## Advances in Differential Equations

- Adv. Differential Equations
- Volume 12, Number 8 (2007), 893-960.

### High regularity of solutions of compressible Navier-Stokes equations

#### Abstract

We study the *barotropic* compressible Navier-Stokes equations in a bounded or an
unbounded domain $\Omega $ of $ \mathbf{R}^3$. The initial density may vanish in an open
subset of $\Omega$ or be positive but vanish at space infinity. We first prove the local
existence of solutions $(\rho^{(j)}, u^{(j)})$ in $C([0,T_* ]; H^{2(k-j)+3} \times D_0^1
\cap D^{2(k-j)+3} (\Omega ) )$, $0 \le j \le k, k \ge 1$ under the assumptions that the
data satisfy compatibility conditions and the initial density is sufficiently small. To
control the non-negativity or decay at infinity of density, we need to establish a
boundary-value problem of a $(k+1)$-coupled elliptic system which may not be, in general,
solvable. The smallness condition of the initial density is necessary for the solvability
of the elliptic system; this is not necessary when the initial density has positive lower
bound. Secondly, we prove the global existence of smooth radially symmetric solutions of
*isentropic* compressible Navier-Stokes equations by controlling every regularity with
$|\rho|_{L^\infty}$.

#### Article information

**Source**

Adv. Differential Equations, Volume 12, Number 8 (2007), 893-960.

**Dates**

First available in Project Euclid: 29 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1367241141

**Mathematical Reviews number (MathSciNet)**

MR2340257

**Zentralblatt MATH identifier**

1146.35072

**Subjects**

Primary: 35Q35: PDEs in connection with fluid mechanics

Secondary: 35B45: A priori estimates 35D05 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 76N10: Existence, uniqueness, and regularity theory [See also 35L60, 35L65, 35Q30]

#### Citation

Cho, Yonggeun. High regularity of solutions of compressible Navier-Stokes equations. Adv. Differential Equations 12 (2007), no. 8, 893--960. https://projecteuclid.org/euclid.ade/1367241141