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

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 $H^N\left(N\ge 3\right)$ solution; in particular, we only require that the $H^4$ norms of the initial data be small when $N\ge 5$ . 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.