Optimal L p – L q -Type Decay Rates of Solutions to the Three-Dimensional Nonisentropic Compressible Euler Equations with Relaxation

Abstract

In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Wu et al. (2021, Adv. Math. Phys. Art. ID 5512285, pp. 1–13), we show the existence and uniqueness of the global small $H^k\left(k⩾3\right)$ solution only under the condition of smallness of the $H^3$ norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal $L^p–L^q$ $\left(1⩽p⩽2,2⩽q⩽\infty \right)$ -type decay rates of the solution and its higher-order derivatives.