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

Shyam Sundar Ghoshal, Animesh Jana

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.

On blow up of $C^1$ solutions of 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. 1985] proved the blow up of smooth solutions when initial data satisfies an `integral condition'. We show that a solution of isentropic Euler equation breaks down if (i) gradient of initial velocity has a negative real eigenvalue at some point 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 to continuous space.
Paper Structure (15 sections, 13 theorems, 158 equations)

This paper contains 15 sections, 13 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Blow up of continuous solution
  4. Blow up of $C^1$ solution
  5. Global existence of smooth solution
  6. Comparison with previous results
  7. Preliminaries
  8. Energy estimates
  9. Proof of Proposition \ref{['proposition:reg']}
  10. Blow up of $C^1$ solutions
  11. Proof of Proposition \ref{['theorem:blow-up-C1-1']}
  12. Proof of Theorem \ref{['theorem:blow-up-C1-2']}
  13. Break down of continuity of a solution
  14. Proof of Theorem \ref{['theorem-cont']}
  15. Global existence of smooth solution

Key Result

Theorem 2.2

Let $(\varrho,\textbf{u})\in C([0,T^*)\times \mathbb{R}^d)$ be an admissible solution to eqn-isen-1--eqn-isen-2 with initial data $(\varrho_0,\textbf{u}_0)$according to Definition defn:admissible. We assume that the initial data $(\varrho_0,\textbf{u}_0)$ satisfies condition-Sideris and assumption-s

Theorems & Definitions (31)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 2.7
  • Remark 2.8
  • Lemma 3.1
  • Lemma 3.2
  • ...and 21 more