Asymptotic Behavior of the 3D Compressible Euler Equations with Nonlinear Damping and Slip Boundary Condition

Huimin Yu

doi:10.1155/2012/584680

Abstract

The asymptotic behavior (as well as the global existence) of classical solutions to the 3D compressible Euler equations are considered. For polytropic perfect gas(P(ρ)=P0ργ), time asymptotically, it has been proved by Pan and Zhao (2009) that linear damping and slip boundary effect make the density satisfying the porous medium equation and the momentum obeying the classical Darcy's law. In this paper, we use a more general method and extend this result to the 3D compressible Euler equations with nonlinear damping and a more general pressure term. Comparing with linear damping, nonlinear damping can be ignored under small initial data.

Highlights

Introduction and Main ResultsWe study the 3D compressible Euler equations with nonlinear damping: ρt ∇ · ρu 0, ρu t ∇ · ρu ⊗ u ∇P ρ −αρu − βρ|u|q−1u.This model represents the compressible flow through porous media with nonlinear external force field
The external term −αρu − βρ|u|q−1u appears in the momentum equation, where α is a positive constant, β is another constant but can be either negative or positive
From 18 , we know that there exists a positive constant T > 0 such that the problem 3.2 and 3.3 has a classical solution ρ x, t for t > T , and ρ∗ ρ x, t > 2 , for t > T

Summary

Introduction and Main Results

We study the 3D compressible Euler equations with nonlinear damping: ρt ∇ · ρu 0, ρu t ∇ · ρu ⊗ u ∇P ρ −αρu − βρ|u|q−1u. The term −βρ|u|q−1u with q > 1 is regarded as a nonlinear source to the linear damping, where the symbol |u| denotes u21 u22 u23 if we assume u u1, u2, u3. The existence as well as approximate behavior of smooth solutions to the initial boundary condition in half line and Cauchy problem are considered. We consider the global existence and asymptotic behavior of classical solutions to the 3D problem 1.1 1.2 with nonlinear damping and slip boundary condition. There exists a constant δ > 0 such that if ρ0 − ρ∗/|Ω| , u0 ∈ H3 Ω and ρ0 − ρ∗/|Ω|, u0 3 ≤ δ, the initial boundary condition 1.1 and 1.2 exists a unique global solution ρ, u in C1 Ω × 0, ∞ ∩ X3 Ω, 0, ∞. U s ≤ C ∇ × u s−1 ∇ · u s−1 u s−1 , 1.13 for s ≥ 1, and the constant C depends only on s and Ω

Global Existence and Asymptotic Behavior

About Porous Medium Equations