Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with L 2 initial data and minimal assumptions on the drift are locally Holder continuous. As an application we show that solutions of the quasigeostrophic equation with initial L 2 data and critical diffusion (−∆) 1/2 , are locally smooth for any space dimension.

In this article, we consider the compressible Navier-Stokes equation with density dependent viscosity coefficients. We focus on the case where those coefficients vanish on vacuum. We prove the stability of weak solutions for periodic domain Ω = T N as well as the whole space Ω = R N , when N = 2 and N = 3. The pressure is given by p = ρ γ , and our result holds for any γ > 1. In particular, we prove the stability of weak solutions of the Saint-Venant model for shallow water.

We study a nonlocal nonlinear parabolic problem with a fractional time derivative. We prove a Krylov-Safonov type result; mainly, we prove Hölder regularity of solutions. Our estimates remain uniform as the order of the fractional time derivative α → 1.

This article is dedicated to the regularity theory for solutions to a class of nonlinear integral variational problems. Those problems are involved in nonlocal image and signal processing.

We extend the De Giorgi-Nash-Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the Hölder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations "of type II", sometimes also called ultraparabolic equations of Kolmogorov type, but with rough coefficients: it combines a first-order skew-symmetric operator with a second-order elliptic operator involving derivatives along only part of the coordinates and with rough coefficients. These general results are then applied to the non-negative essentially bounded weak solutions of the Landau equation with inverse-power law γ ∈ [−d, 1] whose mass, energy and entropy density are bounded and mass is bounded away from 0, and we deduce the Hölder regularity of these solutions.

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Abstract. We consider Navier-Stokes equations for compressible viscous fluids in one dimension. It is a well known fact that if the initial datum are smooth and the initial density is bounded by below by a positive constant, then a strong solution exists locally in time. In this paper, we show that under the same hypothesis, the density remains bounded by below by a positive constant uniformly in time, and that strong solutions therefore exist globally in time. Moreover, while most existence results are obtained for positive viscosity coefficients, the present result holds even if the viscosity coefficient vanishes with the density. Finally, we prove that the solution is unique in the class of weak solutions satisfying the usual entropy inequality. The key point of the paper is a new entropy-like inequality introduced by Bresch and Desjardins for the shallow water system of equations. This inequality gives additional regularity for the density (provided such regularity exists at initial time).

Abstract. In this paper, we prove the existence of global weak solutions for 3D compressible Navier-Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins entropy conservation [2]. The main contribution of this paper is to derive the Mellet-Vasseur type inequality [32] for the weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the compressible Navier-Stokes equations, for any γ > 1 in two dimensional space and for 1 < γ < 3 in three dimensional space, with large initial data possibly vanishing on the vacuum. This solves an open problem proposed by Lions in [27].