## Osaka Journal of Mathematics

### A characterization on breakdown of smooth spherically symmetric solutions of the isentropic system of compressible Navier--Stokes equations

#### Abstract

We study an initial boundary value problem on a ball for the isentropic system of compressible Navier--Stokes equations, in particular, a criterion of breakdown of the classical solution. For smooth initial data away from vacuum, it is proved that the classical solution which is spherically symmetric loses its regularity in a finite time if and only if the concentration of mass forms around the center in Lagrangian coordinate system. In other words, in Euler coordinate system, either the density concentrates or vanishes around the center. For the latter case, one possible situation is that a vacuum ball appears around the center and the density may concentrate on the boundary of the vacuum ball simultaneously.

#### Article information

Source
Osaka J. Math., Volume 52, Number 1 (2015), 271-285.

Dates
First available in Project Euclid: 24 March 2015

https://projecteuclid.org/euclid.ojm/1427202881

Mathematical Reviews number (MathSciNet)
MR3326611

Zentralblatt MATH identifier
1317.35175

#### Citation

Huang, Xiangdi; Matsumura, Akitaka. A characterization on breakdown of smooth spherically symmetric solutions of the isentropic system of compressible Navier--Stokes equations. Osaka J. Math. 52 (2015), no. 1, 271--285. https://projecteuclid.org/euclid.ojm/1427202881

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