Global smooth solutions to Euler equations for a perfect gas

Abstract

We consider Euler equations for a perfect gas in R, 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. Introduction. We consider Euler equations for a perfect gas:  ∂tρ+ div(ρu) = 0, ρ(∂tu+ (u · ∇)u) +∇p = 0, ∂tS + u · ∇S = 0, (1) where t ∈ R+, x ∈ R and u : R × R+ → R stands for the velocity, ρ : R×R+ → R+ for the density, p = (γ−1)ρe for the pressure, with e the internal energy of the gas and S : R×R+ → R+ for the entropy. The adiabatic constant of the gas is denoted by γ > 1 and d ≥ 1 is the dimension of the space. We are interested in the existence of global smooth solutions to the Cauchy problem for (1) with (ρ0, u0, S0) as initial data. There exist few results concerning this problem, especially when d is strictly larger than one. The choice of initial data is decisive for this problem and it depends on whether one wants to prove or to disprove global existence. We aim at finding conditions on (ρ0, u0, S0) as weak as possible which ensure the existence of a global smooth solution. In [4], T. Sideris has shown a result of non global existence: the initial density is close to a constant at infinity—the constant should be different from 0—and some global quantities have to be large. For d = 1, in the isentropic case, we have a 2 × 2 system. In this case, some results can be proved using P. D. Lax’s works [3]. In the same case with less restrictive conditions, J. Y. Chemin [2] has also proved a result of non global existence: the initial velocity has to be smaller than the initial sound speed in each point—this quantity depends mostly on ρ0—. In [1], 1397