ON THE WELL-POSEDNESS OF THE FULL LOW MACH NUMBER LIMIT SYSTEM IN GENERAL CRITICAL BESOV SPACES

Abstract

This work is devoted to the well-posedness issue for the low Mach number limit system obtained from the full compressible Navier–Stokes system, in the whole space ℝd with d ≥ 2. In the case where the initial temperature (or density) is close to a positive constant, we establish the local existence and uniqueness of a solution in critical homogeneous Besov spaces of type [Formula: see text]. If, in addition, the initial velocity is small then we show that the solution exists for all positive time. In the fully nonhomogeneous case, we establish the local well-posedness in nonhomogeneous Besov spaces [Formula: see text] (still with critical regularity) for arbitrarily large data with positive initial temperature. Our analysis strongly relies on the use of a modified divergence-free velocity which allows to reduce the system to a nonlinear coupling between a parabolic equation and some evolutionary Stokes system. As in the recent work by Abidi and Paicu [Existence globale pour un fluide inhomogène, Ann. Inst. Fourier 57(3) (2007) 883–917]. Concerning the density-dependent incompressible Navier–Stokes equations, the Lebesgue exponents of the Besov spaces for the temperature and the (modified) velocity, need not be the same. This enables us to consider initial data in Besov spaces with a negative index of regularity.