Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques

TL;DR

This paper establishes improved upper bounds for the lifespan of solutions to the compressible Euler equations with damping, using Orlicz spaces to handle nonlinear pressure terms, applicable in 2D and 3D.

Contribution

Introduces Orlicz spaces techniques to analyze the lifespan of solutions, providing unified bounds independent of the adiabatic index for high dimensions.

Findings

Unified upper bounds depending only on dimension and damping

Improved lifespan estimates over previous results

Potential applicability to related nonlinear PDE problems

Abstract

In this paper we are interested in the upper bound of the lifespan estimate for the compressible Euler system with time dependent damping and small initial perturbations. We employ some techniques from the blow-up study of nonlinear wave equations. The novelty consists in the introduction of tools from the Orlicz spaces theory to handle the nonlinear term emerging from the pressure $p \equiv p(\rho)$, which admits different asymptotic behavior for large and small values of $\rho-1$, being $\rho$ the density. Hence we can establish, in high dimensions $n\in\{2,3\}$, unified upper bounds of the lifespan estimate depending only on the dimension $n$ and on the damping strength, and independent of the adiabatic index $\gamma>1$. We conjecture our results to be optimal. The method employed here not only improves the known upper bounds of the lifespan for $n\in\{2,3\}$, but has potential application in the study of related problems.