‎In this paper‎, ‎we study the Neumann boundary value problem of a class of nonlinear divergence type diffusion equations‎. ‎By a priori estimates‎, ‎difference and variation techniques‎, ‎we establish the existence and uniqueness of weak solutions of this problem.

On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable Abstract. We show compactness of bounded sets of weak solutions to the isentropic compressible Navier-Stokes equations in three space dimensions under the hypothesis that the adiabatic constant γ > 3/2.

We prove the global-in-time existence of large-data finite-energy weak solutions to an incompressible hybrid Vlasov-magnetohydrodynamic model in three space dimensions. The model couples three essential ingredients of magnetized plasmas: a transport equation for the probability density function, which models energetic rarefied particles of one species; the incompressible Navier–Stokes system fo...

We give several remarks on Strichartz estimates for homogeneous wave equation with special attention to the cases of Lx estimates, radial solutions and initial data from the inhomogeneous Sobolev spaces. In particular, we give the failure of the endpoint estimate L4tL ∞ x for n = 2.

We compare the metric and the Palatini formalism to obtain the Einstein equations in the presence of higher-order curvature corrections that consist of contractions of the Riemann tensor, but not of its derivatives. We find that, with the exception of standard Einsteingravity, the two formalisms are not equivalent, but that the Palatini formalism is contained within the metric formalism, in the...

We study several regularizing methods, stationary phase or averaging lemmas for instance. Depending on the regularity assumptions that are made, we show that they can either be derived one from the other or that they lead to different results. Those are applied to Scalar Conservation Laws to precise and better explain the regularity of their solutions.

We present numerical solutions of a two-dimensional Riemann problem for the unsteady transonic small disturbance equations that provides an asymptotic description of the Mach reflection of weak shock waves. We develop a new numerical scheme to solve the equations in selfsimilar coordinates and use local grid refinement to resolve the solution in the reflection region. The solutions contain a re...

We prove the weak-strong uniqueness for measure-valued solutions of the incompressible Euler equations. These were introduced by R.DiPerna and A.Majda in their landmark paper [10], where in particular global existence to any L initial data was proven. Whether measure-valued solutions agree with classical solutions if the latter exist has apparently remained open. We also show that DiPerna’s mea...

K e y w o r d s D i f f u s i o n c o n v e c t i o n equations, L °° bounds of weak solutions. 1. I N T R O D U C T I O N In this note, we consider weak solutions to diffusion-convection equations with less regular convective field and boundary data. The general form of the problem under consideration is -~u(t,x)O _ t~Au(t,x) + d i v (b(x)u(t ,x)) = f ( t , x ,u ( t , x ) ) , in (0, T) x ~t (1...