Singularity formation for compressible Euler equations with time-dependent damping

2020/08/18 by Ying Sui, Huimin Yu, Sui, Ying +1
Mathematics · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2008.07756

Abstract

In this paper, we consider the compressible Euler equations with time-dependent damping (\a)/((1+t)^λ)u in one space dimension. By constructing 'decoupled' Riccati type equations for smooth solutions, we provide some sufficient conditions under which the classical solutions must break down in finite time. As a byproduct, we show that the derivatives blow up, somewhat like the formation of shock wave, if the derivatives of initial data are appropriately large at a point even when the damping coefficient goes to infinity with a algebraic growth rate. We study the case λ≠1 and λ=1 respectively, moreover, our results have no restrictions on the size of solutions and the positivity/monotonicity of the initial Riemann invariants. In addition, for 1

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