On blow up of $C^1$ solutions of isentropic Euler system

TL;DR

This paper investigates conditions under which solutions to the isentropic Euler system blow up or exist globally, extending previous results to include continuous solutions and providing new criteria based on initial data properties.

Contribution

It introduces new blow-up and global existence criteria for solutions of the isentropic Euler system, including continuous solutions, based on initial velocity and density conditions.

Findings

Blow-up occurs if initial velocity gradient has a negative eigenvalue and density Hessian is small.

Global solutions exist if initial velocity eigenvalues are non-negative and density is small.

Continuous solutions can also break down under certain integral conditions.

Abstract

In this article, we study the break-down of smooth and continuous solutions to isentropic Euler system in multi dimension. Sideris [Comm. Math. Phys. 1985] proved the blow up of smooth solutions when initial data satisfies an `integral condition'. We show that a $C^1$ solution of isentropic Euler equation breaks down if (i) gradient of initial velocity has a negative real eigenvalue at some point $x_0\in\mathbb{R}^d$ and (ii) Hessian of initial density satisfies a smallness condition in Sobolev space. Our proof also works for the data which fails to satisfy the above-mentioned `integral condition'. Furthermore, we prove the global existence of smooth solution when (i) eigenvalues of gradient of initial velocity have non-negative real-part and (ii) initial density satisfies a smallness condition. This extends the global existence result of [Grassin, Indiana Univ. Math. J. 1998]. Another goal of this article is to study the breakdown of continuous weak solutions of isentropic Euler equations. We are able to show that the `integral condition' of Sideris can cause the breakdown of continuous solutions in finite time. This improves the blow up result of Sideris from $C^1$ to continuous space.