Global Existence and Long-Time Behavior of Solutions to the Full Compressible Euler Equations with Damping and Heat Conduction in    ℝ   3

Yunshun Wu,Rong Shen,Yong Wang

doi:10.1155/2021/5512285

Abstract

We study the Cauchy problem of the three-dimensional full compressible Euler equations with damping and heat conduction. We prove the existence and uniqueness of the global small H N N ≥ 3 solution; in particular, we only require that the H 4 norms of the initial data be small when N ≥ 5 . Moreover, we use a pure energy method to show that the global solution converges to the constant equilibrium state with an optimal algebraic decay rate as time goes to infinity.

Highlights

We study the following Cauchy problem of the full compressible Euler equations with damping and heat conduction: for ðx, tÞ ∈ R3 × 1⁄20,∞Þ, >>>>>< ρt + div ðρuÞt +ðρuÞ = 0, div ðρu ⊗ uÞ+∇P = −αρu,>>>>>: ðρEÞt + div ðρuE + uPÞ = ðρ, u, θÞðx, tÞjt=0 = ðρ0, u0, −αρu2 θ0ÞðxÞ+ κΔθ, ⟶ ð1, 1Þ, jxj⟶∞: ð1ÞHere, the unknown variables ρ = ρðx, tÞ,u = uðx, tÞ, θ = θðx, tÞ, P = Pðx, tÞ denote the density, the velocity, the absolute temperature, and the pressure, respectively
We use a pure energy method to show that the global solution converges to the constant equilibrium state with an optimal algebraic decay rate as time goes to infinity
We review the known results about the compressible Euler equations with damping

Summary

Introduction

We study the following Cauchy problem of the full compressible Euler equations with damping and heat conduction: for ðx, tÞ ∈ R3 × 1⁄20,∞Þ,. The global existence and long-time behavior of solutions to the multidimensional compressible isentropic Euler equations with damping were studied by many researchers (cf [15, 24,25,26,27,28,29,30,31,32,33,34,35] and the references cited therein). By Corollary 3, we prove the optimal Lp–Lq-type time-decay rates without the smallness assumption on the Lpð1 ≤ p < 2Þ norm of the initial data. With regard to the initial-boundary value problem of the three-dimensional full compressible damped Euler equations (1), the case of κ = 0 was solved in [42, 43], and the corresponding ðP, u, SÞ-system was adopted. We prove the global solution (Theorem 1) and the time-decay rates (Theorem 2) in Sections 4 and 5, respectively

Preliminaries

Energy Estimates

Global Solution

Time-Decay Rates