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 k ⩾ 3 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 1 ⩽ p ⩽ 2 , 2 ⩽ q ⩽ ∞ -type decay rates of the solution and its higher-order derivatives.
Highlights
In this paper, we shall study the nonisentropic compressible Euler equations with relaxation:>>>>>< >>>>>: ρt + div ðρuÞ = 0, ðρuÞt + div ðρu ⊗ uÞ+∇P = −1 τ ρu, ðρEÞt div ðρuE uPÞ 1 τ ρu2ρ τ ðgðρÞ θÞ: ð1Þ
We review the known research results for the compressible Euler equations with relaxation
The global existence and large-time behavior of solutions to the multidimensional isentropic compressible Euler equations with relaxation were studied by many researchers
Summary
We shall study the nonisentropic compressible Euler equations with relaxation (cf. [1]):. The global existence and large-time behavior of solutions to the multidimensional isentropic compressible Euler equations with relaxation were studied by many researchers (cf [16, 25,26,27,28,29,30,31,32,33,34,35,36] and the references cited therein). In this paper, following the similar discussions in [37], we shall use a delicate energy method to obtain a refined global existence and uniqueness result, in which we only require the initial H3 norm to be small. By Corollary 3, we prove the optimal Lp – Lq-type decay rates without the smallness assumption on the Lp norm of the initial data. The detailed proof of Lemma 15 is given in Appendix B
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