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 $G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$.