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

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.

arXiv comment38 pages

References in corpus (14)

• From $p_0(n)$ to $p_0(n+2)$
• The Threshold between Effective and Noneffective Damping for Semilinear Waves
• Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent
• Blow-up for semilinear damped wave equations with sub-Strauss exponent in the scattering case
• On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping
• On the global existence and blowup of smooth solutions of 3-D compressible Euler equations with time-depending damping
• A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent
• A global existence result for a semilinear wave equation with scale-invariant damping and mass in even space dimension
• Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma
• Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture
• The lifespan of solutions of semilinear wave equations with the scale-invariant damping in two space dimensions
• Nonexistence of global solutions for generalized Tricomi equations with combined nonlinearity
• A blow-up result for a generalized Tricomi equation with nonlinearity of derivative type
• Formation of finite-time singularities for nonlinear elastodynamics with small initial disturbances