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