GLOBAL SMOOTH SOLUTIONS TO EULER EQUATION S FOR AN ISENTROPIC PERFECTGAS

One considers Euler equations for an isentropic perfect gas in R-d whe re d greater than or equal to 1. One shows that there exists global sm ooth solutions, that is in C-j([0: infinity[; Hm-j(R-d)) for j is an e lement of {0, 1}, provided that the initial data satisfy: u(0) is an e lement of H-m(R-d) with m > 1 + d/2, and For All x is an element of R- d, Sp(du(0)(x)) boolean AND R_ = empty set, rho(0) has a compact suppo rt and rho(0)(gamma-1/2) is small enough in H-m(R-d), where u(0) stand s for initial velocity, rho(0) for initial density, and gamma for the adiabatic constant of the gas. One denotes by Sp(du(0)(x)) the spectru m of the jacobian matrix of u(0).