Global existence and time decay estimate of solutions to the compressible Navier–Stokes

Abstract

In this research, we study the global existence of solutions to the compressible Navier–Stokes–Korteweg system around a constant state. This system describes liquid-vapor type two-phase flow with a phase transition with diffuse interface. Previous works assume that pressure is a monotone function for change of density similarly to the usual compressible Navier–Stokes system. On the other hand, due to phase transition the pressure is in fact non-monotone function, and the linearized system loses symmetry in a critical case such that the derivative of pressure is 0 at the given constant state. We show that global $L^{2}$ solutions are available for the critical case of small data, whose momentum is in its derivative form, and obtain parabolic type decay rate of the solutions. This is proved based on the decomposition of solutions to a low frequency part and a high frequency part.

Keywords

Partial differential equations compressible Navier–Stokes–Korteweg system global solution time decay rate

1. Introduction

We study global existence of solutions to the following compressible Navier–Stokes–Korteweg system in $R^{n}$ ( $n ⩾ 3$ ): $\begin{array}{l} (1.1) & \{\begin{matrix} \partial_{t} ρ + div m = 0, \\ \partial_{t} m + div (\frac{m \otimes m}{ρ}) + \nabla P (ρ) = div (S (\frac{m}{ρ}) + K (ρ)), \\ ρ (x, 0) = ρ_{0}, m (x, 0) = m_{0} . \end{matrix} \end{array}$ Here $ρ = ρ (x, t)$ and $m = (m_{1} (x, t), \dots, m_{n} (x, t))$ denote the unknown density and momentum, respectively, at time $t \in R_{+}$ and position $x \in R^{n}$ ; $ρ_{0} = ρ_{0} (x)$ and $m_{0} = m_{0} (x)$ denote given initial data; $S$ and $K$ denote the viscous stress tensor and the Korteweg stress tensor that are given by $\begin{array}{rcl} (1.2) & \{\begin{matrix} S (\frac{m}{ρ}) = (μ^{'} div \frac{m}{ρ}) δ_{i, j} + 2 μ d_{i j} (\frac{m}{ρ}), \\ K (ρ) = \frac{κ}{2} (Δ ρ^{2} - | \nabla ρ |^{2}) δ_{i, j} - κ \frac{\partial ρ}{\partial x_{i}} \frac{\partial ρ}{\partial x_{j}}, \end{matrix} \end{array}$ where $d_{i j} (\frac{M}{ρ}) = \frac{1}{2} (\frac{\partial}{\partial x_{i}} {(\frac{M}{ρ})}_{j} + \frac{\partial}{\partial x_{j}} {(\frac{M}{ρ})}_{i})$ ; μ and $μ^{'}$ are the viscosity coefficients that are assumed to be constants satisfying $\begin{matrix} μ > 0, \frac{2}{n} μ + μ^{'} ⩾ 0 . \end{matrix}$ κ denotes the capillary constant that is assumed to be a positive constant. Note that if $κ = 0$ in the Korteweg tensor, the usual compressible Navier–Stokes equation (the abbreviation is used by “CNS” below) is obtained; $P = P (ρ)$ is the pressure that is assumed to be a smooth function of ρ. Here we assume that P satisfies $\begin{array}{rcl} (1.3) & P^{'} (ρ_{*}) = 0, \end{array}$ where $ρ_{*}$ is a given positive constant and $(ρ_{*}, 0)$ denotes a given constant state. We consider solutions to (1.1) around the constant state.

(1.1) is the system describing motion of two-phase flow between liquid and vapor which allows phase transition in a compressible fluid. The phase transition is described by a diffuse interface, whose phase boundary is regarded as a narrow transition zone. Thus, in this system the fluid state is governed by a phase parameter, density change. Due to the diffuse interface, it is enough to handle one system in a single domain. Difficulties stemming from the topological change of the interface do not occur opposite to the case of a model with a sharp interface. Historically, Van der Waals [20] studied the diffuse interface model as a sharp gradient of the density in the liquid-vapor type two-phase flow. Korteweg [13], who was a student of Van der Waals, provided the Korteweg type stress tensor according to the Van der Waals theory. The stress tensor includes $\nabla ρ \otimes \nabla ρ$ similarly to (1.2). Dunn and Serrin [3] strictly provided the system (1.1) with (1.2). Recently, Heida and Málek [6] derived (1.1) by the entropy production method that does not require difficult concepts such as multipolarity or interstitial working, which are used in [3].

Our aim is to show the global existence of solutions to (1.1) and study the convergence rate of the solution to the given constant state under the condition (1.3). Concerning the global existence of solutions to (1.1) on $R^{n}$ , our research finds that all studies assume that $P^{'} (ρ_{*}) > 0$ , which is the same condition as in CNS [14]. Hattori and Li [4,5] obtain the global existence of $H^{N + 1} \times H^{N}$ solutions with small initial data $u_{0} \in H^{N + 1} \times H^{N}$ , where $H^{N}$ denotes the usual $L^{2}$ Sobolev space and N is an integer satisfying that $N ⩾ [n / 2] + 2$ and $[n / 2]$ denotes the integer part of $n / 2$ . Danchin and Desjardins [1] consider (1.1) in critical spaces. They show the global existence of solutions with small initial data $u_{0} \in (B_{2, 1}^{\frac{n}{2}} \cap B_{2, 1}^{\frac{n}{2} - 1}) \times B_{2, 1}^{\frac{n}{2} - 1}$ , where $B_{2, 1}^{\frac{n}{2}}$ denotes the usual homogeneous Besov space. Tan and R. Zhang, X. Zhang and Tan and Tan, Wang and Xu [17,18,22] show global existence for small initial data in some Sobolev spaces which have lower regularity that that of [4,5] in three dimensional case. In addition, Wang and Tan [21] study the decay rate of the solutions. They show that if initial data satisfy $‖ (ρ_{0}, v_{0}) ‖_{(H^{s + 1} \times H^{s}) \cap L^{1}} < < 1$ ( $s ⩾ 3$ ), where v denotes the velocity field $v = \frac{m}{ρ}$ , it holds that for $t > 0$ $\begin{array}{l} {‖ (ρ (t) - ρ_{*}, v (t)) ‖}_{L^{p}} ⩽ C {(1 + t)}^{- \frac{3}{2} (1 - \frac{1}{p})} (2 ⩽ p ⩽ 6), \\ {‖ \nabla (ρ (t) - ρ_{*}, v (t)) ‖}_{L^{2}} ⩽ C {(1 + t)}^{- \frac{5}{4}} . \end{array}$

However, as shown in J. Daube [2], the pressure is a non-monotone function due to the phase transitions. The Van der Waals equation gives the pressure P as $\begin{matrix} P (ρ) = ρ^{2} φ^{'} (ρ) \end{matrix}$ for a given Helmholtz energy $φ = φ (ρ)$ . In order to model phase transitions, it is assumed that the Helmholtz free energy $\tilde{W} (ρ) = ρ φ (ρ)$ has a double-well sharp (Fig. 1). Hence, as shown in [2], the Helmholtz free energy of double-well sharp together with the relation between P and $\tilde{W} (ρ)$ ; $\begin{matrix} P (ρ) = ρ {\tilde{W}}^{'} (ρ) - \tilde{W} (ρ) \end{matrix}$ show that pressure is a non-monotone function with respect to density (Fig. 2).

Fig. 1.

Typical shape of $\tilde{W} = \tilde{W} (ρ)$ ([2]).

Fig. 2.

Typical shape of the pressure $P = P (ρ)$ .

Consequently, we should consider not only the $P^{'} (ρ_{*}) > 0$ case, but also $P^{'} (ρ_{*}) ⩽ 0$ in (1.1). When $P^{'} (ρ_{*}) < 0$ , Fig. 2 shows that the fluid state is in a phase transition, which is a mixed state between liquid and vapor. In this case, the leading part of the linear system has an accretive operator $+ Δ$ . Hence, we cannot expect the constant state to be stable and have a global existence of solutions around the constant state. On the other hand, if $P^{'} (ρ_{*}) > 0$ , we note above that previous results show that the constant state is stable. Therefore, we propose to study whether or not the critical case $P^{'} (ρ_{*}) = 0$ has a global existence of solutions around the constant state. When $P^{'} (ρ_{*}) = 0$ , the linear system loses symmetry for a linear derivative operator of spacial variables even if we assume that $κ = 0$ . This introduces mathematical difficulties. It is well known that symmetry contributes to stability and $L^{2}$ decay estimate in solutions of general hyperbolic conservation systems with relaxation terms, which includes CNS, as in Kawashima, Shizuta, and Umeda and Kawashima and Shizuta [10,11]. Furthermore, due to $P^{'} (ρ_{*}) = 0$ the momentum part of the fundamental solutions to linear system in a low frequency has worse order terms in the Fourier space, as shown in (3.6) below than that of Kobayashi and Shibata [12] for linearized CNS. This fact prevents us from getting the parabolic type time decay estimate of solutions.

We shall show that for (1.1) there exist global $L^{2}$ solutions for small data $u_{0} =^{⊤} (ρ_{0}, m_{0})$ with a regularity assumption such that $m_{0}$ has the derivative form $m_{0} = \partial_{x} {\tilde{m}}_{0}$ . Furthermore, the solutions decay to the constant state with the parabolic type decay rate; $\begin{matrix} {‖ \nabla^{k} (ρ (t) - ρ_{*}, m (t)) ‖}_{L^{2}} ⩽ C {(1 + t)}^{- \frac{n}{4} - \frac{k}{2}}, \end{matrix}$ for $k = 0, 1$ . This rate coincides with those of [21] for the case $P^{'} (ρ_{*}) > 0$ and [14] for CNS in three dimensional space.

To prove the global existence theorem, we introduce the decomposition of solutions to a low frequency part and a high frequency part as in Okita [15] and Tsuda [19]. For the linear estimate of the low frequency part, we use a similar method to that of [12] for CNS, and Shibata [16] for the linear viscoelastic system. By virtue of combining explicit forms of the fundamental solutions with the regularity assumption, we overcome worse order terms which appear in the density part than that of [12], and we can get the estimate with the same decay order as that of solutions to the heat equation. Note that by using the conservation form, the nonlinearity satisfies the regularity assumption and we can also estimate the nonlinear problem.

On the other hand, as for the high frequency part, we use the $L^{2}$ energy method in the Fourier space. As the linear system loses symmetry in the conservation of momentum, obtaining a closed estimate for $L^{2}$ -norm of the density is a key point. To get the closed estimate, we combine Hattori and Li [4,5] type $L^{2}$ energy method and the Poincaré type estimate, which holds in the high frequency part. Then we can derive the linear estimate for both the density and momentum in the usual $L^{2}$ Sobolev spaces. Concerning nonlinear estimates, note that ρ has the smoothing effect from the Korteweg tensor. Therefore, as opposed to CNS, even if we consider the conservation form (1.1), no derivative loss occurs in the energy method for the nonlinear problem.

Using linear estimates and the iteration argument in time weighted spaces, we show both existence of time global solutions for small data and the decay rate of the solutions as in [14].

This paper is organized as follows. In Section 2 notations and fundamental lemmas used in this paper are stated. In Section 3, the main result is stated. In Section 4, the proof of existence of global solutions and the decay rate is stated.

2. Preliminaries

In this section notations are introduced. We also present some lemmas for the proof of the main result.

Let X be a given Banach space. Then the norm on X is denoted by $‖ \cdot ‖_{X}$ .

Let $1 ≦ p ≦ \infty$ . We denote by $L^{p}$ the usual Lebesgue space which are the set of p-th powered integrable functions on $R^{n}$ with a finite p and essentially bounded functions on $R^{n}$ with $p = \infty$ . Let $k ⩾ 0$ be an integer. We denote by $W^{k, p}$ and $H^{k}$ the usual $L^{p}$ and $L^{2}$ Sobolev space of order k respectively. (As usual, $H^{0}$ is defined by $H^{0} : = L^{2}$ .)

Let $w =^{⊤} (w_{1}, \dots, w_{n})$ on $R^{n}$ . We denote simply by $L^{p}$ the set of all vector fields $\begin{matrix} {w =^{⊤} (w_{1}, \dots, w_{n}); w_{j} \in L^{p} (j = 1, \dots, n)}, \end{matrix}$ with the defined norm $‖ \cdot ‖_{L^{p}} : = ‖ \cdot ‖_{{(L^{p})}^{n}}$ . Similarly we denote by a function space X the linear space of all vector fields $w =^{⊤} (w_{1}, \dots, w_{n})$ on $R^{n}$ satisfying $w_{j} \in X$ ( $j = 1, \dots, n$ ), and $‖ \cdot ‖_{X}$ is the norm $‖ \cdot ‖_{X^{n}}$ if no confusion will occur.

Let $u =^{⊤} (ϕ, m)$ with $ϕ \in H^{k}$ and $m =^{⊤} (m_{1}, \dots, m_{n}) \in H^{j}$ . We denote by $‖ u ‖_{H^{k} \times H^{j}}$ the norm of u on $H^{k} \times H^{j}$ ; $\begin{matrix} ‖ u ‖_{H^{k} \times H^{j}} : = {(‖ ϕ ‖_{H^{k}}^{2} + ‖ w ‖_{H^{j}}^{2})}^{\frac{1}{2}} . \end{matrix}$ If $j = k$ , let $\begin{matrix} H^{k} : = H^{k} \times {(H^{k})}^{2}, ‖ u ‖_{H^{k}} : = ‖ u ‖_{H^{k} \times {(H^{k})}^{2}} (u =^{⊤} (ϕ, m)) . \end{matrix}$ Similarly, for $u =^{⊤} (ϕ, m) \in X \times Y$ with $m =^{⊤} (m_{1}, \dots, m_{n})$ , we put $\begin{matrix} ‖ u ‖_{X \times Y} : = {(‖ ϕ ‖_{X}^{2} + ‖ m ‖_{Y}^{2})}^{\frac{1}{2}} (u =^{⊤} (ϕ, m)), \end{matrix}$ and if $Y = X^{n}$ , simply we define that $\begin{matrix} X : = X \times X^{n}, ‖ u ‖_{X} : = ‖ u ‖_{X \times X^{n}} (u =^{⊤} (ϕ, m)) . \end{matrix}$

We denote the Fourier transform of f for the space variables x using the symbols $\hat{f}$ and $F [f]$ ; $\begin{array}{rcl} \hat{f} (ξ) = F [f] (ξ) : = \int_{R^{n}} f (x) e^{- i x \cdot ξ} d x (ξ \in R^{n}), \end{array}$ and its inverse is $\begin{array}{rcl} F^{- 1} [f] (x) : = {(2 π)}^{- n} \int_{R^{n}} f (ξ) e^{i ξ \cdot x} d ξ (x \in R^{n}) . \end{array}$ Let $k ⩾ 0$ be an integer and let $r_{1}$ be a positive constant. $H_{(\infty)}^{k}$ is defined by the set of all $u \in H^{k}$ with $supp \hat{u} \subset {| ξ | ⩾ r_{1}}$ , and $L_{(1)}^{2}$ is defined by the set of all $u \in L^{2}$ with $supp \hat{u} \subset {| ξ | ⩽ r_{1}}$ .

To decompose into a low frequency part and a high frequency part operators $P_{j}$ ( $j = 1, \infty$ ) on $L^{2}$ are introduced by $\begin{array}{rcl} P_{j} f : = F^{- 1} ({\hat{χ}}_{j} F [f]) (f \in L^{2}, j = 1, \infty), \end{array}$ where $\begin{array}{l} {\hat{χ}}_{1} (ξ) : = \{\begin{matrix} 1 & (| ξ | ⩽ r_{1}), \\ 0 & (| ξ | ⩾ r_{1}), \end{matrix} \\ {\hat{χ}}_{\infty} (ξ) : = 1 - {\hat{χ}}_{1} (ξ), \end{array}$ where $r_{1}$ is any positive number.

Let $0 < T < + \infty$ . A solution space $X^{s} (0, T)$ is defined by $\begin{array}{rcl} X^{s} (0, T) & = & {u =^{⊤} (ϕ, m); u = P_{1} u + P_{\infty} u, \\ P_{1} u \in C ([0, T]; L_{(1)}^{2}), P_{\infty} u =^{⊤} (ϕ_{\infty}, m_{\infty}), \\ ϕ_{\infty} \in C ([0, T]; H_{(\infty)}^{s + 1}) \cap L^{2} (0, T; H_{(\infty)}^{s + 2}), \\ m_{\infty} \in C ([0, T]; H_{(\infty)}^{s}) \cap L^{2} (0, T; H_{(\infty)}^{s + 1})} \end{array}$ with the norm $\begin{array}{rcl} ‖ P_{1} u, P_{\infty} u ‖_{X^{s} (0, T)} & = & ‖ P_{1} u ‖_{C ([0, T]; L_{(1)}^{2})} + ‖ P_{\infty} ϕ ‖_{C ([0, T]; H_{(\infty)}^{s + 1}) \cap L^{2} (0, T; H_{(\infty)}^{s + 2})} \\ + ‖ P_{\infty} m ‖_{C ([0, T]; H_{(\infty)}^{s}) \cap L^{2} (0, T; H_{(\infty)}^{s + 1})} . \end{array}$ Similarly, when $T = \infty$ , $X^{s} (0, \infty)$ is defined by $\begin{array}{rcl} X^{s} (0, \infty) & = & {u =^{⊤} (ϕ, m); u = P_{1} u + P_{\infty} u, \\ P_{1} u \in C ([0, \infty); L_{(1)}^{2}), P_{\infty} u =^{⊤} (ϕ_{\infty}, m_{\infty}), \\ ϕ_{\infty} \in C ([0, \infty); H_{(\infty)}^{s + 1}) \cap L^{2} (0, \infty; H_{(\infty)}^{s + 2}), \\ m_{\infty} \in C ([0, \infty); H_{(\infty)}^{s}) \cap L^{2} (0, \infty; H_{(\infty)}^{s + 1})} . \end{array}$

For operators $L_{1}$ and $L_{2}$ , the commutator of $L_{1}$ and $L_{2}$ is represented as $[L_{1}, L_{2}]$ ; $\begin{array}{rcl} [L_{1}, L_{2}] f : = L_{1} (L_{2} f) - L_{2} (L_{1} f) . \end{array}$

For a nonnegative number s, $[s]$ stands for the Gaussian symbol.

The symbol “∗” stands for the convolution on the space variable x.

Here we state some fundamental Lemmas.

The following Sobolev type inequality is well known.

Lemma 2.1.
Let $n ⩾ 3$ and s be an integer satisfying $s ⩾ [n / 2] + 1$ . Then it holds true that $\begin{array}{rcl} ‖ f ‖_{L^{\infty}} ⩽ C ‖ \nabla f ‖_{H^{s - 1}} \end{array}$ for $f \in H^{s}$ .

For estimates of nonlinearity, some estimates are stated below. The latter lemma is the commutator estimate.
Lemma 2.2 ([8, Lemma A.2]).

Let s be an integer satisfying $s ⩾ [n / 2] + 1$ , $s_{j}$ and $μ_{(j)} ⩾ 0$ ( $j = 1, \dots, ℓ$ ) be integers and multiindices satisfy $0 ⩽ | μ_{(j)} | ⩽ s_{j} ⩽ s + | μ_{(j)} |$ , $μ = μ_{(1)} + \dots + μ_{(ℓ)}$ , $s = s_{1} + \dots + s_{ℓ} ⩽ (ℓ - 1) s + | μ |$ , respectively. Then we have $\begin{array}{rcl} {‖ \partial_{x}^{μ_{(1)}} f_{1} \dots \partial_{x}^{μ_{(ℓ)}} f_{ℓ} ‖}_{L^{2}} ⩽ C \prod_{1 ⩽ j ⩽ ℓ} ‖ f_{j} ‖_{H^{s_{j}}} (f_{j} \in H^{s_{j}}) . \end{array}$

Lemma 2.3 ([7, Lemma 4.3]).

Let s be an integer satisfying $s ⩾ [n / 2] + 1$ . Suppose that F is a smooth function on I, where I is a compact interval of $R$ . Then for a multi-index α with $1 ⩽ | α | ⩽ s$ , there holds that $\begin{matrix} {‖ [\partial_{x}^{α}, F (f_{1})] f_{2} ‖}_{L^{2}} ⩽ C ‖ F ‖_{C^{| α |} (I)} {1 + ‖ \nabla f_{1} ‖_{s - 1}^{| α | - 1}} ‖ \nabla f_{1} ‖_{H^{s - 1}} ‖ f_{2} ‖_{H^{| α |}}, \end{matrix}$ for $f_{1} \in H^{s}$ with $f_{1} (x) \in I$ for all $x \in R^{n}$ and $f_{2} \in H^{| α |}$ ; and $\begin{matrix} {‖ [\partial_{x}^{α}, F (f_{1})] f_{2} ‖}_{L^{2}} ⩽ C ‖ F ‖_{C^{| α |} (I)} {1 + ‖ \nabla f_{1} ‖_{s - 1}^{| α | - 1}} ‖ \nabla f_{1} ‖_{H^{s}} ‖ f_{2} ‖_{H^{| α | - 1}}, \end{matrix}$ for $f_{1} \in H^{s + 1}$ with $f_{1} (x) \in I$ for all $x \in R^{n}$ and $f_{2} \in H^{| α | - 1}$ .

Concerning the projections $P_{1}$ and $P_{\infty}$ , the following properties hold true.

Lemma 2.4 ([15, Lemma 4.2]).

$P_{1}$ is a bounded linear operator from $L^{2}$ to $H^{k}$ with any integer $k ⩾ 0$ and satisfies $\begin{array}{rcl} {‖ \nabla^{k} P_{1} f ‖}_{L^{2}} ⩽ C ‖ f ‖_{L^{2}} (f \in L^{2}) . \end{array}$ Especially $P_{1}$ is bounded from $L^{2}$ to $L^{p}$ ( $2 ⩽ p ⩽ \infty$ ).

Lemma 2.5 ([15, Lemma 4.2], [9, Lemma 4.4]).

Let $k ⩾ 0$ be an integer. Then on $H^{k}$ , $P_{\infty}$ is a bounded linear operator.

It holds true that $\begin{array}{l} ‖ P_{\infty} f ‖_{L^{2}} ⩽ C ‖ \nabla f ‖_{L^{2}} (f \in H^{1}), \\ ‖ F_{(\infty)} ‖_{L^{2}} ⩽ C ‖ \nabla F_{(\infty)} ‖_{L^{2}} (F_{(\infty)} \in H_{(\infty)}^{1}) . \end{array}$

3. Main results

In this section, a main result is stated for (1.1). (1.1) is reformulated as follows. Hereafter we may assume that $ρ_{*} = 1$ without loss of generality. We set $ϕ = ρ - 1$ . Substituting ϕ into (1.1), the following system is obtained; $\begin{array}{rcl} (3.1) & \{\begin{matrix} \partial_{t} ϕ + div m = 0, \\ \partial_{t} m - ν Δ m - \tilde{ν} \nabla div m - κ \nabla Δ ϕ = F (u), \\ ϕ |_{t = 0} = ϕ_{0}, m |_{t = 0} = m_{0}, \end{matrix} \end{array}$ where $u =^{⊤} (ϕ, m)$ , $ν = μ$ , $\tilde{ν} = μ + μ^{'}$ , $ϕ_{0} = ρ_{0} - 1$ , $\begin{array}{l} (3.2) & F (u) = - {div ((1 + P_{(1)} (ϕ) ϕ) m \otimes m) + \nabla (P_{(2)} (ϕ) ϕ^{2}) \\ (3.3) & - ν Δ (P_{(1)} (ϕ) ϕ m) - \tilde{ν} \nabla div (P_{(1)} (ϕ) ϕ m) - div Φ (ϕ)}, \\ P_{(1)} (ϕ) = \int_{0}^{1} f^{'} (1 + τ ϕ) d τ, f (τ) = \frac{1}{τ} (τ \in R), \\ P_{(2)} (ϕ) = \int_{0}^{1} (1 - τ) P^{″} (1 + τ ϕ) d τ, \\ Φ (ϕ) = κ {ϕ Δ ϕ I_{n} + (\nabla ϕ) \cdot (\nabla ϕ) I_{n} - \frac{| \nabla ϕ |^{2}}{2} I_{n} - \nabla ϕ \otimes \nabla ϕ} . \end{array}$ Note that (3.1) is not symmetric in contrast to the usual compressible Navier–Stokes system as in [14]. Therefore, the general theory for symmetric hyperbolic conservation law, as in [10,11], cannot be applied.

(1.1) is linearized and we have $\begin{array}{rcl} (3.4) & \{\begin{matrix} \partial_{t} ϕ + div m = 0, \\ \partial_{t} m - ν Δ m - \tilde{ν} \nabla div m - κ \nabla Δ ϕ = 0, \\ ϕ |_{t = 0} = ϕ_{0}, m |_{t = 0} = m_{0} . \end{matrix} \end{array}$ Taking the Fourier transform of (3.4) with respect to the space variable x yields that $\begin{array}{rcl} (3.5) & \{\begin{matrix} \partial_{t} \hat{ϕ} (t, ξ) + i ξ \cdot \hat{m} (t, ξ) = 0, \\ \partial_{t} \hat{m} (t, ξ) + ν | ξ |^{2} \hat{m} (t, ξ) + \tilde{ν} ξ (ξ \cdot \hat{m} (t, ξ)) + i ξ κ | ξ |^{2} \hat{ϕ} (t, ξ) = 0, \\ \hat{ϕ} (0, ξ) = {\hat{ϕ}}_{0}, \hat{m} (0, ξ) = {\hat{m}}_{0} \end{matrix} \end{array}$ with a parameter ξ. Therefore, the solutions of (3.4) are given by the following formulas. We define that $A = \frac{ν + \tilde{ν}}{2}$ , $K = \frac{2 \sqrt{κ}}{ν + \tilde{ν}}$ . If $| ξ | \neq 0$ and $K \neq 1$ the Fourier transforms of ϕ and m are given by $\begin{array}{l} \hat{ϕ} = \frac{λ_{+} (ξ) e^{λ_{-} (ξ) t} - λ_{-} (ξ) e^{λ_{+} (ξ) t}}{λ_{+} (ξ) - λ_{-} (ξ)} {\hat{ϕ}}_{0} - i \frac{e^{λ_{+} (ξ) t} - e^{λ_{-} (ξ) t}}{λ_{+} (ξ) - λ_{-} (ξ)} ξ \cdot {\hat{m}}_{0}, \\ (3.6) & \hat{m} = e^{- ν | ξ |^{2} t} {\hat{m}}_{0} - i ξ κ | ξ |^{2} (\frac{e^{λ_{+} (ξ) t} - e^{λ_{-} (ξ) t}}{λ_{+} (ξ) - λ_{-} (ξ)}) {\hat{ϕ}}_{0} \\ + (\frac{λ_{+} (ξ) e^{λ_{+} (ξ) t} - λ_{-} (ξ) e^{λ_{-} (ξ) t}}{λ_{+} (ξ) - λ_{-} (ξ)} - e^{- ν | ξ |^{2} t}) \frac{ξ (ξ \cdot {\hat{m}}_{0})}{| ξ |^{2}}, \end{array}$ where $\begin{array}{l} (3.7) & λ_{\pm} (ξ) = - A (| ξ |^{2} \pm | ξ |^{2} \sqrt{1 - K^{2}}) . \end{array}$

If $K = 1$ , $\begin{array}{l} \hat{ϕ} = (1 + ν | ξ |^{2} t) e^{- ν | ξ |^{2} t} {\hat{ϕ}}_{0} - t e^{- ν | ξ |^{2} t} (i^{⊤} ξ {\hat{m}}_{0}), \\ \hat{m} = A^{2} | ξ |^{2} t e^{- A | ξ |^{2} t} i ξ {\hat{ϕ}}_{0} + e^{- ν | ξ |^{2} t} {\hat{m}}_{0} \\ + ((A | ξ |^{2} t - 1) e^{- A | ξ |^{2} t} - e^{- ν | ξ |^{2} t}) \frac{ξ^{⊤} ξ {\hat{m}}_{0}}{| ξ |^{2}} . \end{array}$

Set $\begin{array}{rcl} u =^{⊤} (ϕ, m), u_{0} =^{⊤} (ϕ_{0}, m_{0}), A = (\begin{matrix} 0 & div \\ - κ \nabla Δ & - ν Δ - \tilde{ν} \nabla div \end{matrix}) . \end{array}$ Then (3.1) is written as follows. $\begin{array}{rcl} (3.8) & \partial_{t} u + A u = F (u), u |_{t = 0} = u_{0}, \end{array}$ where $F (u) =^{⊤} (0, f (u))$ . Due to the Duhamel principle, (3.8) is transformed to the integral equations; $\begin{array}{rcl} (3.9) & u (t) = S (t) u_{0} + \int_{0}^{t} S (t, τ) F (u (τ)) d τ, \end{array}$ where $S = S (t)$ denotes the solution operator of the system whose definition is given by (3.6); $\begin{array}{rcl} (3.10) & S (t) (\begin{matrix} ϕ_{0} \\ m_{0} \end{matrix}) = F^{- 1} \{(\begin{matrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{matrix}) (\begin{matrix} {\hat{ϕ}}_{0} \\ {\hat{m}}_{0} \end{matrix})\}, \end{array}$ where for $K \neq 1$ $\begin{array}{l} S_{11} = \frac{λ_{+} (ξ) e^{λ_{-} (ξ) t} - λ_{-} (ξ) e^{λ_{+} (ξ) t}}{λ_{+} (ξ) - λ_{-} (ξ)}, \\ S_{12} = - i \frac{e^{λ_{+} (ξ) t} - e^{λ_{-} (ξ) t}}{λ_{+} (ξ) - λ_{-} (ξ)}^{⊤} ξ, \\ S_{21} = - i ξ κ | ξ |^{2} (\frac{e^{λ_{+} (ξ) t} - e^{λ_{-} (ξ) t}}{λ_{+} (ξ) - λ_{-} (ξ)}), \\ S_{22} = e^{- ν | ξ |^{2} t} I_{n} + (\frac{λ_{+} (ξ) e^{λ_{+} (ξ) t} - λ_{-} (ξ) e^{λ_{-} (ξ) t}}{λ_{+} (ξ) - λ_{-} (ξ)} - e^{- ν | ξ |^{2} t}) \frac{ξ^{⊤} ξ}{| ξ |^{2}} \end{array}$ and for $K = 1$ $\begin{array}{l} S_{11} = (1 + ν | ξ |^{2} t) e^{- ν | ξ |^{2} t}, \\ S_{12} = t e^{- ν | ξ |^{2} t} (i^{⊤} ξ), \\ S_{21} = A^{2} | ξ |^{2} t e^{- A | ξ |^{2} t} i ξ, \\ S_{22} = e^{- ν | ξ |^{2} t} I_{n} + ((A | ξ |^{2} t - 1) e^{- A | ξ |^{2} t} - e^{- ν | ξ |^{2} t}) \frac{ξ^{⊤} ξ}{| ξ |^{2}} . \end{array}$ We obtain global $L^{2}$ solutions to (3.9) for small data with a regularity assumption of m and decay rate of the solutions. The primary result is stated as follows.

Theorem 3.1.
Let $u_{0} =^{⊤} (ϕ_{0}, m_{0}) \in H^{s + 1} \times H^{s}$ where s is an integer satisfying $s ⩾ [n / 2] + 1$ and $m_{0} = \partial_{x} {\tilde{m}}_{0}$ . We also assume that $^{⊤} (ϕ_{0}, {\tilde{m}}_{0}) \in L^{1}$ . We set $\begin{matrix} E_{0} = ‖ u_{0} ‖_{H^{s + 1} \times H^{s}} + {‖^{⊤} (ϕ_{0}, {\tilde{m}}_{0}) ‖}_{L^{1}} . \end{matrix}$ Then there exists a positive constant $δ_{0}$ such that if $E_{0} ⩽ δ_{0}$ there exists a global solution $u \in L^{2} (0, \infty; H^{s + 2} \times H^{s + 1})$ to ( 3.9 ) and we obtain the following decay rate of the solution: $\begin{array}{rcl} (3.11) & {‖ \nabla^{k} u ‖}_{L^{2}} ⩽ C {(1 + t)}^{- n / 4 - k / 2} (k = 0, 1) . \end{array}$ In addition, the uniqueness of the solution holds in the class $\begin{matrix} {u =^{⊤} (ϕ, m); u = P_{1} u + P_{\infty} u, ‖ P_{1} u, P_{\infty} u ‖_{X^{s} (0, \infty)} ⩽ C δ_{0}} . \end{matrix}$
Remark 3.2.
Concerning the norm of $X^{s} (0, \infty)$ , note that as $n ⩾ 3$ , if $P_{1} u$ satisfies the decay estimate (3.11) then $P_{1} u \in L^{2} (0, \infty; L_{(1)}^{2})$ ; therefore, we have that $X^{s} (0, \infty) \subset L^{2} (0, \infty; H^{s + 2} \times H^{s + 1})$ .

4. Existence of global solutions

In this section, we show the existence of global solutions to (3.9) and the decay rate of the solutions. Let $\begin{matrix} Γ [u] = S (t) u_{0} + \int_{0}^{t} S (t, τ) F (u (τ)) d τ \end{matrix}$ To solve (3.9), we look for a fixed point of Γ for a given $u_{0}$ . Since $\begin{matrix} P_{j} Γ [F (u)] = Γ [P_{j} F (u)] (j = 1, \infty) \end{matrix}$ and $\begin{array}{l} supp \hat{P_{1} F} (u) \subset {| ξ | ⩽ r_{1}}, \\ supp \hat{P_{\infty} F} (u) \subset {| ξ | ⩾ r_{1}} \end{array}$ for each $t \in [0, T]$ , we shall estimate projections of Γ on $L_{(1)}^{2}$ and $L_{(\infty)}^{2}$ respectively.

4.1. Estimates of Γ; the low frequency part

In this subsection, we estimate Γ on the low frequency part. In the low frequency part, solution operators $S_{1} (t)$ and $S_{1} (t)$ are defined by $\begin{matrix} S_{1} (t) = S (t) |_{L_{(1)}^{2}}, S_{1} (t) F = \int_{0}^{t} S_{1} (t - τ) F (τ) d τ . \end{matrix}$ We show that the solution operator $S_{1} (t)$ is a bounded linear operator on $L_{(1)}^{2}$ for an initial data $u_{01} =^{⊤} (ϕ_{01}, m_{01})$ with $m_{01} = \partial_{x} {\tilde{m}}_{01}$ with decay estimate of $S_{1} (t)$ . We have

Proposition 4.1.

Let $u_{01} =^{⊤} (ϕ_{01}, m_{01})$ and $m_{01} = \partial_{x} {\tilde{m}}_{01}$ . For each $^{⊤} (ϕ_{01}, {\tilde{m}}_{01}) \in L_{(1)}^{2}$ and all $T > 0$ , $S_{1} (t)$ satisfies $\begin{matrix} S_{1} (t) u_{01} \in C ([0, T]; L_{(1)}^{2}), \end{matrix}$ and it holds true that $\begin{matrix} {‖ S_{1} (\cdot) u_{1} ‖}_{C ([0, T]; L_{(1)}^{2})} ⩽ C {‖^{⊤} (ϕ_{01}, {\tilde{m}}_{01}) ‖}_{L^{2}}, \end{matrix}$ where $T > 0$ is any given positive number and C is a positive constant independent of T.

If $^{⊤} (ϕ_{01}, {\tilde{m}}_{01}) \in L^{1}$ under the assumption of (i), $S_{1} (t)$ satisfies the decay estimate $\begin{matrix} {‖ \nabla^{k} S_{1} (t) u_{1} ‖}_{L_{(1)}^{2}} ⩽ C {(1 + t)}^{- n / 4 - k / 2} {‖^{⊤} (ϕ_{01}, {\tilde{m}}_{01}) ‖}_{L^{1}} \end{matrix}$ for $t > 0$ and $k \in Z_{+}$ , where C is a positive constant independent of t.

Proof.
We consider the case $K \neq 1$ . Due to (3.10), we see that $\begin{array}{rcl} (4.1) & S_{1} (t) (\begin{matrix} ϕ_{01} \\ m_{01} \end{matrix}) = F^{- 1} \{(\begin{matrix} S_{11} & S_{12} ξ_{j} \\ S_{21} & S_{22} ξ_{j} \end{matrix}) (\begin{matrix} {\hat{ϕ}}_{01} \\ {\hat{\tilde{m}}}_{01} \end{matrix})\} \end{array}$ for some $j \in {1, \dots n}$ . We prove (ii) before (i). Concerning $S_{12}$ part in $S_{1} (t)$ , we set $\begin{matrix} K_{12} (t, x) {\tilde{m}}_{0} = F^{- 1} (\frac{e^{λ_{+} (ξ) t} - e^{λ_{-} (ξ) t}}{λ_{+} (ξ) - λ_{-} (ξ)} ξ_{j}^{⊤} ξ {\hat{\tilde{m}}}_{01}) . \end{matrix}$ A cut-off function $χ_{0} = F^{- 1} {\hat{χ}}_{0}$ with ${\hat{χ}}_{0}$ is introduced by $\begin{array}{l} (4.2) & \begin{aligned} {\hat{χ}}_{0} \in C^{\infty} (R^{n}), 0 ⩽ {\hat{χ}}_{0} ⩽ 1, {\hat{χ}}_{0} = 1 on {| ξ | ⩽ r_{1}}, \\ supp {\hat{χ}}_{0} \subset {| ξ | ⩽ 2 r_{1}} . \end{aligned} \end{array}$ Since ${\tilde{m}}_{01} \in L_{(1)}^{2}$ , we see that ${\tilde{m}}_{01} = F^{- 1} ({\hat{χ}}_{0} {\hat{\tilde{m}}}_{01})$ . We estimate $‖ K_{12} (t, \cdot) ‖_{L^{2}}$ by (3.7) as follows. $\begin{array}{rcl} {‖ K_{12} (t, \cdot) ‖}_{L^{2}}^{2} & ⩽ & C \int_{| ξ | ⩽ 2 r_{\infty}} \frac{| e^{λ_{+} (ξ) t} - e^{λ_{-} (ξ) t} |^{2} | ξ |^{4} | {\hat{χ}}_{0} (ξ) |^{2}}{| λ_{+} (ξ) - λ_{-} (ξ) |^{2}} d ξ \\ ⩽ & C \int_{| ξ | ⩽ 2 r_{1}} e^{- c | ξ |^{2} t} d ξ \\ ⩽ & C \int_{0}^{2 r_{1}} e^{- c r^{2} t} r^{n - 1} d r \\ ⩽ & C {(\frac{1}{\sqrt{t}})}^{n} \int_{0}^{2 \sqrt{t} r_{\infty}} e^{- c η^{2}} η^{n - 1} d η ⩽ C t^{- n / 2}, \end{array}$ where we used change of variables as $r \sqrt{t} = η$ . Hence we get that $\begin{matrix} {‖ K_{12} (t, \cdot) ‖}_{L^{2}} ⩽ t^{- n / 4} . \end{matrix}$ Similarly, we see that $\begin{matrix} {‖ \nabla^{k} K_{12} (t, \cdot) ‖}_{L^{2}} ⩽ {(1 + t)}^{- n / 4 - k / 2} . \end{matrix}$ Therefore, it holds by the Young inequality that $\begin{matrix} {‖ F^{- 1} (S_{12} ξ_{j} {\hat{\tilde{m}}}_{01}) ‖}_{L^{2}} ⩽ {(1 + t)}^{- n / 4 - k / 2} ‖ {\tilde{m}}_{01} ‖_{L^{1}} . \end{matrix}$ As the other part of $S_{1} (t)$ and the case $K = 1$ can be similarly estimated, we have (ii). The estimate of (i) can also be estimated as in (ii) and we omit this proof. Note that by definition of $S_{1} (t)$ and the Lebesgue convergence theorem, we obtain that $S_{1} (t) u_{01} \in C ([0, T]; L_{(1)}^{2})$ . This completes the proof. □

We set $\begin{matrix} Γ_{1} [F] = S_{1} (t) u_{01} + S_{1} (t) F \end{matrix}$ for $F = \partial_{x} f$ . By direct application of Proposition 4.1, we can derive the estimate of $Γ_{1}$ . Proposition 4.2.

Let $u_{01} =^{⊤} (ϕ_{01}, m_{01})$ and $m_{01} = \partial_{x} {\tilde{m}}_{01}$ . For each $^{⊤} (ϕ_{01}, {\tilde{m}}_{01}) \in L_{(1)}^{2}$ , $F = \partial_{x} f$ with $f \in L^{2} (0, T; L_{(1)}^{2})$ and all $T > 0$ , $Γ_{1}$ satisfies $\begin{matrix} Γ_{1} [F] \in C ([0, T]; L_{(1)}^{2}), \end{matrix}$ and $\begin{matrix} {‖ Γ_{1} [F] (t) ‖}_{L^{2}} ⩽ C {‖^{⊤} (ϕ_{01}, {\tilde{m}}_{01}) ‖}_{L^{2}} + C \int_{0}^{t} {‖ f (τ) ‖}_{L^{2}} d τ, \end{matrix}$ where C is a positive constant independent of t.

If in addition, $^{⊤} (ϕ_{01}, {\tilde{m}}_{01}) \in L^{1}$ and $f \in C ([0, T]; L^{1})$ then $Γ_{1}$ is estimated by $\begin{array}{l} {‖ \nabla^{k} Γ_{1} [F] (t) ‖}_{L^{2}} \\ ⩽ C {(1 + t)}^{- n / 4 - k / 2} {‖^{⊤} (ϕ_{01}, {\tilde{m}}_{01}) ‖}_{L^{1}} + C \int_{0}^{t} {(1 + t - τ)}^{- n / 4 - k / 2} {‖ f (τ) ‖}_{L^{1}} d τ \end{array}$ for $t > 0$ and $k \in Z_{+}$ , where C is a positive constant independent of t.

4.2. Estimates of Γ; the high frequency part

In this subsection, we estimate Γ on the high frequency part. In the high frequency, the solution operators, $S_{\infty} (t)$ and $S_{\infty} (t)$ , are defined by $\begin{matrix} S_{\infty} (t) = S (t) |_{L_{(\infty)}^{2}}, S_{\infty} (t) F = \int_{0}^{t} S_{\infty} (t - τ) F (τ) d τ . \end{matrix}$

We first show that ${S_{\infty} (t)}_{t ⩾ 0}$ is a $C^{0}$ semigroup on $H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1}$ .

Proposition 4.3.
Let $n ⩾ 3$ and s be an integer satisfying $s ⩾ [n / 2] + 1$ . Then ${S_{\infty} (t)}_{t ⩾ 0}$ is a $C^{0}$ semigroup on $H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1}$ with $\begin{matrix} S_{\infty} (\cdot) u_{0 \infty} \in C ([0, T]; H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1}) \end{matrix}$ for $u_{0 \infty} \in C ([0, T]; H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1})$ and $\begin{array}{rcl} {‖ S_{\infty} (t) u_{0 \infty} ‖}_{H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1}} ⩽ ‖ u_{0} ‖_{H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1}} \end{array}$ for $t \in [0, T]$ , where $T > 0$ is any positive number and C is independent of t.
Proof.
Let $F_{\infty} =^{⊤} (F_{\infty}^{1}, F_{\infty}^{2}) \in H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1}$ . The resolvent problem is $\begin{array}{rcl} (4.3) & λ u_{\infty} + A u_{\infty} = F_{\infty} \end{array}$ for $u_{\infty} =^{⊤} (ϕ_{\infty}, m_{\infty})$ , where $λ \in C$ . Taking the Fourier transform of (4.3), $\begin{array}{rcl} (4.4) & λ {\hat{u}}_{\infty} + {\hat{A}}_{ξ} {\hat{u}}_{\infty} = {\hat{F}}_{\infty}, \end{array}$ where $\begin{array}{rcl} {\hat{A}}_{ξ} = (\begin{matrix} 0 & i^{⊤} ξ \\ i κ | ξ | ξ & ν | ξ |^{2} I_{n} + \tilde{ν} ξ^{⊤} ξ \end{matrix}) . \end{array}$ A similar manner to the proof of Proposition 4.4 below yields that $\begin{array}{l} Re λ {\sum_{| α | = 0}^{s} (κ_{1} {| {(i ξ)}^{α} {\hat{m}}_{\infty} |}^{2} + κ_{1} κ {| {(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty} |}^{2} + {(i ξ)}^{α} {\hat{m}}_{\infty} \cdot \overline{{(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty}})} \\ + d_{1} (\sum_{| α | = 0}^{s} {| {(i ξ)}^{α} (i ξ) {\hat{m}}_{\infty} |}^{2} + \sum_{| α | = 0}^{s} {| {(i ξ)}^{α} | ξ |^{2} {\hat{ϕ}}_{\infty} |}^{2}) \\ (4.5) & ⩽ C {\sum_{| α | = 0}^{s} {| {(i ξ)}^{α} {\hat{F}}_{\infty}^{1} |}^{2} + \sum_{| α | = 0}^{s - 1} ({| {(i ξ)}^{α} {\hat{F}}_{\infty}^{2} |}^{2}} \end{array}$ for $ξ \in R^{n}$ , where $κ_{1}$ and $d_{1}$ are the same constants in (4.8). Thus ${(λ + {\hat{A}}_{ξ})}^{- 1}$ is well defined on $Re λ > 0$ for each $ξ \in R^{n}$ and ${\hat{u}}_{\infty}$ is represented by ${\hat{u}}_{\infty} = {(λ + {\hat{A}}_{ξ})}^{- 1} {\hat{F}}_{\infty}$ . Here the norm $| | | \cdot | | |_{s}$ stands for $\begin{matrix} | | | u_{\infty} | | |_{s} : = {(\sum_{| α | = 0}^{s} κ {‖ \partial_{x}^{α} ϕ_{\infty} ‖}_{L_{(\infty)}^{2}}^{2} + \sum_{| α | = 0}^{s - 1} {‖ \partial_{x}^{α} m_{\infty} ‖}_{L_{(\infty)}^{2}}^{2})}^{\frac{1}{2}} \end{matrix}$ on the $L^{2}$ Sobolev space $H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1}$ . We see from (4.5) and definition of $L_{(\infty)}^{2}$ that $\begin{matrix} Re λ | | | u_{\infty} | | |_{s} ⩽ C | | | F_{\infty} | | |_{s} \end{matrix}$ and if $Re λ > 0$ , $\begin{matrix} ‖ u_{\infty} ‖_{H^{s} \times H^{s - 1}} ⩽ \frac{C}{Re λ} | | | F_{\infty} | | |_{s} . \end{matrix}$ Hence $\begin{matrix} {λ; Re λ > 0} \subset ρ (- A), \end{matrix}$ where $ρ (- A)$ is the resolvent set of the operator $- A$ and it holds true that $\begin{matrix} | | | {(λ + A)}^{- 1} F_{\infty} | {| |}_{s} ⩽ \frac{C}{Re λ} | | | F_{\infty} | | |_{s} . \end{matrix}$ This together with the Hille–Yoshida theorem imply that $- A$ generates $S_{\infty} (t) = e^{- t A}$ , $C^{0}$ semigroup on $H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1}$ , and we obtain (4.3). This completes the proof. □

Set $\begin{matrix} S_{\infty} (t) F_{\infty} = \int_{0}^{t} S_{\infty} (t - τ) F_{\infty} (τ) d τ . \end{matrix}$ By the Duhamel principle $S_{\infty}$ is a solution operator for the linearized problem $\begin{array}{rcl} (4.6) & \partial_{t} u_{\infty} + A u_{\infty} = F_{\infty}, u_{\infty} |_{t = 0} = 0 \end{array}$ for $u_{\infty} =^{⊤} (ϕ_{\infty}, m_{\infty})$ . Furthermore, we obtain the following
Proposition 4.4.
$\begin{matrix} S_{\infty} (\cdot) F_{\infty} \in C ([0, T]; H_{(\infty)}^{s + 1} \times H_{(\infty)}^{s}) \cap L^{2} (0, T; H_{(\infty)}^{s + 2} \times H_{(\infty)}^{s + 1}) \end{matrix}$ for $F_{\infty} \in L^{2} (0, T; H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1})$ and it holds that $\begin{array}{l} {‖ S_{\infty} (t) F_{\infty} ‖}_{H_{(\infty)}^{s + 1} \times H_{(\infty)}^{s}} + {‖ S_{\infty} (\cdot) F_{\infty} ‖}_{L^{2} (0, T; H_{(\infty)}^{s + 2} \times H_{(\infty)}^{s + 1})} \\ (4.7) & ⩽ C ‖ F_{\infty} ‖_{L^{2} (0, T; H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1})} . \end{array}$ for $t \in [0, T]$ , where $T > 0$ is any positive number and C is independent of t and T.
Proof.
We use the energy estimate in the Fourier space. Our claim is to show $\begin{array}{l} \sum_{| α | = 0}^{s} \frac{\partial}{\partial t} (κ_{1} κ {‖ {(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty} ‖}_{L_{(\infty)}^{2}}^{2} + κ_{1} {‖ {(i ξ)}^{α} {\hat{m}}_{\infty} ‖}_{L_{(\infty)}^{2}}^{2}) + \sum_{| α | = 0}^{s} \frac{\partial}{\partial t} ({(i ξ)}^{α} {\hat{m}}_{\infty} \cdot \overline{{(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty}}) \\ + d_{1} \sum_{| α | = 0}^{s} ({‖ {(i ξ)}^{α} | ξ |^{2} {\hat{ϕ}}_{\infty} ‖}_{L_{(\infty)}^{2}}^{2} + {‖ {(i ξ)}^{α} (i ξ) {\hat{m}}_{\infty} ‖}_{L_{(\infty)}^{2}}^{2}) \\ (4.8) & ⩽ C ‖ F_{\infty} ‖_{H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1}}^{2}, \end{array}$ where $κ_{1}$ and $d_{1}$ are positive constants. Let $ξ \in R^{n}$ and $| ξ | ⩾ r_{1}$ . Taking the Fourier transform of (4.6), we see that $\begin{array}{rcl} (4.9) & \{\begin{matrix} \frac{\partial}{\partial t} {\hat{ϕ}}_{\infty} + i ξ \cdot {\hat{m}}_{\infty} = {\hat{F}}_{1 \infty}, \\ \frac{\partial}{\partial t} {\hat{m}}_{\infty} + ν | ξ |^{2} {\hat{m}}_{\infty} + \tilde{ν} ξ ξ \cdot {\hat{m}}_{\infty} + i κ ξ | ξ |^{2} {\hat{ϕ}}_{\infty} = {\hat{F}}_{2 \infty}, \end{matrix} \end{array}$ where ${\hat{F}}_{\infty} =^{⊤} ({\hat{F}}_{1 \infty}, {\hat{F}}_{2 \infty})$ . For a multi-index α satisfying $| α | ⩽ s$ , taking the complex inner product of ${(i ξ)}^{α} {(4.9)}_{2}$ with ${(i ξ)}^{α} {\hat{m}}_{\infty}$ and taking the sum of α and from the real part for $| ξ | ⩾ r_{1}$ we have that $\begin{array}{l} \frac{1}{2} \sum_{| α | = 0}^{s} \frac{\partial}{\partial t} {| {(i ξ)}^{α} {\hat{m}}_{\infty} |}^{2} + κ Re ({(i ξ)}^{α} (i ξ) | ξ |^{2} {\hat{ϕ}}_{\infty} \cdot \overline{{(i ξ)}^{α} {\hat{m}}_{\infty}}) \\ + \frac{ν}{2} \sum_{| α | = 0}^{s} {| {(i ξ)}^{α} (i ξ) {\hat{m}}_{\infty} |}^{2} + \tilde{ν} \sum_{| α | = 0}^{s} {| {(i ξ)}^{α} (i ξ)^{⊤} {\hat{m}}_{\infty} |}^{2} \\ (4.10) & ⩽ C \sum_{| α | = 0}^{s - 1} | {(i ξ)}^{α} {\hat{F}}_{2 \infty} |, \end{array}$ where C is some positive constant. Note that due to (4.9) we obtain that $\begin{array}{rcl} (i ξ) | ξ |^{2} {\hat{ϕ}}_{\infty} \cdot \overline{{\hat{m}}_{\infty}} & = & - (| ξ |^{2} {\hat{ϕ}}_{\infty} \overline{i ξ \cdot {\hat{m}}_{\infty}}) \\ = & | ξ |^{2} {\hat{ϕ}}_{\infty} \overline{\frac{\partial}{\partial t} {\hat{ϕ}}_{\infty}} - | ξ |^{2} {\hat{ϕ}}_{\infty} \overline{{\hat{F}}_{1 \infty}} . \end{array}$ Therefore, we derive the inequality $\begin{array}{l} \frac{1}{2} \sum_{| α | = 0}^{s} \frac{\partial}{\partial t} {{| {(i ξ)}^{α} {\hat{m}}_{\infty} |}^{2} + κ | {(i ξ)}^{α} | ξ | {\hat{ϕ}}_{\infty} |^{2}} + \frac{ν}{2} \sum_{| α | = 0}^{s} {| {(i ξ)}^{α} (i ξ) {\hat{m}}_{\infty} |}^{2} + \tilde{ν} \sum_{| α | = 0}^{s} {| {(i ξ)}^{α} (i ξ) \cdot {\hat{m}}_{\infty} |}^{2} \\ (4.11) & ⩽ ϵ \sum_{| α | = 0}^{s} {| {(i ξ)}^{α} | ξ |^{2} {\hat{ϕ}}_{\infty} |}^{2} + C (\sum_{| α | = 0}^{s} {| {(i ξ)}^{α} {\hat{F}}_{1 \infty} |}^{2} + {| \sum_{| α | = 0}^{s - 1} | {(i ξ)}^{α} {\hat{F}}_{2 \infty} |}^{2}), \end{array}$ where ϵ is a positive constant satisfying $ϵ ⩽ \frac{c_{2}}{2 κ_{1}}$ , $κ_{1}$ and $c_{2}$ are defined below. On the other hand, we take the complex inner product of ${(i ξ)}^{α} {(4.9)}_{2}$ with ${(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty}$ to obtain $\begin{array}{l} \sum_{| α | = 0}^{s} \frac{\partial}{\partial t} ({(i ξ)}^{α} {\hat{m}}_{\infty} \cdot \overline{{(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty}}) - {(i ξ)}^{α} {\hat{m}}_{\infty} \cdot \overline{{(i ξ)}^{α} (i ξ) \frac{\partial}{\partial t} {\hat{ϕ}}_{\infty}} \\ + ν ({(i ξ)}^{α} | ξ |^{2} {\hat{m}}_{\infty} \cdot \overline{{(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty}}) + \tilde{ν} ({(i ξ)}^{α} (ξ^{⊤} ξ {\hat{m}}_{\infty}) \cdot \overline{{(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty}}) \\ + κ \sum_{| α | = 0}^{s} {| {(i ξ)}^{α} | ξ |^{2} {\hat{ϕ}}_{\infty} |}^{2} \\ (4.12) & = \sum_{| α | = 0}^{s} ({(i ξ)}^{α} {\hat{F}}_{2 \infty} \cdot \overline{{(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty}}) . \end{array}$ Since $\begin{array}{rcl} (4.13) & {\hat{m}}_{\infty} \cdot \overline{(i ξ) \frac{\partial}{\partial t} {\hat{ϕ}}_{\infty}} = (- i ξ \cdot {\hat{m}}_{\infty}) \cdot \overline{(- i ξ \cdot {\hat{m}}_{\infty} + {\hat{F}}_{1 \infty})} \end{array}$ by ${(4.9)}_{2}$ , we see from (4.12) that $\begin{array}{l} \sum_{| α | = 0}^{s} \frac{\partial}{\partial t} ({(i ξ)}^{α} {\hat{m}}_{\infty} \cdot \overline{{(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty}}) + c_{2} \sum_{| α | = 0}^{s} {| {(i ξ)}^{α} | ξ |^{2} {\hat{ϕ}}_{\infty} |}^{2} \\ ⩽ c_{3} \sum_{| α | = 0}^{s} ({| {(i ξ)}^{α} (i ξ) {\hat{m}}_{\infty} |}^{2} + {| {(i ξ)}^{α} i ξ \cdot {\hat{m}}_{\infty} |}^{2}) \\ (4.14) & + C (\sum_{| α | = 0}^{s} {| {(i ξ)}^{α} {\hat{F}}_{1 \infty} |}^{2} + {| \sum_{| α | = 0}^{s - 1} | {(i ξ)}^{α} {\hat{F}}_{2 \infty} |}^{2}) . \end{array}$ Let $κ_{1}$ be suitable large constant satisfying that $\frac{κ_{1} ν}{2} ⩾ 2 c_{3}$ , $κ_{1} \tilde{ν} ⩾ 2 c_{3}$ and $\begin{array}{rcl} | \sum_{| α | = 0}^{s} ({(i ξ)}^{α} {\hat{m}}_{\infty} \cdot \overline{{(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty}}) | ⩽ \frac{1}{2} (κ_{1} κ {‖ {(i ξ)}^{α} (i ξ) {\hat{ϕ}}_{\infty} ‖}_{L_{(\infty)}^{2}}^{2} + κ_{1} {‖ {(i ξ)}^{α} {\hat{m}}_{\infty} ‖}_{L_{(\infty)}^{2}}^{2}) . \end{array}$ Considering $κ_{1} \times (4.11) + (4.14)$ we get (4.8). Integrating (4.8) on time and by the Plancherel theorem and Lemma 2.5 it holds that $\begin{matrix} {‖ ϕ_{\infty} (t) ‖}_{H^{s + 1}}^{2} + {‖ m_{\infty} (t) ‖}_{H^{s}}^{2} + d_{1} \int_{0}^{t} ‖ \nabla m_{\infty} ‖_{H^{s}}^{2} + ‖ \nabla ϕ_{\infty} ‖_{H^{s + 1}}^{2} d τ ⩽ C ‖ F_{\infty} ‖_{L^{2} (0, T; H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1})}^{2} . \end{matrix}$ This implies (4.7). This completes the proof. □
Remark 4.5.
In the proof of Proposition 4.4, we obtain the following energy estimate.
Proposition 4.6.
Let $s ⩾ 0$ be an integer with $s ⩾ [n / 2] + 1$ . Suppose that $\begin{array}{l} u_{0 \infty} =^{⊤} (ϕ_{0 \infty}, m_{0 \infty}) \in H_{(\infty)}^{s + 1} \times H_{(\infty)}^{s}, \\ F_{\infty} =^{⊤} (F_{\infty}^{1}, F_{\infty}^{2}) \in L^{2} (0, T^{'}; H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1}) \end{array}$ for all $T^{'} > 0$ . We suppose that $u_{\infty} =^{⊤} (ϕ_{\infty}, m_{\infty})$ satisfies $\begin{array}{rcl} (4.15) & \{\begin{matrix} \partial_{t} u_{\infty} + A u_{\infty} = F_{\infty}, \\ u_{\infty} |_{t = 0} = u_{0 \infty} \end{matrix} \end{array}$ and $\begin{array}{rcl} ϕ_{\infty} \in C ([0, T^{'}]; H_{(\infty)}^{s + 1}) \cap L^{2} (0, T^{'}; H_{(\infty)}^{s + 2}), m_{\infty} \in C ([0, T^{'}]; H_{(\infty)}^{s}) \cap L^{2} (0, T^{'}; H_{(\infty)}^{s + 1}) \end{array}$ for all $T^{'} > 0$ . Then an energy functional $E [u_{\infty}]$ exists and it holds true that $\begin{array}{rcl} (4.16) & \frac{d}{d t} E [u_{\infty}] (t) + d ({‖ \nabla ϕ_{\infty} (t) ‖}_{H^{s + 1}}^{2} + {‖ \nabla m_{\infty} (t) ‖}_{H^{s}}^{2}) ⩽ C {‖ F_{\infty} (t) ‖}_{H^{s} \times H^{s - 1}}^{2} \end{array}$ on $(0, T^{'})$ for all $T^{'} > 0$ . Here d is a positive constant; C is a positive constant independent of $T^{'}$ ; $E [u_{\infty}]$ is equivalent to $‖ u_{\infty} ‖_{H^{s + 1} \times H^{s}}^{2}$ , i.e, $\begin{matrix} C^{- 1} ‖ u_{\infty} ‖_{H^{s + 1} \times H^{s}}^{2} ⩽ E [u_{\infty}] ⩽ C ‖ u_{\infty} ‖_{H^{s + 1} \times H^{s}}^{2}; \end{matrix}$ and $E [u_{\infty}] (t)$ is absolutely continuous in $t \in [0, T^{'}]$ for all $T^{'} > 0$ .

We set $\begin{matrix} Γ_{\infty} [F] = S_{\infty} (t) u_{0 \infty} + S_{\infty} (t) F . \end{matrix}$ By direct application of Propositions 4.3–4.6, we can derive the estimate of $Γ_{\infty}$ . Proposition 4.7.
Let $u_{0 \infty} =^{⊤} (ϕ_{0 \infty}, m_{0 \infty})$ and s be an integer satisfying $s ⩾ [n / 2] + 1$ . For each $^{⊤} (ϕ_{0 \infty}, m_{0 \infty}) \in H_{(\infty)}^{s + 1} \times H_{(\infty)}^{s}$ , $F \in L^{2} (0, T; H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1})$ and all $T > 0$ , $Γ_{\infty}$ satisfies $\begin{matrix} Γ_{\infty} [F] \in C ([0, T]; H_{(\infty)}^{s + 1} \times H_{(\infty)}^{s}) \cap L^{2} (0, T; H_{(\infty)}^{s + 2} \times H_{(\infty)}^{s + 1}), \end{matrix}$ and $\begin{array}{l} {‖ Γ_{\infty} [F] ‖}_{C ([0, T]; H_{(\infty)}^{s + 1} \times H_{(\infty)}^{s}) \cap L^{2} (0, T; H_{(\infty)}^{s + 2} \times H_{(\infty)}^{s + 1})} \\ (4.17) & ⩽ C {‖^{⊤} (ϕ_{0 \infty}, m_{0 \infty}) ‖}_{H_{(\infty)}^{s + 1} \times H_{(\infty)}^{s}} + C ‖ F ‖_{L^{2} (0, T; H_{(\infty)}^{s} \times H_{(\infty)}^{s - 1})}, \end{array}$ where C is a positive constant independent of T. Furthermore, $Γ_{\infty} [F]$ satisfies the estimate ( 4.16 ), i.e., $\begin{array}{rcl} (4.18) & \frac{d}{d t} E [Γ_{\infty} [F]] (t) + d ({‖ \nabla ϕ_{\infty} (t) ‖}_{H^{s + 1}}^{2} + {‖ \nabla m_{\infty} (t) ‖}_{H^{s}}^{2}) ⩽ C {‖ F (t) ‖}_{H^{s} \times H^{s - 1}}^{2}, \end{array}$ where $Γ_{\infty} [F] =^{⊤} (ϕ_{\infty}, m_{\infty})$ .

4.3. Iteration argument to show the existence of global solutions

In this subsection, we show the existence of global solutions for small data by the iteration argument. Recall that $\begin{matrix} Γ [u] = S (t) u_{0} + \int_{0}^{t} S (t, τ) F (u (τ)) d τ, \end{matrix}$ where $u_{0} =^{⊤} (ϕ_{0}, m_{0})$ , $m_{0} = \partial_{x} {\tilde{m}}_{0}$ and $F (u)$ denotes the nonlinearity terms of (1.1). The iteration scheme is given as follows. We define $u^{(k)}$ ( $k = 1, \dots$ ) by $\begin{array}{rcl} (4.19) & u^{(k)} = Γ [u^{(k - 1)}] \end{array}$ and $u^{(0)}$ is given by $u^{(0)} = S (t) u_{0}$ . Applying $P_{j}$ ( $j = 1, \infty$ ) to (4.19) respectively, we obtain that $\begin{array}{rcl} (4.20) & u_{j}^{(k)} = Γ_{j} [u_{1}^{(k - 1)} + u_{\infty}^{(k - 1)}] \end{array}$ where $u_{j}^{(k)} = P_{j} u^{(k)}$ , $u_{0 j} = P_{j} u_{0}$ and $u_{j}^{(0)} = S_{j} (t) u_{0 j}$ ( $j = 1, \infty$ ).

For any $0 < T < + \infty$ we define a time weighted function space $Z_{a}^{s} (0, T)$ by $\begin{array}{rcl} Z_{a}^{s} (0, T) & = & {u =^{⊤} (ϕ, m); u = P_{1} u + P_{\infty} u = u_{1} + u_{\infty}, \\ P_{1} u \in C ([0, T]; L_{(1)}^{2}), P_{\infty} u =^{⊤} (ϕ_{\infty}, m_{\infty}), \\ ϕ_{\infty} \in C ([0, T]; H_{(\infty)}^{s + 1}) \cap L^{2} (0, T; H_{(\infty)}^{s + 2}), m_{\infty} \in C ([0, T]; H_{(\infty)}^{s}) \cap L^{2} (0, T; H_{(\infty)}^{s + 1}), \\ ‖ u_{1}, u_{\infty} ‖_{Z^{s} (0, T)} ⩽ a} \end{array}$ and the norm $‖ u_{1}, u_{\infty} ‖_{Z^{s} (0, T)}$ is defined by $\begin{array}{rcl} ‖ u_{1}, u_{\infty} ‖_{Z^{s} (0, T)} & = & sup_{0 ⩽ t ⩽ T} \sum_{j = 0}^{1} {(1 + t)}^{\frac{n}{4} + \frac{j}{2}} {‖ \nabla^{j} u_{1} ‖}_{L^{2}} \\ + sup_{0 ⩽ t ⩽ T} {(1 + t)}^{\frac{n}{4} + \frac{1}{2}} ‖ u_{\infty} ‖_{H^{s + 1} \times H^{s}} \\ + ‖ \nabla u_{\infty} ‖_{L^{2} (0, T; H_{(\infty)}^{s + 1} \times H_{(\infty)}^{s})} \\ + sup_{0 ⩽ t ⩽ T} {(1 + t)}^{\frac{n}{4} + \frac{1}{2}} {(\int_{0}^{t} e^{- C_{2} (t - τ)} {(1 + τ)}^{- \frac{n}{2} - 1} ‖ \nabla u_{\infty} ‖_{H^{s + 1} \times H^{s}}^{2} d τ)}^{\frac{1}{2}}, \end{array}$ where a and $C_{2}$ are positive constants independent of k and T and are defined below respectively. Note that the space $Z_{a}^{s} (0, T)$ has completeness with the norm $‖ u_{1}, u_{\infty} ‖_{Z^{s} (0, T)}$ . By Proposition 4.1(ii), for $j = 0, 1$ and $k ⩾ 1$ it holds that $\begin{array}{l} (4.21) & {‖ \nabla^{j} u^{(k)} (t) ‖}_{L^{2}} ⩽ {‖ \nabla^{j} S_{1} (t) u_{01} ‖}_{L^{2}} + \int_{0}^{t} {‖ \nabla^{j} S_{1} (t, τ) P_{1} F (u^{(k - 1)}) ‖}_{L^{2}} d τ, \\ (4.22) & {‖ \nabla^{j} S_{1} (t) u_{01} ‖}_{L^{2}} ⩽ C {(1 + t)}^{- \frac{n}{4} - \frac{j}{2}} {‖^{⊤} (ϕ_{01}, {\tilde{m}}_{01}) ‖}_{L^{1}} . \end{array}$ We estimate the second term of right hand side in (4.21). Due to the conservation form $F (u)$ can be represented by the divergence form, that is, $\begin{matrix} F (u) =^{⊤} (0, \sum_{ℓ = 1}^{n} \partial_{x_{ℓ}} f_{ℓ} (u)) \end{matrix}$ where $f_{ℓ} (u)$ is suitable nonlinear terms given from (3.3). (For example, each component of $m \otimes m$ .) Hence we see from Theorem 4.1(ii), the fact $n ⩾ 3$ and direct computation for $L^{1}$ norm of the nonlinearity that for $‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖_{Z^{s} (0, T)} ⩽ 1$ $\begin{array}{l} \int_{0}^{t} {‖ \nabla^{j} S_{1} (t, τ) P_{1} F (u^{(k - 1)}) ‖}_{L^{2}} d τ \\ ⩽ C \int_{0}^{t} {(1 + t - τ)}^{- \frac{n}{4} - \frac{j}{2}} {‖ f_{ℓ} (u^{(k - 1)}) ‖}_{L^{1}} d τ \\ ⩽ C \int_{0}^{t} {(1 + t - τ)}^{- \frac{n}{4} - \frac{j}{2}} {(1 + τ)}^{- \frac{2 n}{4}} d τ {‖ P_{1} u^{(k - 1)}, P_{\infty} u^{(k - 1)} ‖}_{Z^{s} (0, T)}^{2} \\ (4.23) & ⩽ C {(1 + t)}^{- \frac{n}{4} - \frac{j}{2}} {‖ P_{1} u^{(k - 1)}, P_{\infty} u^{(k - 1)} ‖}_{Z^{s} (0, T)}^{2} . \end{array}$ Owing to (4.21), (4.22) and (4.23) we get that for $k ⩾ 1$ , $‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖_{Z^{s} (0, T)} ⩽ 1$ and $0 < T < + \infty$ $\begin{array}{rcl} (4.24) & sup_{0 ⩽ t ⩽ T} \sum_{j = 0}^{1} {(1 + t)}^{\frac{n}{4} + \frac{j}{2}} {‖ \nabla^{j} u_{1}^{(k)} ‖}_{L^{2}} ⩽ C_{0} E_{0} + C_{1} {‖ P_{1} u^{(k - 1)}, P_{\infty} u^{(k - 1)} ‖}_{Z^{s} (0, T)}^{2}, \end{array}$ where constants $C_{0}$ and $C_{1}$ are independent of k and T. Obviously, it holds that $\begin{array}{rcl} sup_{0 ⩽ t ⩽ T} \sum_{j = 0}^{1} {(1 + t)}^{\frac{n}{4} + \frac{j}{2}} {‖ \nabla^{j} u_{1}^{(0)} ‖}_{L^{2}} ⩽ C_{0} E_{0} . \end{array}$

We use the estimate (4.18) for $u_{\infty}$ . Note that the following estimate related to the nonlinearity $P_{\infty} F (u)$ is obtained by direct computations based on Lemmas 2.1–2.5.

Lemma 4.8.

It holds that for $t \in [0, T]$ and $‖ u_{1}, u_{\infty} ‖_{Z^{s} (0, T)} ⩽ 1$ $\begin{array}{l} {‖ P_{\infty} F (u) ‖}_{H^{s} \times H^{s - 1}} \\ ⩽ C {(1 + t)}^{- \frac{n}{2} - 1} ‖ u_{1}, u_{\infty} ‖_{Z^{s} (0, T)}^{2} + C {(1 + t)}^{- \frac{n}{4} - \frac{1}{2}} ‖ u_{1}, u_{\infty} ‖_{Z^{s} (0, T)} ‖ \nabla u_{\infty} ‖_{H^{s + 1} \times H^{s}} . \end{array}$

Proof.

We estimate $P_{\infty} (ϕ \nabla Δ ϕ)$ which is one of the nonlinear terms. For $| α | ⩽ s - 1$ we see from Lemmas 2.1, 2.3, and 2.5 that $\begin{array}{rcl} {‖ \partial_{x}^{α} P_{\infty} (ϕ \nabla Δ ϕ) ‖}_{L^{2}} & ⩽ & {‖ ϕ \partial_{x}^{α} \nabla Δ ϕ ‖}_{L^{2}} + {‖ [\partial_{x}^{α}, ϕ] \nabla Δ ϕ ‖}_{L^{2}} \\ ⩽ & ‖ ϕ ‖_{L^{\infty}} {‖ \partial_{x}^{α} \nabla Δ ϕ ‖}_{L^{2}} + ‖ \nabla ϕ ‖_{H^{s}} ‖ \nabla Δ ϕ ‖_{H^{| α | - 1}} \\ ⩽ & ‖ \nabla ϕ ‖_{H^{s - 1}} ‖ \nabla Δ ϕ ‖_{H^{s - 1}} + ‖ \nabla ϕ ‖_{H^{s}} ‖ \nabla Δ ϕ ‖_{H^{s - 2}} . \end{array}$ Hence, it is derived that $\begin{matrix} {‖ P_{\infty} (ϕ \nabla Δ ϕ) ‖}_{H^{s - 1}} ⩽ C {(1 + t)}^{- \frac{n}{2} - 1} ‖ u_{1}, u_{\infty} ‖_{Z^{s} (0, T)}^{2} + C {(1 + t)}^{- \frac{n}{4} - \frac{1}{2}} ‖ u_{1}, u_{\infty} ‖_{Z^{s} (0, T)} ‖ \nabla u_{\infty} ‖_{H^{s + 1} \times H^{s}} . \end{matrix}$ As another nonlinear term can be estimated similarly, we get Lemma 4.8. This completes the proof. □

Let $D [u_{\infty}^{(k)}] = ‖ \nabla ϕ_{\infty}^{(k)} (t) ‖_{H^{s + 1}}^{2} + ‖ \nabla m_{\infty}^{(k)} (t) ‖_{H^{s}}^{2}$ . By (4.18) and Lemma 4.8, there exists a positive constant $C_{2}$ such that for $t \in [0, T]$ and $‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖_{Z^{s} (0, T)} ⩽ 1$ $\begin{array}{l} E [u_{\infty}^{(k)}] (t) + d \int_{0}^{t} e^{- C_{2} (t - τ)} D [u_{\infty}^{(k)}] (τ) d τ \\ ⩽ e^{- C_{2} t} E [u_{\infty}^{(k)}] (0) \\ + C {‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖}_{Z^{s} (0, T)}^{4} \int_{0}^{t} e^{- C_{2} (t - τ)} {(1 + τ)}^{- n - 2} d τ \\ + C {‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖}_{Z^{s} (0, T)}^{2} \int_{0}^{t} e^{- C_{2} (t - τ)} {(1 + τ)}^{- \frac{n}{2} - 1} D [u_{\infty}^{(k - 1)}] (τ) d τ \\ ⩽ e^{- C_{2} t} E [u_{\infty}^{(k)}] (0) \\ + C {‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖}_{Z^{s} (0, T)}^{4} {(1 + t)}^{- n - 2} \\ (4.25) & + C {‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖}_{Z^{s} (0, T)}^{2} \int_{0}^{t} e^{- C_{2} (t - τ)} {(1 + τ)}^{- \frac{n}{2} - 1} D [u_{\infty}^{(k - 1)}] (τ) d τ . \end{array}$ $D [u_{\infty}^{(k)}]$ and $\tilde{E} [u_{\infty}^{(k)}]$ are defined by $\begin{array}{l} D [u_{\infty}^{(k)}] (t) = {(1 + t)}^{\frac{n}{2} + 1} \int_{0}^{t} e^{- C_{2} (t - τ)} {(1 + τ)}^{- \frac{n}{2} - 1} D [u_{\infty}^{(k)}] (τ) d τ, \\ \tilde{E} [u_{\infty}^{(k)}] (t) = sup_{0 ⩽ τ ⩽ t} {(1 + τ)}^{\frac{n}{2} + 1} E [u_{\infty}^{(k)}] (τ) . \end{array}$ We see from (4.25) that $\begin{array}{l} \tilde{E} [u_{\infty}^{(k)}] (t) + d D [u_{\infty}^{(k)}] (t) \\ ⩽ C (\tilde{E} [u_{\infty}^{(k)}] (0) + {‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖}_{Z^{s} (0, T)}^{4} + C {‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖}_{Z^{s} (0, T)}^{2} D [u_{\infty}^{(k - 1)}] (t)) \\ (4.26) & ⩽ C_{0}^{2} E_{0}^{2} + C_{3}^{2} {‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖}_{Z^{s} (0, T)}^{4}, \end{array}$ where the constant $C_{3}$ is independent of k and T. Furthermore, due to Proposition 4.7 and Lemma 4.8 we derive that $\begin{array}{rcl} (4.27) & {‖ u_{\infty}^{(k)} ‖}_{L^{2} (0, T; H^{s + 2} \times H^{s + 1})} ⩽ C_{0} E_{0} + C_{4} {‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖}_{Z^{s} (0, T)}^{2} \end{array}$ when $‖ u_{1}^{(k - 1)}, u_{\infty}^{(k - 1)} ‖_{Z^{s} (0, T)} ⩽ 1$ , where $C_{4}$ is a positive constant independent of k and T.

Now we are in a position to prove the main result. Let $E_{0} ⩽ min {\frac{1}{2 C_{0}}, \frac{1}{4 C_{0} C_{1}}, \frac{1}{4 C_{0} C_{3}}, \frac{1}{4 C_{0} C_{4}}}$ and $a = min {1, \sqrt{C_{0} E_{0} / C_{1}}, \sqrt{C_{0} E_{0} / C_{3}}, \sqrt{C_{0} E_{0} / C_{4}}}$ . We see from (4.24), (4.26) and (4.27) that there holds that $\begin{matrix} {‖ P_{1} u^{(k)}, P_{\infty} u^{(k)} ‖}_{Z^{s} (0, T)} ⩽ 2 C_{0} E_{0} ⩽ min {1, \sqrt{C_{0} E_{0} / C_{1}}, \sqrt{C_{0} E_{0} / C_{3}}, \sqrt{C_{0} E_{0} / C_{4}}}, \end{matrix}$ for $u^{(k)} \in Z_{a}^{s} (0, T)$ and $k = 0, \dots$ inductively. Furthermore, $\begin{array}{l} {‖ P_{1} (u^{(k + 1)} - u^{(k)}), P_{\infty} (u^{(k + 1)} - u^{(k)}) ‖}_{Z^{s} (0, T)} \\ (4.28) & ⩽ C_{5} E_{0} {‖ u_{1}^{(k)} - u_{1}^{(k - 1)}, u_{\infty}^{(k)} - u_{\infty}^{(k - 1)} ‖}_{Z^{s} (0, T)}, \end{array}$ where $C_{5} = max {2 C_{0} C_{1}, 2 C_{0} C_{3}, 2 C_{0} C_{4}}$ . Therefore, when in addition $E_{0} ⩽ \frac{1}{2 C_{5}}$ it holds that $\begin{array}{l} {‖ P_{1} (u^{(k + 1)} - u^{(k)}), P_{\infty} (u^{(k + 1)} - u^{(k)}) ‖}_{Z_{a}^{s} (0, T)} \\ (4.29) & ⩽ \frac{1}{2} {‖ u_{1}^{(k)} - u_{1}^{(k - 1)}, u_{\infty}^{(k)} - u_{\infty}^{(k - 1)} ‖}_{Z_{a}^{s} (0, T)} . \end{array}$ Since T is any number satisfying $0 < T < + \infty$ and the constants which appear in (4.24), (4.26), (4.27), (4.28) and (4.29) do not depend on T, from the iteration there exists a unique global solution u in the class $\begin{matrix} {u =^{⊤} (ϕ, m); u = P_{1} u + P_{\infty} u \in X^{s} (0, \infty), ‖ P_{1} u, P_{\infty} u ‖_{X^{s} (0, \infty)} ⩽ a} \end{matrix}$ and u satisfies the decay estimate $\begin{matrix} {‖ \nabla^{k} u ‖}_{L^{2}} ⩽ C {(1 + t)}^{- n / 4 - k / 2} (k = 0, 1) . \end{matrix}$ This completes the proof.

Footnotes

Acknowledgements

The first author is partly supported by Grants-in-Aid for Scientific Research with the Grant number: 16H03945. The second author is partly supported by Grant-in-Aid for JSPS Fellows with the Grant number: A17J047780.

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Global existence and time decay estimate of solutions to the compressible Navier–Stokes–Korteweg system under critical condition

Abstract

Keywords

1. Introduction

Lemma 2.1. Let n ⩾ 3 and s be an integer satisfying s ⩾ [ n / 2 ] + 1 . Then it holds true that ‖ f ‖ L ∞ ⩽ C ‖ ∇ f ‖ H s − 1 for f ∈ H s . For estimates of nonlinearity, some estimates are stated below. The latter lemma is the commutator estimate. Lemma 2.2 ([8, Lemma A.2]).

Lemma 2.3 ([7, Lemma 4.3]).

Lemma 2.4 ([15, Lemma 4.2]).

Lemma 2.5 ([15, Lemma 4.2], [9, Lemma 4.4]).

3. Main results

4.1. Estimates of Γ; the low frequency part

Footnotes

Acknowledgements

References