Some mathematical results on gases with small Mach number
This thesis is devoted to the study of the dynamics of the gases with small Mach number. The model comes from the complete Navier-Stokes equations when the Mach number goes to zero, and we aim at showing that it is well-posed. The viscous and inviscid cases are both considered. The physical coefficients may depend on the unknown density (or on the unknown temperature).In particular, we consider the effects of the thermal conductivity and hence large variations of entropy are allowed. Recall that if there is no thermal diffusion, then the low Mach number limit just implies the incompressibility condition. In the framework considered here, by introducing a new solenoidal velocity field, the system becomes a nonlinear coupling between a quasi-parabolic equation for the density and an evolutionary Stokes (or Euler) system for the velocity and the pressure. For the case with viscosity, we establish classical results, namely the strong solutions exist locally (resp. globally) in time for big (resp. small) initial data. We consider the Cauchy problem in the critical Besov spaces with the lowest regularity. Under a special relationship between the two physical coefficients, the system recasts in a simpler form and one may prove that there exist weak solutions with finite energy. In dimension two, this implies that strong solutions with finite energy exist for all positive times. In the inviscid case, we first prove the well-posedness result in endpoint Besov spaces, which can be embedded into the set of Lipschitzian functions. Continuation criterions and estimates for the lifespan are both established.If we suppose the initial data to be in the borderline Besov spaces with infinite Lebesgue exponent and to be of finite energy, we also have a local existence result. In dimension two, the lifespan goes to infinity when the density tends to a positive constant. Estimates for products and commutators, together with a priori estimates for the parabolic equations and the Stokes (or Euler) system with variable coefficients, are postponed in the appendix. These estimates are based on the Littlewood-Paley theory and the paradifferential calculus