Global smooth solutions to Euler equations for a perfect gas

Abstract

We consider Euler equations for a perfect gas in R d , where d ≥ 1. We state that global smooth solutions exist under the hypotheses (H1)-(H3) on the initial data. We choose a small smooth initial density, and a smooth enough initial velocity which forces particles to spread out. We also show a result of global in time uniqueness for these global solutions.

From gas dynamics to pressureless gas dynamics

“…There are few examples of global smooth solutions to the multidimensional compressible Euler equations; some are given in [2,5]. In this paper, we shall consider some initial data for which M. Grassin has shown in [2] that the corresponding solution is global and enjoys suitable decay properties.…”

“…In this paper, we shall consider some initial data for which M. Grassin has shown in [2] that the corresponding solution is global and enjoys suitable decay properties.…”

“…Let us remark that an initial density of order O(1) for the system (1.1) corresponds to an initial density of order O(ε 1/(γ−1) ) for the usual Euler equations (that is, when ε = 1) that were considered in [2]. Consequently, we can apply the result of [2] and obtain a global smooth solution (ρ ε , u ε ) of (1.1), provided the initial velocity field satisfies the assumptions of [2].…”

“…Consequently, we can apply the result of [2] and obtain a global smooth solution (ρ ε , u ε ) of (1.1), provided the initial velocity field satisfies the assumptions of [2]. This is because the existence result of [2] relies on a smallness assumption for the initial density. Moreover, the analysis of [2] yields L ∞ bounds for the solution.…”

“…This is because the existence result of [2] relies on a smallness assumption for the initial density. Moreover, the analysis of [2] yields L ∞ bounds for the solution. In particular, these bounds show the convergence of u ε towards the velocity field u solution to (1.3).…”