On the isentropic compressible Euler equation with adiabatic index γ = 1

Dong Li, Changxing Miao and Xiaoyi Zhang

Vol. 262 (2013), No. 1, 109–128

DOI: 10.2140/pjm.2013.262.109

Abstract

We consider the isentropic compressible Euler equations with polytropic gamma law P(ρ) = ρ^γ in dimensions d ≤ 3. We address the borderline case when adiabatic index γ = 1 and establish local theory in the Sobolev space C_t⁰L_x^p ∩ C_t⁰Ḣ_x^k for d < p ≤ 4. This covers a class of physical solutions which can decay to vacuum at spatial infinity and are not compact perturbations of steady states. We construct a blowup scenario where initially the fluid is quiet in a neighborhood of the origin but is supersonic near the spatial infinity. For this special class of noncompact initial data, we prove the formation of singularities in finite time.

Keywords

compressible Euler equation, blowup solutions

Mathematical Subject Classification 2010

Primary: 35Q35

Secondary: 76N10

Milestones

Received: 8 January 2012

Revised: 1 August 2012

Accepted: 16 October 2012

Published: 27 March 2013