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

Abstract

We consider the isentropic compressible Euler equations with polytropic gamma law P.(ρ) = ργ in dimensions d ≤ 3. We address the borderline case when adiabatic index γ = 1 and establish local theory in the Sobolev space C0t Lpx C0t Hkx for d lt; p le; 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. © 2013 Mathematical Sciences Publishers.