfill \end{array}$$

(32)

$$\begin{array}{ccc}& & +\left({\displaystyle \frac{1}{2\pi i}{\oint }_{\Gamma }\frac{z{e}^{zt}}{z^2+(\nu +\tilde{\nu }){\left|\xi \right|}^{2}z+{\kappa }_0{\gamma \left|\xi \right|}^4+{\gamma }^2{\left|\xi \right|}^2}dz}\right)\frac{\xi (\xi ·{\widehat{m}}_0)}{{\left|\xi \right|}^2},\hfill \end{array}$$

(33)

where Γ is a closed pass surrounding ${\lambda }_{\pm }$ and included in set $\{z\in \mathbb{C}|\mathrm{Re}z\le -c_0\}$, and $c_0$ is a positive number satisfying

$$\underset{min\left\{\frac{1}{2},\frac{B}{2\sqrt{1-K^2}}\right\}\le \left|\xi \right|\le 2\frac{B}{\sqrt{1-K^2}}}{max}\mathrm{Re}{\lambda }_{\pm }\le -2c_0.$$

Cut-off functions ${\phi }_1$, ${\phi }_{\infty }$ and ${\phi }_M$ in $C^{\infty }\left({\mathbb{R}}^2\right)$ are defined by [11] as follows: in the case such that $K\ne 1$, ${\phi }_1$ is given by

$$\begin{array}{ccc}& & {\phi }_1\left(\xi \right)=\left\{\begin{array}{c}1\mathrm{for}\left|\xi \right|\le \frac{1}{2}\hfill \\ 0\mathrm{for}\left|\xi \right|\ge 1\hfill \end{array}\right.if\frac{B}{2\sqrt{|1-K^2|}}>1;\hfill \\ & & {\phi }_1\left(\xi \right)=\left\{\begin{array}{c}{\displaystyle 1\mathrm{for}\left|\xi \right|\le \frac{B}{2\sqrt{|1-K^2|}}}\hfill \\ 0\mathrm{for}\left|\xi \right|\ge 1\hfill \end{array}\right.if\frac{B}{2\sqrt{|1-K^2|}}\le 1<\frac{B}{\sqrt{2|1-K^2|}};\hfill \\ & & {\phi }_1\left(\xi \right)=\left\{\begin{array}{c}{\displaystyle 1\mathrm{for}\left|\xi \right|\le \frac{B}{2\sqrt{|1-K^2|}}}\hfill \\ {\displaystyle 0\mathrm{for}\left|\xi \right|\ge \frac{B}{\sqrt{2|1-K^2|}}}\hfill \end{array}\right.if\frac{B}{\sqrt{2|1-K^2|}}\le 1.\hfill \end{array}$$

${\phi }_{\infty }$ and ${\phi }_M$ are

$$\begin{array}{ccc}& & {\phi }_{\infty }\left(\xi \right)=\left\{\begin{array}{c}{\displaystyle 1\mathrm{for}\left|\xi \right|\ge \frac{2B}{\sqrt{|1-K^2|}}}\hfill \\ {\displaystyle 0\mathrm{for}\left|\xi \right|\le \frac{\sqrt{2}B}{\sqrt{|1-K^2|}},}\hfill \end{array}\right.{\phi }_M\left(\xi \right)=1-{\phi }_1\left(\xi \right)-{\phi }_{\infty }\left(\xi \right).\hfill \end{array}$$

In the case that $K=1$, ${\phi }_1$ and ${\phi }_{\infty }$ are

$$\begin{array}{ccc}& & {\phi }_1\left(\xi \right)=\left\{\begin{array}{c}1\mathrm{for}\left|\xi \right|\le \frac{1}{2}\hfill \\ 0\mathrm{for}\left|\xi \right|\ge 1,\hfill \end{array}\right.{\phi }_{\infty }\left(\xi \right)=\left\{\begin{array}{c}0\mathrm{for}\left|\xi \right|\le \frac{1}{2}\hfill \\ 1\mathrm{for}\left|\xi \right|\ge 1,\hfill \end{array}\right.\hfill \\ & & {\phi }_1\left(\xi \right)+{\phi }_{\infty }\left(\xi \right)=1.\hfill \end{array}$$

We define solution operators $E_1$ and $E_{\infty }$ on a low- and high-frequency part of (6), respectively, as follows:

$$\begin{array}{ccc}& & E_1\left(t\right)=(E_{1,\varphi }\left(t\right),E_{1,m}\left(t\right)),\hfill \\ & & E_{\infty }\left(t\right)=(E_{\infty ,\varphi }\left(t\right),E_{\infty ,m}\left(t\right)),\hfill \end{array}$$

(34)

where

$$\begin{array}{ccc}& & E_{1,\varphi }\left(t\right)({\varphi }_0,m_0)\left(x\right)={\mathcal{F}}^{-1}\left[{\phi }_1\left(\xi \right)\widehat{\varphi }(t,\xi )\right]\left(x\right),\hfill \end{array}$$

(35)

$$\begin{array}{ccc}& & E_{1,m}\left(t\right)({\varphi }_0,m_0)\left(x\right)={\mathcal{F}}^{-1}\left[{\phi }_1\left(\xi \right)\widehat{m}(t,\xi )\right]\left(x\right),\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& & E_{\infty ,\varphi }\left(t\right)({\varphi }_0,m_0)\left(x\right)={\mathcal{F}}^{-1}\left[({\phi }_M\left(\xi \right)+{\phi }_{\infty }\left(\xi \right))\widehat{\varphi }(t,\xi )\right]\left(x\right),\hfill \end{array}$$

(37)

$$\begin{array}{ccc}& & E_{\infty ,m}\left(t\right)({\varphi }_0,m_0)\left(x\right)={\mathcal{F}}^{-1}\left[({\phi }_M\left(\xi \right)+{\phi }_{\infty }\left(\xi \right))\widehat{m}(t,\xi )\right]\left(x\right).\hfill \end{array}$$

(38)

In [11], the solution operator is shown to have an exponential decay in time on the high-frequency part (34). In fact we have

Theorem 8.

([11] Theorem 3.2) Let $1\le p\le \infty$. Then, it holds that

$$\begin{array}{c}\parallel {\partial }_t^k{\partial }_x^{\alpha }{E}_{\infty }\left(t\right)({\varphi }_0,m_0){\parallel }_{L^p}\hfill \\ \le C_{k,\alpha }{e}^{-ct}\left\{(1+t^{-{\delta }_1-\frac{\left|\alpha \right|}{2}-k})\left[\parallel {\varphi }_0{\parallel }_{L^p}+{\parallel m_0\parallel }_{L^p}\right]+t^{-{\delta }_2-\frac{\left|\alpha \right|}{2}-k}{\parallel {\varphi }_0\parallel }_{L^p}\right\}\hfill \end{array}$$

(39)

for $t>0$, $k\ge 0$ and $\left|\alpha \right|\ge 0$, where $({\delta }_1,{\delta }_2)=(1/2,1)$ and $(1,3/2)$ for $K\ne 1$ and $K=1$, respectively.

Therefore, in order to show Theorem 6, it is enough to consider the low-frequency part. We estimate the Green function. We put

$$\begin{array}{ccc}\hfill L_{11,j}(t,x)& =& {\displaystyle {\mathcal{F}}^{-1}\left\{\frac{{\lambda }_{+}\left(\xi \right)e^{{\lambda }_{-}\left(\xi \right)t}-{\lambda }_{-}\left(\xi \right)e^{{\lambda }_{+}\left(\xi \right)t}}{{\lambda }_{+}\left(\xi \right)-{\lambda }_{-}\left(\xi \right)}{\phi }_j\left(\xi \right)\right\}\left(x\right),}\hfill \\ \hfill L_{12,j}(t,x)& =& {\mathcal{F}}^{-1}(-i\gamma {}^{\top }\xi {\widehat{L}}_j),\hfill \\ \hfill {\widehat{L}}_j(t,\xi )& =& {\displaystyle \frac{e^{{\lambda }_{+}\left(\xi \right)t}-e^{{\lambda }_{-}\left(\xi \right)t}}{{\lambda }_{+}\left(\xi \right)-{\lambda }_{-}\left(\xi \right)}{\phi }_j\left(\xi \right),}\hfill \\ \hfill L_{21,j}(t,x)& =& {\mathcal{F}}^{-1}\{-\xi (i\gamma +k_0{\left|\xi \right|}^2\left){\widehat{L}}_j\right\},\hfill \\ \hfill L_{22,j}(t,x)& =& K_{1,j}(t,x)+K_{2,j}(t,x)-K_{3,j}(t,x),\hfill \\ \hfill K_{1,j}(t,x)& =& {\mathcal{F}}^{-1}\left[e^{-{\nu \left|\xi \right|}^{2}t}{\phi }_j\left(\xi \right)\right]\left(x\right)I_n,\hfill \\ \hfill K_{2,j}(t,x)& =& {\displaystyle {\mathcal{F}}^{-1}\left\{\frac{{\lambda }_{+}\left(\xi \right)e^{{\lambda }_{+}\left(\xi \right)t}-{\lambda }_{-}\left(\xi \right)e^{{\lambda }_{-}\left(\xi \right)t}}{{\lambda }_{+}\left(\xi \right)-{\lambda }_{-}\left(\xi \right)}\frac{\xi {}^{\top }\xi }{{\left|\xi \right|}^2}{\phi }_j\left(\xi \right)\right\}\left(x\right),}\hfill \\ \hfill K_{3,j}(t,x)& =& {\mathcal{F}}^{-1}\left[e^{-{\nu \left|\xi \right|}^{2}t}\frac{\xi {}^{\top }\xi }{{\left|\xi \right|}^2}{\phi }_j\left(\xi \right)\right]\left(x\right)\hfill \end{array}$$

for $j=1,\infty$. We see from (30) that

$$\begin{array}{c}\hfill E_j\left(t\right)({\varphi }_0,m_0)=\left(\begin{array}{cc}{L}_{11,j}(t,·)& L_{12,j}(t,·)\\ L_{21,j}(t,·)& L_{22,j}(t,·)\end{array}\right)\ast \left(\begin{array}{c}{\varphi }_0\\ m_0\end{array}\right)\end{array}$$

(40)

for $j=1,\infty$.

We set

$$K_1{m}_0={\mathcal{F}}^{-1}\left[\frac{{\lambda }_{+}\left(\xi \right)e^{{\lambda }_{+}\left(\xi \right)t}-{\lambda }_{-}\left(\xi \right)e^{{\lambda }_{-}\left(\xi \right)t}}{{\lambda }_{+}\left(\xi \right)-{\lambda }_{-}\left(\xi \right)}\frac{\xi {}^{\top }\xi }{{\left|\xi \right|}^2}{\phi }_1\left(\xi \right){\widehat{m}}_0\right].$$

$K_1{m}_0$ is a part of the Green matrix and corresponds to the diffusion-wave part $m-{\mathcal{K}}_{\nu }\ast m_{0,in}$. Our claim is the following estimate.

Proposition 1.

Let $m_0\in L^2\cap L^1$, $\left|x\right|m_0\in L^1$ and ${\widehat{m}}_0\ne 0$. Then, it holds that

$${\displaystyle \underset{t\to \infty }{lim\; sup}|{\int }_0^t{\parallel K_1{m}_0\left(\tau \right)\parallel }_{L^2}^{2}d\tau -logt|\le C_1,}$$

where $C_1=C_1\left(u_0\right)$ is a positive constant independent of t.

Proof. 

By the Plancherel theorem and (31), we see that there exists a positive constant C, such that

$$C\parallel e^{-{\left|\xi \right|}^{2}t}\widehat{m_0}{\left(\xi \right)\parallel }_{L^2\left(\right|\xi |\le c_1)}^2\le \parallel K_1{m}_0{\left(t\right)\parallel }_{L^2}^2\le \frac{1}{C}{\parallel e^{-{\left|\xi \right|}^{2}t}\widehat{m_0}\left(\xi \right)\parallel }_{L^2\left(\right|\xi |\le c_1)}^2,$$

where $c_1=1$ when $\frac{B}{2\sqrt{|1-K^2|}}>1$ or $\frac{B}{2\sqrt{|1-K^2|}}\le 1<\frac{B}{\sqrt{2|1-K^2|}}$ and $c_1=\frac{B}{\sqrt{2|1-K^2|}}$ when $\frac{B}{\sqrt{2|1-K^2|}}\le 1$. Hence, we have to estimate $\parallel e^{-{\left|\xi \right|}^{2}t}\widehat{m_0}{\left(\xi \right)\parallel }_{L^2\left(\right|\xi |\le c_1)}^2$. It follows from the polar coordinates that

$$\begin{array}{c}\hfill {\displaystyle \parallel e^{-{\left|\xi \right|}^{2}t}\widehat{m_0}{\left(\xi \right)\parallel }_{L^2\left(\right|\xi |\le c_1)}^2=C_{\omega }{\int }_0^{c_1}{\left|e^{-r^{2}t}{\widehat{m}}_0\left(r\omega \right)\right|}^{2}rdr,}\end{array}$$

where $r=\left|\xi \right|$, $\omega =\xi /\left|\xi \right|$, and $C_{\omega }$ is some positive constant that appears in the polar coordinates. Changing variables $r\sqrt{t}=s$, we have

$${\displaystyle {\int }_0^{c_1}|e^{-r^{2}t}{\widehat{m}}_0{\left(\omega r\right)|}^{2}rdr=t^{-1}{\int }_0^{c_1\sqrt{t}}{e}^{-2s^2}{\left|\widehat{m_0}(s\omega /\sqrt{t})\right|}^{2}sds.}$$

This, together with the fundamental theorem of calculus for ${\widehat{m}}_0$, implies that

$$\begin{array}{c}\hfill |\parallel e^{-{\left|\xi \right|}^{2}t}\widehat{m_0}{\left(\xi \right)\parallel }_{L^2\left(\right|\xi |\le c_1)}^2-C_{\omega }{t}^{-1}{\left|{\widehat{m}}_0\left(0\right)\right|}^2{\int }_0^{c_1\sqrt{t}}{e}^{-2s^2}sds|\\ =C_{\omega }\left|t^{-1}{\int }_0^{c_1\sqrt{t}}\left(\right|{\widehat{m}}_0(s\omega /\sqrt{t}){|}^2-|{\widehat{m}}_0\left(0\right){|}^2)e^{-2s^2}sds\right|\hfill \\ {\displaystyle \le C{t}^{-1}{\int }_0^{c_1\sqrt{t}}{|{\widehat{m}}_0(s\omega /\sqrt{t})-{\widehat{m}}_0\left(0\right)|}^2{e}^{-2s^2}sds}\hfill \\ =C{t}^{-1}{\int }_0^{c_1\sqrt{t}}|{\int }_0^1{\nabla }_{\xi }{\widehat{m}}_0\left(\theta \frac{s\omega }{\sqrt{t}}\right)d\theta \frac{s\omega }{\sqrt{t}}{|}^2{e}^{-2s^2}sds\hfill \\ \le C{t}^{-1}{\int }_0^{c_1\sqrt{t}}{\parallel {\nabla }_{\xi }{\widehat{m}}_0\parallel }_{L^{\infty }}^2|\frac{s\omega }{\sqrt{t}}{|}^2{e}^{-2s^2}sds\hfill \\ \le C\parallel \left|x\right|m_0{\parallel }_{L^1}^2{t}^{-2}{\int }_0^{c_1\sqrt{t}}{e}^{-2s^2}{s}^{3}ds\hfill \\ \le C_2{t}^{-2}\hfill \end{array}$$

(41)

for a positive constant $C_2=C_2\left(u_0\right)$ independent of t. ${\widehat{m}}_0\left(0\right)\ne 0$ by our assumption. Since

$${\displaystyle {\int }_0^1\parallel e^{-{\left|\xi \right|}^2\tau }\widehat{m_0}{\left(\xi \right)\parallel }_{L^2}^{2}d\tau \le C{\parallel m_0\parallel }_{L^2}^2,}$$

we estimate ${\displaystyle {\int }_1^t{\parallel e^{-{\left|\xi \right|}^2\tau }\widehat{m_0}\left(\xi \right)\parallel }_{L^2}^{2}d\tau }$. We set

$${\displaystyle I_1\left(t\right):=C_{\omega }{\left|{\widehat{m}}_0\left(0\right)\right|}^2{t}^{-1}{\int }_0^{c_1\sqrt{t}}{e}^{-2s^2}sds}$$

Applying (41) yields that, for $t\ge 1$,

$$\begin{array}{c}\hfill {\displaystyle \left|{\int }_1^t\left\{\parallel e^{-{\left|\xi \right|}^2\tau }\widehat{m_0}{\left(\xi \right)\parallel }_{L^2\left(\right|\xi |\le c_1)}^2-I_1\left(\tau \right)\right\}d\tau \right|\le C_2{\int }_1^t{\tau }^{-2}d\tau \le C_2.}\end{array}$$

(42)

Let positive constants $Q_1$ and $Q_2$ be defined by

$${\displaystyle Q_1:={\int }_0^{c_1}{e}^{-2s^2}{s}^{3}ds,Q_2:={\int }_0^{\infty }{e}^{-2s^2}{s}^{3}ds.}$$

It obviously follows that, for $t\ge 1$,

$${\displaystyle Q_1\le {\int }_0^{c_1\sqrt{t}}{e}^{-2s^2}{s}^{3}ds\le Q_2.}$$

This implies that, for $t\ge 1$,

$${\displaystyle {\int }_1^t{C}_{\omega }|{\widehat{m}}_0{\left(0\right)|}^2{Q}_1{\tau }^{-1}d\tau \le {\int }_1^t{I}_1\left(\tau \right)d\tau \le {\int }_1^t{C}_{\omega }{\left|{\widehat{m}}_0\left(0\right)\right|}^2{Q}_2{\tau }^{-1}d\tau }$$

and thus,

$$\begin{array}{c}\hfill {\displaystyle C_{\omega }|{\widehat{m}}_0{\left(0\right)|}^2{Q}_{1}logt\le {\int }_1^t{I}_1\left(\tau \right)d\tau \le C_{\omega }{\left|{\widehat{m}}_0\left(0\right)\right|}^2{Q}_{2}logt.}\end{array}$$

(43)

(42) and (43) yield that

$$\begin{array}{c}\hfill {\displaystyle \underset{t\to \infty }{lim\; sup}|{\int }_1^t{\parallel e^{-{\left|\xi \right|}^2\tau }\widehat{m_0}\left(\xi \right)\parallel }_{L^2\left(\right|\xi |\le c_1)}^{2}d\tau -logt|\le C{C}_2.}\end{array}$$

Therefore, there exists a positive constant $C_1=C_1\left(u_0\right)$ independent of t, such that

$${\displaystyle \underset{t\to \infty }{lim\; sup}|{\int }_0^t{\parallel K_1{m}_0\left(\tau \right)\parallel }_{L^2}^{2}d\tau -logt|\le C_1.}$$

Since other parts of diffusion wave $m-{\mathcal{K}}_{\nu }\ast m_{0,in}$ that appear in the Green matrix on the low-frequency part are estimated similarly to Proposition 1, we obtain the estimation in Theorem 6. The proof is completed. □

5. Conclusions

We studied the asymptotic behavior of solutions to the compressible Navier–Stokes–Korteweg system in ${\mathbb{R}}^2$. Concerning the linearized system for (1), under some initial conditions given by Hardy space ${\mathcal{H}}^1$, we showed some space–time $L^2$ estimates for the density and Stokes flow parts of the momentum. The potential flow part of the momentum was also shown to grow at the rate of logarithmic order in space–time $L^2$ norm. The asymptotic behaviors in space–time $L^2$ of the diffusion-wave parts were shown to be essentially different between density and the potential flow part of the momentum. Nonlinear terms are expected to decay faster than the linear terms in the Duhamel formula do, as in [7,11]. As a consequence, the leading terms of the asymptotic expansion of solution u for (1) were given by

$$\begin{array}{c}\hfill u\left(t\right)\sim \underset{\mathrm{solutions\; to\; linearized\; system}}{\underbrace{\left(\begin{array}{c}0\\ M\left(t\right)-{\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}\end{array}\right)+\left(\begin{array}{c}\rho \left(t\right)-{\rho }_{\ast }\\ {\mathcal{K}}_{\nu }\left(t\right)\ast M_{0,in}\end{array}\right)}}+\underset{\mathrm{nonlinear\; parts}}{\underbrace{\cdots }}\mathrm{in}{L}^2(0,\infty ;{\mathbb{R}}^2).\end{array}$$

(44)

Analysis for asymptotic behavior in the two-dimensional case is difficult because the time decay of solutions to the linear system in two-dimensional cases is slower than that in higher-dimensional cases. To overcome this difficulty, we used a gain of regularity by the Hardy space; by the $L^2$ energy estimate of the Morawetz type, we succeeded to derive the asymptotic behavior (44).

Concerning future works, it is important to consider how the pressure term has an effect on the asymptotic behavior of (1). As shown in our paper [19], since (1) governs the motion of two-phase fluids, pressure is a nonmonotone function. When pressure decreases, solutions are expected to be unstable due to positive eigenvalues in linear systems. Hence, we will study the asymptotic behavior of solutions with relation to a critical value, such that $P^{\prime }\left(\rho \right)=0$ holds or an initial condition of pressure. Furthermore, from the point of view of engineering, analysis of two-phase fluids in bounded or unbounded domains with boundaries is more important. On the basis of our analysis of the Caucy problem, we will study the asymptotic behavior of solutions to CNSK under boundary conditions.