Isentropic Approximation and Gevrey Regularity for the Full Compressible Euler Equations in $${\mathbb {R}}^{N}$$

Abstract

The article is devoted to the study of isentropic approximation and Gevrey regularity for the full compressible Euler system in $${\mathbb {R}}^{N}$$ (or $${\mathbb {T}}^{N}$$ ) with any dimension $$N\ge 1$$ . We first establish the existence and uniqueness of solution in Gevrey function spaces $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$ , then with the definition modulus of continuity, we show that the solution of Euler system is continuously dependent of the initial data $$v_{0}$$ in $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$ . Finally, the isentropic approximation is investigated in Banach spaces $${\mathcal {B}}_{T}^\nu ({\mathbb {R}}^{N})$$ , provided the initial entropy $$S_{0}(x)$$ changes closing a constant $${\bar{S}}$$ in Gevrey function spaces $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$ .