Global smooth solutions of Euler equations for Van der Waals gases

TL;DR

This paper proves the global existence of smooth solutions to the Euler equations for Van der Waals gases with small density, extending previous results for perfect gases by developing a new symmetrisation method.

Contribution

It introduces a novel symmetrisation technique for Van der Waals gases and establishes global solutions in Sobolev spaces, generalizing prior work on perfect gases.

Findings

Global existence of smooth solutions for small density Van der Waals gases

Development of a specific symmetrisation method for null density regions

Extension of previous results from perfect gases to Van der Waals gases

Abstract

We prove global in time existence of solutions of the Euler compressible equations for a Van der Waals gas when the density is small enough in $\H{m}$, for $m$ large enough. To do so, we introduce a specific symmetrisation allowing areas of null density. Next, we make estimates in $\H{m}$, using for some terms the estimates done by M. Grassin, who proved the same theorem in the easier case of a perfect polytropic gas. We treat the remaining terms separately, due to their non-linearity.