## Communications in Mathematical Sciences

- Commun. Math. Sci.
- Volume 7, Number 3 (2009), 627-634.

### On the formation of shocks to the compressible Euler equations

#### Abstract

We present an explicit upper-bound for the life-span of C3-smooth solutions to the multi-dimensional compressible Euler equations with a certain class of initial data containing compression vacuum states. We also show that the divergence of a fluid velocity will blow up along the particle trajectories issued from the compression vacuum states, which represent a shock formation at the vacuum states.

#### Article information

**Source**

Commun. Math. Sci., Volume 7, Number 3 (2009), 627-634.

**Dates**

First available in Project Euclid: 26 October 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.cms/1256562816

**Mathematical Reviews number (MathSciNet)**

MR2569026

**Zentralblatt MATH identifier**

1183.35225

**Subjects**

Primary: 35Q35: PDEs in connection with fluid mechanics 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]

**Keywords**

Compressible Euler equations blow-up shock wave particle trajectory vacuum boundary

#### Citation

Chae, Dongho; Ha, Seung-Yeal. On the formation of shocks to the compressible Euler equations. Commun. Math. Sci. 7 (2009), no. 3, 627--634. https://projecteuclid.org/euclid.cms/1256562816