53

数理解析研究所講究録
第2009巻 2016年 53-64

Some global well‐posedness results for thecompressible barotropic viscous fluid flow
Yuko Enomoto

Department of Mathematical Sciences, Shibaura Institute of Technology
e‐yuko@shibaura‐it.ac.jp

1

Introduction

This article is a brief survey of the paper [3], which is ajoint work with Yoshihiro Shibata,

Professor of Waseda University.
We consider the compressible Navier‐Stokes equations:\left{bginary}{l $\rho_{t}+mahr{d}\mathr{i}\mathr{v}($\ho+r$_{*})\mathr{u})=0&\mathr{i}\mathr{n}$\Omegatis(0,T)\ ($rho+\ $_{*})(mathr{u}_+\mathr{u}\cdotnabl\mthr{u})-$\m triangle\mthr{u}-$\n abl\mthr{d}\mathr{i}\mathr{v}\mathr{u}+\nablmthfrak{p}=0&\mathr{i}\mathr{n}$\Omegatis(0,T)\ mathr{u}=0&\mathr{o}\mathr{n}$\Gam ties(0,T)\ ($rho,\matr{u})|_=0($\rho_{0},\mathr{u}_0)&\mathr{i}\mathr{n}$\Omega \nd{ray}\ight.

(1.1)

where $\Omega$ is an exterior domain of the 3‐dimensional Euclidean space \mathb {R}^{3} , that is $\Omega$=\mathbb{R}^{3}\backslash \mathcal{O}with some bounded domain \mathcal{O} in \mathbb{R}^{3} and $\Gamma$ the boundary of $\Omega$ which is assume to be asmooth compact hyper‐surface.

Here, $\rho$_{*} is a positive constant describing the mass densityof the reference body, and $\mu$ and $\nu$ are positive constants describing the first and second

viscosity coefficients, respectively. Moreover, $\rho$= $\rho$(x, t) with x=(x_{1}, x_{2}, x_{3})\in $\Omega$ is
a unknown function, $\rho$_{*}+ $\rho$ being thedensity field, \mathrm{u}=(u_{1}(x, t), u_{2}(x, t), u_{3}(x, t)) the
unknown velocity field, \mathfrak{p} the unknown pressure field, ($\rho$_{0}, \mathrm{u}_{0}) prescribed initial data, and\mathrm{u}|_{ $\Gamma$}=0 is non‐slip boundary condition. We consider only the barotropic case, that is,\mathfrak{p}=P($\rho$_{*}+ $\rho$),

where P(s) is a C^{\infty} function defined for s>0 satisfying the condition:$\rho$_{1}\leq P'( $\rho$+$\rho$_{*})\leq$\rho$_{2}for | $\rho$|\leq$\rho$_{*}/2

with some positive constants $\rho$_{1} and $\rho$_{2}.

The purpose of this article is to prove the global well‐posedness of (1.1).

Moreover, it

is proved the optimal decay property.

When $\Omega$=\mathbb{R}^{3} or $\Omega$ is a 3‐dimensional exterior domain, Matsumura and Nishida [6, 7]
proved the global well‐posedness of problem (1.1) under the assumption that the H^{3} normof initial data $\rho$_{0} and \mathrm{u}_{0} are small enough. And also, Matsumura and Nishida [6, 7] andlater on Deckelnick [1, 2] proved some convergence rate of solutions to the stationary
solutions by using the energy method. The optimal decay properties were obtained by

54[8] when $\Omega$=\mathbb{R}^{N} and by Kobayashi and Shibata [5] when $\Omega$ is a 3‐dimensional
L_{p}-L_{q} decay estimate for the linearized equations.
In 2002, Kawashita [4] proved the global well‐posedness of (1.1) when $\Omega$=\mathbb{R}^{N}(N\geq 2)
under the assumption that the H^{s} norm of initial data with s=[N/2]+1 are small enough,in particular, s=2 when N=2 and 3. Later, Wang and Tan [10] proved the optimal
decay estimate of Kawashitas solution when $\Omega$=\mathbb{R}^{3} with the help of the L_{p}-L_{q} decay
estimate due to Ponce [8]. The purpose of this paper is to prove the same results for the
Ponceexterior domain with the help of theinitial boundary value problem in 3‐dimensional exterior domains as that for the Cauchy

problem obtained by Kawashita [4] and Wang and Tan [10].
The following theorem shows the global well‐posedness of (1.1).
Theorem 1.1. Let $\Omega$ becompact hyper‐surface.
dataa3‐dimensional exterior domain whose

Then,

there existsaboundary $\Gamma$ is a C^{2}

small positive number $\delta$ such that if initial($\rho$_{0}, \mathrm{u}_{0})\in H^{2}( $\Omega$)^{4} satisfy the smallness assumption: \Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2}( $\Omega$)}\leq $\delta$ and thecompatibility condition: \mathrm{u}_{0}|_{ $\Gamma$}=0 then problem (1.1) with T=\infty admits unique solutions,$\rho$ and \mathrm{u} with$\rho$\in C^{0}([(0, \infty), H^{2}( $\Omega$))\cap C^{1}([0, \infty), H^{1}( $\Omega$))$\rho$_{t}\in L_{2}((0, \infty), H^{1}( $\Omega$)) , \nabla $\rho$\in L_{2}((0, \infty), H^{1}( $\Omega$)^{3})\mathrm{u}\in C^{0}([0, \infty), H^{2}( $\Omega$)^{3})\cap C^{1}([0, \infty), L_{2}( $\Omega$)^{3})\mathrm{u}_{t}\in L_{2}((0, \infty), H^{1}( $\Omega$)^{3}) , \nabla \mathrm{u}\in L_{2}((0, \infty), H^{2}( $\Omega$)^{9})
,

,

(1.2)

possessing the estimate:\displaystyle \sup_{0<s<t}\Vert( $\rho$, \mathrm{u} s)\Vert^{2}+\sup_{0<s<t}\Vert(\nabla $\rho,\ \rho$_{s}, \nabla \mathrm{u} s)\Vert^{2}+\sup_{0<s<t}\Vert(\nabla^{2}$\rho$_{)}\nabla$\rho$_{s}, \nabla^{2}\mathrm{u} s)\Vert^{2}+\displaystyle \int_{0}^{t}(\Vert\nabla \mathrm{u}(\cdot, s)\Vert^{2}+\Vert$\rho$_{s}(\cdot, s)\Vert^{2}+\Vert $\rho$(\cdot, s)\Vert_{L_{2}($\Omega$_{R}) ^{2})ds+\displaystyle \int_{0}^{t}(\Vert\nabla^{2}\mathrm{u}(\cdot, s)\Vert^{2}+\Vert \mathrm{u}_{s}(\cdot, s)\Vert^{2}+\Vert\nabla $\rho$(\cdot, s)\Vert^{2}+\Vert\nabla$\rho$_{s}(\cdot, s)\Vert^{2})ds+\displaystyle \int_{0}^{t}(\Vert\nabla^{3}\mathrm{u}(\cdot, s)\Vert^{2}+\Vert\nabla \mathrm{u}_{s}(\cdot, s)\Vert^{2}+\Vert\nabla^{2} $\rho$(\cdot, s)\Vert^{2})ds\leq C\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2}( $\Omega$)}^{2}(1.3)

for any t>0 with some constant C independent of $\delta$.
Moreover, we proved the optimal decay property.
Theorem 1.2. Let $\Omega$ becompact hyper‐surface.

a3‐dimensional exterior domain whose

Then,

there existsaboundary $\Gamma$ is a C^{3}
if ini‐

small positive number $\delta$ such thattial data ($\rho$_{0}, \mathrm{u}_{0})\in L_{1}( $\Omega$)^{4}\cap H^{2}( $\Omega$)^{4}

satisfy the smallness assumption: \Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{L_{1}( $\Omega$)}+\Vert'($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2}( $\Omega$)}\leq $\delta$ as well as the compatibility condition, then problem (1.1) with T=\infty

admits unique solutions $\rho$ and \mathrm{u} satisfying the same regularity conditions (1.2) and (1.3)

55

in Theorem 1.1 and possessing the decay estimate:\displaystyle \sup_{0<s<t}(1+s)^{3/4}\Vert( $\rho$, \mathrm{u} s +\sup_{0<s<t}(1+s)^{5/4}\Vert(\nabla $\rho$, \nabla\mathrm{u}+\displaystyle \sup_{0<s<t}(1+s)^{5/4}\Vert(\nabla^{2} $\rho$, \nabla^{2}\mathrm{u} ss\leq C(\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{L_{1}( $\Omega$)}+\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2}( $\Omega$)})for any t>0 with some constant C independent of $\delta$.

Outline of proof of Theorem 1.1

2

In this section we consider the following equation:\left{bginary}{l $\rho_{t}+\mahr{u}\cdotnabl$\rho+ $_{*}\mathr{d}\mathr{i}\mathr{v}\mathr{u}=f_n&\mathr{i}\mathr{n}$\Omegati s(0,T)\ mathr{u}_-$\mu_{*}triangle\mthr{u}-$\n_{*}abl\mthr{d}\mathr{i}\mathr{v}\mathr{u}+$\gam $_{*}\nabl$rho=\mathr{g}_n&\mathr{i}\mathr{n}$\Omegati s(0,T)\ mathr{u}|_$\Gam $}=0,(\rho$,mathr{u})|_t=0($\rho_{0},\mathr{u}_0)&\mathr{i}\mathr{n}$\Omega,\nd{ary}\ight.

(2.1)

where

$\mu$_{*}= $\mu$/$\rho$_{*}, $\nu$_{*}= $\nu$/$\rho$_{*)} $\gamma$_{*}=P'($\rho$_{*})/$\rho$_{*},
f_{n}= $\rho$ \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u},\displaystyle\mathrm{g}_{n}=(\frac{1}{$\rho$+$\rho$_{*} -\frac{1}{$\rho$_{*} )($\mu$\triangle\mathrm{u}+$\nu$\nabla\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u})-(\frac{P'($\rho$+$\rho$_{*}) {$\rho$+$\rho$_{*} -\frac{P'($\rho$_{*}) {$\rho$_{*} )\nabla$\rho$.
For the sake of simplicity, we use the abbreviation:(q\neq 2) and \Vert\cdot\Vert_{H^{s}( $\Omega$)}=\Vert\cdot\Vert_{H^{s}}(s=1,2).

We set\Vert\cdot\Vert_{L_{2}( $\Omega$)}=\Vert.

\Vert\cdot\Vert_{L_{q}( $\Omega$)}=\Vert\cdot\Vert_{q}\displaystyle \mathrm{I}_{1}(t)=\sup_{0<s<t}\Vert($\rho$_{)}\mathrm{u} s)\Vert^{2}+\int_{0}^{t}(\Vert\nabla \mathrm{u}(_{)}s)\Vert^{2}+\Vert$\rho$_{s}(\cdot, s)\Vert^{2}+\Vert $\rho$(\cdot, s)\Vert_{L_{2}($\Omega$_{R}) ^{2})ds,\displaystyle \mathrm{I}_{2}(t)=\sup_{0<s<t}\Vert(\nabla $\rho,\ \rho$_{s}, \nabla \mathrm{u} s) \Vert^{2}
I3+\displaystyle \int_{0}^{t}(\Vert\nabla^{2}\mathrm{u}(\cdot, s)\Vert^{2}+\Vert \mathrm{u}_{s}(\cdot, s)\Vert^{2}+\Vert\nabla $\rho$(\cdot, s)\Vert^{2}+\Vert\nabla$\rho$_{s}(\cdot, s)\Vert^{2})ds,(t)=\displaystyle \sup_{0<s<t}\Vert(\nabla^{2} $\rho$, \nabla$\rho$_{s}, \nabla^{2}\mathrm{u} s) \Vert^{2}+\displaystyle \int_{0}^{t}(\Vert\nabla^{3}\mathrm{u}(\cdot, s)\Vert^{2}+\Vert\nabla \mathrm{u}_{s}(\cdot, s)\Vert^{2}+\Vert\nabla^{2} $\rho$(\cdot, s)\Vert^{2})ds,

I (t)=\mathrm{I}_{1}(t)+\mathrm{I}_{2}(t)+\mathrm{I}_{3}(t).

To prove Theorem 1.1, it suffices to prove that\mathrm{I}(t)\leqq K_{1}(\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2} ^{2}+\mathrm{I}(t)^{3/2}+\mathrm{I}(t)^{2})
provided that \Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2} \leq 1.

(2.2)

56

Multiplying the first equation in (2.1) by $\rho$_{*}^{-1} $\rho$ and the second one in (2.1) by $\gamma$_{*}^{-1}\mathrm{u},
we

have

\displaystyle \frac{1}{2}\frac{d}{dt}\{$\rho$_{*}^{-1}\Vert $\rho$(\cdot, )\Vert^{2}+$\gam a$_{*}^{-1}\Vert \mathrm{u}(\cdot, )\Vert^{2}\ +$\mu$_{*}\Vert\nabla \mathrm{u}(\cdot, )\Vert^{2}+$\nu$_{*}\Vert \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u}(\cdot, )\Vert^{2}=\displaystyle\frac{1}{$\rho$_{*} (f_{n}, $\rho$)+\frac{1}{$\gam a$_{*} (\mathrm{g}_{n},\mathrm{u})+( \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u})p, $\rho$)

(2.3)

.

In what follows, to estimate the non‐linear term we use the following estimates:\Vert u\Vert_{6}\leq C\Vert\nabla u\Vert,\Vert u\Vert_{L_{2}($\Omega$_{R})}\leq C_{R}\Vert\nabla u\Vert,\Vert u\Vert_{3}\leq C\Vert u\Vert_{H^{1}},\Vert u\Vert_{\infty}\leq C\Vert\nabla u\Vert_{H^{1}

(2.4)

.

Integrating (2.3) and using (2.4), we have\displaystyle \Vert( $\rho$, \mathrm{u} t)\Vert^{2}+\int_{0}^{t}\Vert\nabla \mathrm{u}(\cdot, s)\Vert^{2}ds\leq C(\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert^{2}+\mathrm{I}(t)^{3/2})

.

(2.5)

By the first equation in (2.1) and (2.4), we have

\Vert$\rho$_{t}\Vert\leq C(1+\Vert\nabla $\rho$\Vert_{H^{1} )\Vert\nabla \mathrm{u}\Vert
so

(2.6)

,

that

\displaystyle \int_{0}^{t}\Vert$\rho$_{8}(\cdot, s)\Vert^{2}ds\leq C(\int_{0}^{t}\Vert\nabla \mathrm{u}(\cdot, s)\Vert^{2}ds+\mathrm{I}(t)^{2})

.

(2.7)

To estimate \Vert $\rho$\Vert_{L_{2}($\Omega$_{R})} , we use the following lemma:
Lemma 2.1. Let $\rho$ and \mathrm{u} be solutions to problem (1.1) with$\rho$\in C^{1}([0, T), H^{1}( $\Omega$))\cap C^{0}([0, T), H^{2}( $\Omega$))\mathrm{u}\in C^{1}([0, T), L_{2}( $\Omega$)^{N})\cap C^{0}([0, T), H^{2}( $\Omega$)^{N}),.

Then, we have\displaystyle\int_{0}^{t}\Vert$\rho$(\cdot,s)\Vert_{L_{2}($\Omega$_{R}) ^{2}d_{S}\leqC\int_{0}^{t}\Vert\nabla$\rho$\Vert\{(1+\Vert\nabla$\rho$\Vert_{H^{1} )\Vert\mathrm{V}\mathrm{u}\Vert(2.8)+\Vert \mathrm{u}_{s}\Vert+\Vert\nabla \mathrm{u}\Vert\Vert\nabla \mathrm{u}\Vert_{H^{1} +\Vert\nabla \mathrm{u}\Vert^{1/2}\Vert\nabla^{2}\mathrm{u}\Vert^{1/2}\}ds.

By (2.8), we see that for any $\epsilon$>0 there exists a constant C_{ $\epsilon$}>0 depending on $\epsilon$ suchthat\displaystyle \int_{0}^{t}\Vert $\rho$(\cdot, s)\Vert_{L_{2}($\Omega$_{R}) ^{2}d_{S}\leq $\epsilon$\int_{0}^{t}(\Vert\nabla $\rho$(\cdot, s)\Vert^{2}+\Vert\nabla^{2}\mathrm{u}(\cdot, s)\Vert^{2})ds+C_{ $\epsilon$}\displaystyle \int_{0}^{t}(\Vert\nabla \mathrm{u}(\cdot, s)\Vert^{2}+\Vert \mathrm{u}_{s}(\cdot, s)\Vert^{2})ds+C_{$\epsilon$}(\displaystyle\sup_{0,s<t}\Vert\nabla\mathrm{u}(\cdot,s)\Vert)^{2}\int_{0}^{t}\Vert\nabla\mathrm{u}(\cdot,s)\Vert^{2}ds.

(2.9)

57

Next, we estimate \Vert \mathrm{u}_{t}(\cdot, t)\Vert^{2} Multiplying the second equation in (2.1) by \mathrm{u}_{t}.

\displaystyle \Vert \mathrm{u}_{t}(_{)}t)\Vert^{2}+\frac{1}{2}\frac{d}{dt}\{$\mu$_{*}\Vert\nabla \mathrm{u}(_{)}t)\Vert^{2}+$\nu$_{*}\Vert \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u}(\cdot, ) \Vert^{2}\}+$\gam a$_{*}(\nabla$\rho$_{)}\mathrm{u}_{t})=(\mathrm{g}_{n}, \mathrm{u}_{t})

we

,

have

(2.10).

By the first equation in (2.1), we have(\displaystyle\nabla$\rho$,\mathrm{u}_{t})=-\frac{d}{dt}($\rho$,\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u})+ ($\rho$_{t} , div u),which, combined with (2.10), furnishes that\displaystyle\frac{1}{2}\frac{d}{dt}\{$\mu$_{*}\Vert\nabla\mathrm{u}(\cdot, )\Vert^{2}+$\nu$_{*}\Vert\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u}(\cdot, )\Vert^{2}-2($\rho$,\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u})\}+\frac{1}{2}\Vert\mathrm{u}_{t}(\cdot, )\Vert^{2}(2.11)<(\Vert \mathrm{g}_{n}(\cdot, t)\Vert^{2}+\Vert$\rho$_{t}(\cdot, t)\Vert^{2}+\Vert\nabla \mathrm{u}(\cdot, t)\Vert^{2})\underline{1}.
-2

By (2.4) and (2.6), we have\displaystyle\frac{1}{2}\frac{d}{dt}\{$\mu$_{*}\Vert\nabla\mathrm{u}(.`t)\Vert^{2}+$\nu$_{*}\Vert\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u}t)\Vert^{2}-2($\rho$,\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u})\}+\frac{1}{2}\Vert\mathrm{u}_{t}(.`t)\Vert^{2}(2.12)\leq C\{\Vert\nabla $\rho$\Vert_{H^{1} ^{2}(\Vert\nabla \mathrm{u}\Vert_{H^{1} ^{2}+\Vert\nabla $\rho$\Vert^{2})+\Vert\nabla \mathrm{u}\Vert^{2}\}.

Integrating (2.12), we have\displaystyle \Vert\nabla \mathrm{u}(\cdot, )\Vert^{2}+\int_{0}^{t}\Vert \mathrm{u}_{s}(\cdot, s)\Vert^{2}ds\leq C\{\Vert\nabla \mathrm{u}_{0}\Vert^{2}+\Vert$\rho$_{0}\Vert^{2}+\Vert $\rho$(\cdot, )\Vert^{2}+\int_{0}^{t}\Vert\nabla \mathrm{u}(\cdot, s)\Vert^{2}ds+ (\displaystyle \sup_{0<s<t}\Vert\nabla $\rho$(., s) \Vert_{H^{1} )^{2}\int_{0}^{t}(\Vert\nabla \mathrm{u}(., s) \Vert_{H^{1} ^{2}+\Vert\nabla $\rho$(\cdot, s)\Vert^{2})ds\} (2.13).

By (2.7), (2.9) and (2.13), we have\displaystyle \int_{0}^{t}(\Vert$\rho$_{s}(\cdot, s)\Vert^{2}+\Vert $\rho$(\cdot, s)\Vert_{L_{2}($\Omega$_{R}) ^{2})ds\leq $\epsilon$\int_{0}^{t}(\Vert\nabla^{2}\mathrm{u}(\cdot, s)\Vert^{2}+\Vert\nabla $\rho$(\cdot, s)\Vert^{2})ds+C_{ $\epsilon$}(\displaystyle \Vert(\nabla \mathrm{u}_{0}, $\rho$_{0})\Vert^{2}+\Vert $\rho$(\cdot, )\Vert^{2}+\int_{0}^{t}\Vert\nabla \mathrm{u}(\cdot, s)\Vert^{2}ds+\mathrm{I}(t)^{2}),

which, combined with (2.5), furnishes that\mathrm{I}_{1}(t)\leq C_{ $\epsilon$}(\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{1} ^{2}+\mathrm{I}(t)^{3/2}+\mathrm{I}(t)^{2})+ $\epsilon$ \mathrm{I}_{2}(t)We estimate $\rho$_{t}, \mathrm{u}_{t} and \nabla \mathrm{u}_{t}.

.

(2.14)

Differentiating the equations in (2.1) once with respect

to t , we have\left{bginary}{l \partil_{}($\rho_{t})+\mahr{u}\cdotnabl($\rho_{t})+$\rho_{*}mathr{d}\mathr{i}\mathr{v}\mathr{u}_=\partil_{}fn-\mathr{u}_\cdotnabl$\rho&mathr{i}\mathr{n}$\Omegati s(0,T)\ partil_{}(\mathr{u}_)-$\mu_{*}triangle\mthr{u}_-$\n_{*}abl\mthr{d}\mathr{i}\mathr{v}\mathr{u}_+$\gam _{*}\nabl$rho_{t}=\paril_{t}\mahr{g}_n&\mathr{i}\mathr{n}$\Omegati s(0,T)\ mathr{u}_|$\Gam $}=0,(\rho$_{t},\mahr{u}_t)|=0}($\rho_{1},\mathr{u}_1)&\mathr{i}\mathr{n}$\Omega,\nd{ary}\ight.

58

where we have set$\rho$_{1}=-\mathrm{d}\mathrm{i}\mathrm{v}( $\rho$_{*}+$\rho$_{0})\mathrm{u}_{0}),\displaystyle\mathrm{u}_{1}=-\mathrm{u}_{0}\cdot\nabla\mathrm{u}_{0}+\frac{1}{$\rho$_{*}+$\rho$_{0} ($\mu$\triangle\mathrm{u}_{0}+$\nu$\nabla\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u}_{0}-P'($\rho$_{*}+$\rho$)\nabla$\rho$).

Analogously to (2.3), we have\displaystyle\frac{1}{2}\frac{d}{dt}\{$\rho$_{*}^{-1}\Vert$\rho$_{t}(\cdot, )\Vert^{2}+$\gam a$_{*}^{-1}\Vert\mathrm{u}_{t}(\cdot, )\Vert^{2}\ +$\mu$_{*}\Vert\nabla\mathrm{u}_{t}(\cdot, )\Vert^{2}+$\nu$_{*}\Vert\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u}_{t} t) \Vert^{2}=\displaystyle\frac{1}{$\rho$_{*} (\partial_{t}f_{n}-\mathrm{u}_{t}\cdot\nabla$\rho,\ rho$_{t})+\frac{1}{$\gam a$_{*} (\partial_{t}\mathrm{g}_{n},\mathrm{u}_{t})+( \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u})$\rho$_{t},$\rho$_{t})(2.15).

By (2.4),\displaystyle \frac{1}{2}\frac{d}{dt}\{$\rho$_{*}^{-1}\Vert$\rho$_{t}(\cdot, )\Vert^{2}+$\gam a$_{*}^{-1}\Vert \mathrm{u}_{t}(_{)}t)\Vert^{2}\}+$\mu$_{*}|\nabla \mathrm{u}_{t} ( t) \Vert^{2}+$\nu$_{*}\Vert\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u}_{t} ( t) \Vert^{2}
\cdot

\cdot

,

,

\leq C(\Vert\nabla \mathrm{u}_{t}\Vert^{2}\Vert\nabla $\rho$\Vert_{H^{1} +\Vert\nabla \mathrm{u}_{t}\Vert\Vert\nabla $\rho$\Vert\Vert$\rho$_{t}\Vert_{H^{1}+\Vert \mathrm{u}_{t}\Vert\Vert\nabla $\rho$\Vert\Vert\nabla$\rho$_{t}\Vert+\Vert\nabla \mathrm{u}\Vert\Vert$\rho$_{t}\Vert_{H^{1} ^{2}).

(2.16)

Integrating (2.16), we have

\Vert($\rho$_{t}, \mathrm{u}_{t} t) \displaystyle \Vert^{2}+\int_{0}^{t})\displaystyle \leq C\{\Vert($\rho$_{1} , \mathrm{u}_{1})\Vert^{2}+(\sup_{0<s<t}\Vert\nabla $\rho$(\cdot, s)\Vert_{H^{1} )\int_{0}^{t}(\Vert \mathrm{u}_{s}(\cdot, s)\Vert_{H^{1} ^{2}+\Vert\nabla$\rho$_{s}(_{)}s)\Vert^{2})ds+(\displaystyle \sup_{0<s<t}\Vert\nabla \mathrm{u}(\cdot, s \int_{0}^{t}\Vert$\rho$_{s}(\cdot, s)\Vert_{H^{1} ^{2}ds\}.

(2.17)

To prove the estimate of the higher order derivatives, we localize the problem in the\cdotwhole space and near the boundary.

Then, we consider the whole space problem:\left{\begin{ar y}{l $\rho$_{t}+\mathrm{u}\cdotnabl$\rho$+\rho$_{*}\mathrm{d}\mathrm{i}\ athrm{v}\ athrm{v}=f&\mathrm{i}\ athrm{n}\mathb{R}^3,\ mathrm{v}_t-$\mu_{*}\triangle\mathrm{v}-$\nu_{*}\nabl\mathrm{d}\mathrm{i}\ athrm{v}\ athrm{v}+$\gam $_{*}\nabl$\rho$=\mathrm{g}&\mathrm{i}\ athrm{n}\mathb{R}^3 \end{ar y}\right.

(2.18)and the half space problem:\left{bginary}{l $\rho_{t}+\mahr{u}\cdotnabl$\rho+ $_{*}\mathr{d}\mathr{i}\mathr{v}\mathr{v}=f&\mathr{i}\mathr{n}\mathb{R}_+^3,\ mathr{v}_t-$\mu_{*}triangle\mathr{v}-$\nu_{*}abl\mthr{d}\mathr{i}\mathr{v}\mathr{v}+$\gam $_{*}\nabl$rho=\mathr{g}&\mathr{i}\mathr{n}\mathb{R}_+^3,\ mathr{v}|_x3=0} & \end{ary}\ight.

(2.19)

where \mathbb{R}_{+}^{3}=\{x=(x_{1}, x_{2}, x_{3})\in \mathbb{R}^{3}|x_{3}>0\}.

In the half space problem, the points are estimations of \partial_{3} $\rho$ and \partial_{3}^{2} $\rho$use

the idea due to Matsumura‐Nishida

[7].

.

To do this, we

From the third component in the sedondequation in (2. 19) we have\displaystyle\partial_{3}^{2}v_{3}=\frac{1}{$\mu$_{*}+$\nu$_{*}\{(v_{3})_{t}-$\mu$_{*}\sum_{i=1}^{2}\partial_{i}^{2}v_{3}-$\nu$_{*}\sum_{i=1}^{2}\partial_{i}\partial_{3}v_{i}-g_{3}\+\frac{$\gam a$_{*}{$\mu$_{*}+$\nu$_{*}\partial_{3}$\rho$.

(2.20)

59

Differentiating the first equation in (2.19) with respact to X3 and inserting the formula(2.20), we have\partial_{3}($\rho$_{t}+\mathrm{u}\cdot\nabla $\rho$)+$\delta$_{*}\partial_{3} $\rho$=\displaystyle\partial_{3}f-\rac{$\rho$_{*}$\nu$_{*} $\mu$_{*}+$\nu$_{*}\sum_{i=1}^{2}\partial_{i}\partial_{3}v_{i}+\frac{$\rho$_{*}$\mu$_{*} $\mu$_{*}+$\nu$_{*}\sum_{i=1}^{2}\partial_{i}^{2}v_{3}-\frac{$\rho$_{*} $\mu$_{*}+$\nu$_{*}(v_{3})_{t}+\frac{$\rho$_{*} $\mu$_{*}+$\nu$_{*}g_{3}where $\delta$_{*}=$\rho$_{*}$\gamma$_{*}/($\mu$_{*}+$\nu$_{*}).

(2.21)

Multiplying (2.21) by \partial_{3} $\rho$ we have,\displaystyle\frac{1}{2}\frac{d}{dt}\Vert\partial_{3}$\rho$(\cdot, )\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2^{*}+\frac{$\delta$}{2}\Vert\partial_{3}$\rho$(\cdot )\Vert_{L_{2}(\mathrm{R}_{+}^{3})^{2}\leq C(\Vert\nabla f\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2}+\Vert \mathrm{g}\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2}+\Vert\nabla\nabla'\mathrm{v}\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2}+\Vert \mathrm{v}_{t}\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2}+\Vert\nabla \mathrm{u}\cdot\nabla $\rho$\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2})

where \nabla\nabla'w=(\partial_{i}\partial_{j}w|i=1,2,3, j=1,2)
we.

Differentiating (2.21) with respacet to X3,

have\partial_{3}^{2}($\rho$_{t}+\mathrm{u}\cdot\nabla $\rho$)+$\delta$_{*}\partial_{3}^{2} $\rho$=\displayst le\partial_{3}^{2}f-\partial_{3}(\frac{$\rho$_{*}$\nu$_{*} $\mu$_{*}+$\nu$_{*}\sum_{i=1}^{2}\partial_{i}\partial_{3}v_{i}+\frac{$\rho$_{*}$\mu$_{*} $\mu$_{*}+$\nu$_{*}\sum_{i=1}^{2}\partial_{i}^{2}v_{3}-\frac{$\rho$_{*} $\mu$_{*}+$\nu$_{*}(v_{3})_{t}+\frac{$\rho$_{*} $\mu$_{*}+$\nu$_{*}g_{3})(2.22)and therefore,\displaystyle\frac{1}{2}\frac{d}{dt}\Vert\partial_{3}^{2}$\rho$(\cdot, )\Vert_{L_{2}(\mathrm{R}_{+}^{3})^{2}+$\delta$_{*}\Vert\partial_{3}^{2}$\rho$(\cdot, )\Vert_{L_{2}(\mathrm{R}_{+}^{3})^{2}\leq C(\Vert(\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u})\cdot\partial_{3}^{2} $\rho$\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2}+\Vert\nabla \mathrm{u}\cdot\nabla^{2} $\rho$\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2}+\Vert\nabla^{2}\mathrm{u}\cdot\nabla $\rho$\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2}+\Vert\nabla^{2}f\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2}+\Vert\nabla^{2}\nabla'\mathrm{v}\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2}+\Vert\nabla \mathrm{v}_{t}\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2}+\Vert\nabla \mathrm{g}\Vert_{L_{2}(\mathrm{R}_{+}^{3}) ^{2})

where \nabla^{2}\nabla'\mathrm{v}=(\nabla^{2}\partial_{1}\mathrm{v}, \nabla^{2}\partial_{2}\mathrm{v}).

Using Sobolevs embedding theorem and (2.4), we have\displaystyle\int_{0}^{t}\Vert\nabla^{2}$\rho$(_{)}s \nabla\mathrm{u} ( ) \displaystyle\Vert^{2}ds\leq\int_{0}^{t}\Vert\nabla^{2}$\rho$ ( ) \Vert^{2}\Vert\nabla\mathrm{u} ( ) \Vert_{\infty}^{2}ds\displaystyle\leqC\int_{0}^{t}\Vert\nabla^{2}$\rho$(\cdot,s)\Vert^{2}\Vert\nabla\mathrm{u}(\cdot,s)\Vert_{H^{2} ^{2}ds\displaystyle \leq C\sup_{0<s<t}\Vert\nabla^{2} $\rho$(\cdot, s)\Vert^{2}\int_{0}^{t}\Vert\nabla \mathrm{u}(\cdot, s)\Vert_{H^{2} ^{2}ds.
\cdot

,

s

\cdot

,

s

\cdot

,

s

In this way, we can enclose the estimation as the gradient \mathrm{u} belongs to L_{2}((0, \infty), H^{2}( $\Omega$))and $\rho$ belongs to

L_{\infty}((0, \infty), H^{2}( $\Omega$)).

Applying the same argument as in the above and using the cut‐off technique, we have\displaystyle \Vert(\nabla $\rho$, \nabla \mathrm{u} t)\Vert^{2}+\int_{0}^{t}\Vert(\nabla^{2}\mathrm{u}, \mathrm{u}_{s}, \nabla $\rho$,\nabla$\rho$_{s})(\cdot, s)\Vert^{2}ds\leq C\{\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{1} ^{2}+\mathrm{I}_{1}(t)+\mathrm{I}(t)^{2}\},

60

which, combined with (2.17), furnishes that\mathrm{I}_{2}(t)\leq C\{\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{1} ^{2}+\mathrm{I}_{1}(t)+\mathrm{I}(t)^{2}\}(2.23).

Therefore, choosing $\epsilon$>0 small enough in (2.14), by (2.23) we have\mathrm{I}_{1}(t)+\mathrm{I}_{2}(t)\leq C\{\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{1} ^{2}+\mathrm{I}(t)^{3/2}+\mathrm{I}(t)^{2}\}.

(2.24)

Analogously to (2.24), we have\mathrm{I}_{3}(t)\leq C\{\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2} ^{2}+\mathrm{I}_{1}(t)+\mathrm{I}_{2}(t)+\mathrm{I}(t)^{2}\}provided that \Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2} \leq 1 Combining (2.24) and (2.25), we have (2.2).
pletes the proof of Theorem 1.1.
.

(2.25)
This com‐

Outline of proof of Theorem 1.2

3

We set\displaystyle \mathrm{D}_{0}(t)=\sup_{0<s<t}(1+s)^{3/4}\Vert( $\rho$, \mathrm{u} s\displaystyle \mathrm{D}_{1}(t)=\sup_{0<s<t}(1+s)^{5/4}\Vert(\nabla $\rho$, \nabla \mathrm{u} s\displaystyle \mathrm{D}_{2}(t)=\sup_{0<s<t}(1+s)^{5/4}\Vert(\nabla^{2} $\rho$, \nabla^{2}\mathrm{u} s\mathrm{D}(t)=\mathrm{D}_{0}(t)+\mathrm{D}_{1}(t)+\mathrm{D}_{2}(t).

To prove Theorem 1.2, it suffices to prove that\mathrm{D}(t)\leq K_{2}(\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{1}+\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2} +\mathrm{D}(t)^{2})(3.1)

with some constant K_{2} with the help of L_{p}-L_{q} decay estimate for the linearized problem.

First of all, we introduce L_{p}-L_{q} decay estimate. To do this, we consider the following

linearized problem:\left{bginary}{l $\rho_{t}+$\gam $\athrm{d}\athrm{i}\athrm{v}\athrm{v}=0&\mathr{i}\mathr{n}$\Omega$\times(0,\infty)\ mathr{v}_t-$\alph triangle\mathr{v}-$\beta nbl\mathr{d}\mathr{i}\mathr{v}\mathr{v}+$\gam $\nabl$\rho=0&\mathr{i}\mathr{n}$\Omega$\times(0,\infty)\ mathr{v}|_$\Gam $}=0,(\rho$,mathr{v})|_t=0($\rho_{0},\mathr{v}_0)\mathr{i}\mathr{n}$\Omega$,& \end{ary}\ight.
where $\alpha$,$\beta$ and $\gamma$ are positive constants.

Let

m

(3.2)

be a non‐negative integer. We assume

that $\Gamma$ is a compact C^{m+1,1} hyper‐surface. Let A be an operator defined by the formula:A( $\rho$, \mathrm{v})=( $\gamma$ \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{v}, - $\alpha$\triangle \mathrm{v}- $\beta$\nabla \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{v}+ $\gamma$\nabla $\rho$)
for any element ( $\rho$, \mathrm{v}) of the domain

W_{p,0}^{m+1,m+2}( $\Omega$)=\{( $\rho$, \mathrm{v})\in W_{p}^{m+1}( $\Omega$)\times W_{p}^{m+2}( $\Omega$)|\mathrm{v}|_{ $\Gamma$}=

By Shibata and Tanaka [9] wé know the generation of C_{0} semigroup \{T(t)\}_{t\geq 0} on
W_{p}^{m+1,m}( $\Omega$)=W_{p}^{m+1}( $\Omega$)\times W_{p}^{m}( $\Omega$) which is analytic. For the solution $\rho$ and \mathrm{v} of theequation (3.2), the following L_{p}-L_{q} decay estimate holds.

0\}

.

,

61

Theorem 3.1. Let $\Omega$ bea3‐dimensional exterior domain whoseboundary $\Gamma$ is a C^{3}

compact hyper‐surface. Let p and q be indices such that 1\leq q\leq 2\leq p<\infty.

Let[(f, \mathrm{g})]_{p,q}=\Vert(f, \mathrm{g})\Vert_{L_{\mathrm{q} ( $\Omega$)}+\Vert(f, \mathrm{g})\Vert_{W_{p}^{1,0}( $\Omega$)}.
Then, for any (f, \mathrm{g})\in W_{p}^{1,0}( $\Omega$)\cap L_{q}( $\Omega$)^{4} and t\geq 1 we have

\Vert T(t)(f, \mathrm{g})\Vert_{L_{p}( $\Omega$)}\leq Ct^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{p})}[(f, \mathrm{g})]_{p,q},\Vert\nabl T(t)f,\mathrm{g})\Vert_{L p}($\Omega$)}\leqC\left\{ begin{ar y}{l t^{-\frac{3}2(\frac{1}\mathrm{q}-\frac{1}p)-\frac{1}2}[(f,\mathrm{g})]_{p,q}&p\leq3,\ t^{-\frac{3}2\mathrm{q} [(f,\mathrm{g})]_{p,q}&p\geq3, \end{ar y}\right.

\Vert\nabla^{2}P_{\mathrm{v} T(t)(f, \mathrm{g})\Vert_{L_{p}( $\Omega$)}\leq Ct^{-\frac{3}{2\mathrm{q} }[(f, \mathrm{g})]_{p,q}where P_{\mathrm{v} is the projection acting on ( $\rho$, \mathrm{v})defined by P_{\mathrm{v} ( $\rho$, \mathrm{v})=\mathrm{v}.

We go back to the proof of the inequality (3.1).

First, we estimate \mathrm{D}_{0}(t) and \mathrm{D}_{1}(t)with the help of L_{p}-L_{q} decay estimate for the linearized problem.

Let\{T(t)\}_{t\geq 0} be theanalytic semigroup associated with the linearized problem:\left{bginary}{l $\rho_{t}+$\rho_{*}mathr{d}\mathr{i}\mathr{v}\mathr{u}=f_n-\mathr{u}\cdotnabl$\rho&mathr{i}\mathr{n}$\Omegati s(0,T)\ mathr{u}_-$\mu_{*}triangle\mthr{u}-$\n_{*}mathr{V}\mathr{d}\mathr{i}\mathr{v}\mathr{u}+$\gam _{*}\nabl$rho=\mathr{g}_n&\mathr{i}\mathr{n}$\Omegati s(0,T)\ mathr{u}|_$\Gam $}=0,(\rho$mathr{u})|_t=0($\rho_{0},\mathr{u}_0)&\mathr{i}\mathr{n}$\Omega.\nd{ary}\ight.

(3.3)

Then, we have ( $\rho$, \mathrm{u} t)=T(t)($\rho$_{0}, \mathrm{u}_{0})+\mathrm{U}(t) with\displaystyle \mathrm{U}(t)=\int_{0}^{t}T(t-s)( f_{n}-\mathrm{u}\cdot\nabla $\rho$ s), \mathrm{g}_{n} s) ds.
Here, we write H^{1,0}=W_{2}^{1,0}( $\Omega$) and \Vert\cdot\Vert_{H^{S} =\Vert\cdot\Vert_{H^{s}( $\Omega$)} for s=1 2. By Theorem 3.1, we,

have\Vert T(t)($\rho$_{0}, \mathrm{u}_{0} \leq C(1+t)^{-\frac{3}{4} (\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{1}+\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{1,0} )\Vert\nabla T(t)($\rho$_{0}, \mathrm{u}_{0} \leq C(1+t)^{-\frac{5}{4} (\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{1}+\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{1} )for any t>0

.

,

(3.4)

To estimate \mathrm{U}(t) , we observe that

\Vert(f_{n}-\mathrm{u}\cdot\nabla $\rho$ s) \Vert_{1}+\Vert(f_{n}-\mathrm{u}\cdot\nabla $\rho$ s) \Vert_{H^{1}}\leq C(1+s)^{-2}\mathrm{D}(s)^{2},\Vert \mathrm{g}_{n}(s)\Vert_{1}+\Vert \mathrm{g}_{n}(s)\Vert\leq C(1+s)^{-2}\mathrm{D}(s)^{2}.
Thus, applying the L_{p}-L_{q} decay estimate and the usual analytic semi‐group estimate, we

have\displaystyle \Vert \mathrm{U}(t)\Vert\leq C\{\int_{0}^{t-1}(t-s)^{-\frac{3}{4} (1+s)^{-2}ds+\int_{t-1}^{t}(1+s)^{-2}ds\}\mathrm{D}(t)^{2}\leq Ct^{-\frac{3}{4} \mathrm{D}(t)^{2},\displaystyle \Vert\nabla \mathrm{U}(t)\Vert\leq C\{\int_{0}^{t-1}(t-s)^{-\frac{5}{4} (1+s)^{-2}ds+\int_{t-1}^{t}(t-s)^{-\frac{1}{2} (1+s)^{-2}ds\}\mathrm{D}(t)^{2}\leq Ct^{-\frac{5}{4} \mathrm{D}(t)^{2}

(3.5)

62

for t\geq 1.

Since \Vert( $\rho$, \mathrm{u} t) \Vert_{H^{2} \leq C\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2} as follows from (1.3), by (3.4) and (3.5)\mathrm{D}_{0}(t)+\mathrm{D}_{1}(t)\leq C\{\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{1}+\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2} +\mathrm{D}(t)^{2}\}Next, we estimate \Vert($\rho$_{t}, \mathrm{u}_{t}

.

(3.6)

For this purpose, we usetLemma 3.2. Let f(t) be a non‐negative C^{1}([0, \infty)) function and let g_{i}(t)(i=1 , 2, 3, 4 )benon‐negative functions such that g_{i}\in C^{0}((0, \infty))(i=1,2,3) and g_{4}\in L_{2}((0, \infty

Assume that\displaystyle \frac{d}{dt}f(t)+cf(t)\leq g_{1}(t)+g_{2}(t)+g_{3}(t)g_{4}(t)for any t>T_{0} with some constant c>0.

Then, for any $\alpha$>0 there exists a T_{1}\geq T_{0} suchthat(1+t)^{ $\alpha$}f(t)\displaystyle \leq(1+T_{1})^{ $\alpha$}f(T_{1})+(2/c)(\sup_{T_{1}<s<t}(1+s)^{ $\alpha$}g_{1}(s)+\displaystyle \int_{T_{1} ^{t}(1+s)^{ $\alpha$}g_{2}(s)ds+\sqrt{1/c}(\sup_{T_{1}<s<t}(1+s)^{ $\alpha$}g_{3}(s) (\int_{T_{1} ^{t}g_{4}(s)^{2}ds)^{1/2}
In view of Theorem 1.1, we may assume that\displaystyle \int_{0}^{t}(\Vert\nabla \mathrm{u}(\cdot, s)\Vert_{H^{2} ^{2}+\Vert\nabla $\rho$(\cdot, s)\Vert_{H^{1} ^{2}+\Vert$\rho$_{s}(\cdot, s)\Vert_{H^{1} ^{2}+\Vert \mathrm{u}_{s}(\cdot, s)\Vert_{H^{1} ^{2})ds+\Vert$\rho$_{t}(_{)}t)\Vert_{H^{1} ^{2}+\Vert( $\rho$, \mathrm{u} t)\Vert_{H^{2} ^{2}\leq C\Vert($\rho$_{0)}\mathrm{u}_{0})\Vert_{H^{2} ^{2}\leq $\epsilon$ (3.7)

for any t>0 with some small $\epsilon$>0 which is decided later. We choose $\epsilon$>0 small enougheventually, so that we may assume that 0< $\epsilon$<1 By (2.6) and (3.7), we have.

\Vert$\rho$_{t}(

.

,

t

\leq C\Vert\nabla \mathrm{u}(

.

,

(3.8)

t

Let $\kappa$ be a small positive number > $\epsilon$ determined later.

By (2.11) and (2.15),\displaystyle \frac{d}{dt}\{$\mu$_{*}\Vert\nabla \mathrm{u}(\cdot, )\Vert^{2}+$\nu$_{*}\Vert \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u}(\cdot, )\Vert^{2}+$\rho$_{*}^{-1}\Vert$\rho$_{t}(\cdot, )\Vert^{2}+$\gam a$_{*}^{-1}\Vert \mathrm{u}_{t}(\cdot, )\Vert^{2}\}+ $\kappa$\{$\mu$_{*}\Vert\nabla \mathrm{u}(\cdot, t)| ^{2}+$\nu$_{*}\Vert \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u} t)\Vert^{2}+$\rho$_{*}^{-1}\Vert$\rho$_{t}(\cdot, t)\Vert^{2}+$\gamma$_{*}^{-1}\Vert \mathrm{u}_{t}(\cdot, t)\Vert^{2}\}\displaystyle \leq C $\kap a$(\Vert\nabla \mathrm{u}(\cdot, t)\Vert^{2}+\Vert$\rho$_{t}(\cdot, t)\Vert^{2}+\Vert \mathrm{u}_{t}(\cdot, t)\Vert^{2})-\min(\frac{1}{2}, $\mu$_{*})\Vert \mathrm{u}_{t}(\cdot, t)\Vert_{H^{1} ^{2}

+C(\Vert \mathrm{g}_{n}(\cdot, t)\Vert^{2}+\Vert$\rho$_{t}(\cdot, t)\Vert^{2}+\Vert\nabla \mathrm{u}(\cdot, t)| ^{2}+|(\partial_{t}f_{n}, $\rho$_{t})|(3.9)+|(\mathrm{u}_{t}\cdot\nabla $\rho,\ \rho$_{t})|+|(\partial_{t}\mathrm{g}_{n}, \mathrm{u}_{t})|+|( \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u})$\rho$_{t}, $\rho$_{t}Combining (3.7), (3.8), (3.9)and(2.4), choosing$\kappa$>0 in sucha

way that CK\displaystyle \min(1/2, $\mu$_{*}) with some constant independent of $\kappa$ , and setting

f(t)=\{$\mu$_{*}\Vert\nabla \mathrm{u}(\cdot, t)\Vert^{2}+$\nu$_{*}\Vert \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{u} t)\Vert^{2}+$\rho$_{*}^{-1}\Vert$\rho$_{t}(., t)\Vert^{2}+$\gamma$_{*}^{-1}\Vert \mathrm{u}_{t}(_{)}t)\Vert^{2}\},g_{1}(t)=C_{ $\kappa$}(\Vert\nabla \mathrm{u}(_{)}t)\Vert^{2}+\Vert\nabla $\rho$(\cdot, t)\Vert^{2})g_{2}(t)=C_{ $\kappa$}\{(| \nabla \mathrm{u}(\cdot, t)\Vert_{H^{2} ^{2}+\Vert\nabla $\rho$(\cdot, t)\Vert_{H^{1} ^{2})\Vert\nabla $\rho$(\cdot, t)\Vert^{2}+\Vert\nabla \mathrm{u}(\cdot, t)\Vert_{H^{2} ^{2}\Vert$\rho$_{t}(\cdot, t)\Vert^{2}\},
,

<

63

we

have

\displaystyle \frac{d}{dt}f(t)+ $\kappa$ f(t)\leq g_{1}(t)+g_{2}(t).

Thus, applying Lemma 3.2 and using Theorem 1.1 to estimate the term corresponding to(1+T_{1})^{ $\alpha$}f(T_{1})

,

we

have

(1+t)^{5/2}\Vert($\rho$_{t}, \mathrm{u}_{t} t) \displaystyle \Vert^{2}\leq C_{ $\kappa$}\{(\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2} ^{2}+\sup_{0<s<t}(1+s)^{5/2}\Vert(\nabla $\rho$, \nabla \mathrm{u} s) \Vert^{2}+(\displaystyle \sup_{0<s<t}(1+s)^{5/2}\Vert\nabla $\rho$(\cdot, s)\Vert^{2})\int_{0}^{t})+(\displaystyle \sup_{0<s<t}(1+s)^{5/2}\Vert$\rho$_{s}(\cdot, s)\Vert^{2})\int_{0}^{t}\Vert\nabla \mathrm{u}(\cdot, s)\Vert_{H^{2} ^{2}ds\}.
Since

C_{ $\kappa$}(\displaystyle \sup_{0<s<t}(1+s)^{5/2}\Vert$\rho$_{s}(\cdot, s)\Vert^{2})\int_{0}^{t})as

follows from (3.7), after fixing $\kappa$ , we choose $\epsilon$< $\kappa$ in such a way that

C_{ $\kappa$} $\epsilon$\leq 1/2

,

we

have

(1+t)^{5/4}\Vert($\rho$_{t}, \mathrm{u}_{t} t \leq C(\Vert($\rho$_{0}, \mathrm{u}_{0} +\mathrm{D}_{1}(t)).

(3.10)

Applying the elliptic estimate to the second equation in (3.3) and using (3.10), we have(1+t)^{5/4}\Vert\nabla^{2}\mathrm{u}(. , t \leq C(\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2}}+\mathrm{D}_{1}(t))
Finally, we estimate \Vert\nabla^{2} $\rho$\Vert.

.

(3.11)

To do this, we consider the whole space problem (2.18)

and the half space problem (2.19) and analogously to (3.11), we have(1+t)^{5/4}\Vert\nabla^{2} $\rho$(\cdot, t \leq C(\Vert($\rho$_{0}, \mathrm{u}_{0})\Vert_{H^{2}}+\mathrm{D}_{1}(t)).

(3.12)

Thus, by (3.6), (3.10), (3.11) and (3.12), we have (3.1), which completes the proof of
Theorem 1.2.

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