On the time continuity of entropy solutions

We show that any entropy solution $u$ of a convection diffusion equation $\partial_t u + \div F(u)-\Delta\phi(u) =b$ in \OT belongs to C([0,T),L^1_{Loc}(\o\O)). The proof does not use the uniqueness of the solution

Error estimates for a numerical approximation to the compressible barotropic Navier-Stokes equations

We present here a general method based on the investigation of the relative energy of the system, that provides an unconditional error estimate for the approximate solution of the barotropic Navier Stokes equations obtained by time and space discretization. We use this methodology to derive an error estimate for a specific DG/finite element scheme for which the convergence was proved in [27]. This is an extended version of the paper submitted to IMAJNA

Entropy estimates for a class of schemes for the euler equations

In this paper, we derive entropy estimates for a class of schemes for the Euler equations which present the following features: they are based on the internal energy equation (eventually with a positive corrective term at the righ-hand-side so as to ensure consistency) and the possible upwinding is performed with respect to the material velocity only. The implicit-in-time first-order upwind scheme satisfies a local entropy inequality. A generalization of the convection term is then introduced, which allows to limit the scheme diffusion while ensuring a weaker property: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L $\infty$ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme, with the above-mentioned generalization of the convection operator, the same result only holds if the ratio of the time to the space step tends to zero

Convergence of the MAC scheme for the compressible stationary Navier-Stokes equations

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three-dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the approximate solutions, up to a subsequence, and in an appropriate sense. We then prove that the limit of the approximate solutions satisfies the mass and momentum balance equations, as well as the equation of state, which is the main difficulty of this study

W-0(1,1) solutions in some borderline cases of Calderon-Zygmund theory

In this paper we study the existence of w(0)(1,1)(Omega) distributional solutions of Dirichlet problems whose simplest example is {-div (|del u|(p-2)del u) = f (x), in Omega; u = 0, on partial derivative Omega. (C) 2012 Elsevier Inc. All rights reserved

Solutions of nonlinear parabolic equations without growth restrictions on the data

The purpose of this paper is to prove the existence of solutions for certain types of nonlinear parabolic partial differential equations posed in the whole space, when the data are assumed to be merely locally integrable functions, without any control of their behaviour at infinity. A simple representative example of such an equation is $$u_t-Delta u + |u|^{s-1}u=f,$$ which admits a unique globally defined weak solution $u(x,t)$ if the initial function $u(x,0)$ is a locally integrable function in $mathbb{R}^N$, $Ngeq 1$, and the second member $f$ is a locally integrable function of $xinmathbb{R}^N$ and $tin [0,T]$ whenever the exponent $s$ is larger than 1. The results extend to parabolic equations results obtained by Brezis and by the authors for elliptic equations. They have no equivalent for linear or sub-linear zero-order nonlinearities