Advances in Differential Equations

High regularity of solutions of compressible Navier-Stokes equations

Yonggeun Cho

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.


Export citation