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Abstract: Fourier analysis methods and in particular techniques based on Littlewood-Paley decomposition and paraproduct have known a growing interest recently for the study of nonlinear evolutionary equations. In this survey paper, we explain how these methods may be implemented so as to study the compresible Navier-Stokes equations in the whole space. We shall investigate both the initial value problem in critical Besov spaces and the low Mach number asymptotics.

In this survey paper, we are concerned with the zero Mach number limit for compressible viscous flows. For the sake of (mathematical) simplicity, we restrict ourselves to the case of barotropicfluids and we assume that the flow evolves in the whole space or satisfies periodic boundary conditions. We focus on the case of ill-prepared data. Hence highly oscillating acoustic waves are likely to propagate through the fluid. We nevertheless state the convergence to the incompressible Navier-Stokes equations when the Mach number ϵgoes to 0. Besides, it is shown that the global existence for the limit equations entails the global existence for the compressible model with small ϵ. The reader is referred to [R. Danchin, Ann. Sci. Éc. Norm. Sup.(2002)] for the detailed proof in the whole space case, and to [R. Danchin, Am. J. Math.124(2002) 1153–1219] for the case of periodic boundary conditions.

Abstract: We are concerned with global existence and uniqueness of strong solutions for a general model of viscous and heat-conductive gases. The initial data are supposed to be close to a stable equilibrium with constant density and temperature. Using uniform estimates for the linearized system with a convection term, we get global well-posedness in a functional setting invariant with respect to the scaling of the associated equations (in space dimension N≧3). We also show a smoothing effect on the velocity and the temperature, and a decay on the difference between the density and the constant reference state. These results extend a previous paper devoted to the barotropic case (see [5]).

The purpose of this work is to prove existence and uniqueness results of suitably smooth solutions for an isothermal model of capillary compressible fluids derived by J.E. Dunn and J. Serrin (1985), which can be used as a phase transition model.

We study here regularity of solutions of the two dirnensional incompressible Euler system whose initial data are vortez patches with singular boundary. We show the persislance of conormal regularity apart from a closed subset. As a consequence, we obtain the following result for vories patches : if the initial boundary is regular apart from a closed subset, it remairw regular for all time apart from the closed subset transported by the pow.