On the blow up of \(C^1\) solutions of the isentropic Euler system

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 101(4):475–478, 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 47(4):1397–1432, 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.