Parabolic Capacity and Soft Measures for Nonlinear Equations

We first introduce, using a functional approach, the notion of capacity related to the parabolic p-Laplace operator. Then we prove a decomposition theorem for measures (in space and time) that do not charge the sets of null capacity. We apply this result to prove existence and uniqueness of renormalized solutions for nonlinear parabolic initial boundary-value problems with such measures as right-hand side.     

Fractal First-Order Partial Differential Equations

The present paper is concerned with semi-linear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton–Jacobi equations. The idea is to combine an integral representation of the operator and Duhamel's formula to prove, on the one hand, the key a priori estimates for the scalar conservation law and the Hamilton–Jacobi equation and, on the other hand, the smoothing effect of the operator. As far as Hamilton–Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows us to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law.     

Non-coercive Linear Elliptic Problems

We study here some linear elliptic partial differential equations (with Dirichlet, Fourier or mixed boundary conditions), to which convection terms (first order perturbations) are added that entail the loss of the classical coercivity property. We prove the existence, uniqueness and regularity results for the solutions to these problems.     

A mixed finite volume scheme for anisotropic diffusion problems on any grid

We present a new finite volume scheme for anisotropic heterogeneous diffusion problems on unstructured irregular grids, which simultaneously gives an approximation of the solution and of its gradient. The approximate solution is shown to converge to the continuous one as the size of the mesh tends to 0, and an error estimate is given. An easy implementation method is then proposed, and the efficiency of the scheme is shown on various types of grids and for various diffusion matrices.     

Unified Convergence Analysis of Numerical Schemes for a Miscible Displacement Problem

Foundations of Computational Mathematics - This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible...     

Analysis of miscible displacement through porous media with vanishing molecular diffusion and singular wells

This article proves the existence of solutions to a model of incompressible miscible displacement through a porous medium, with zero molecular diffusion and modelling wells by spatial measures. We obtain the solution by passing to the limit on problems indexed by vanishing molecular diffusion coefficients. The proof employs cutoff functions to excise the supports of the measures and the discontinuities in the permeability tensor, thus enabling compensated compactness arguments used by Y. Amirat and A. Ziani for the analysis of the problem with $L^2$ wells (Amirat and Ziani, 2004 [1]). We give a novel treatment of the diffusion–dispersion term, which requires delicate use of the Aubin–Simon lemma to ensure the strong convergence of the pressure gradient, owing to the troublesome lower-order terms introduced by the localisation procedure.     

Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature

We investigate the singular limit, as ${\varepsilon \to 0}$ , of the Allen-Cahn equation ${u^\varepsilon_t=\Delta u^\varepsilon+\varepsilon^{-2}f(u^\varepsilon)}$ , with f a balanced bistable nonlinearity. We consider rather general initial data u ₀ that is independent of ${{\varepsilon}}$ . It is known that this equation converges to the generalized motion by mean curvature ?? in the sense of viscosity solutions??defined by Evans, Spruck and Chen, Giga, Goto. However, the convergence rate has not been known. We prove that the transition layers of the solutions ${u^{\varepsilon}}$ are sandwiched between two sharp ??interfaces?? moving by mean curvature, provided that these ??interfaces?? sandwich at t?=?0 an ${\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}$ neighborhood of the initial layer. In some special cases, which allow both extinction and pinches off phenomenon, this enables to obtain an ${\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}$ estimate of the location and the thickness measured in space-time of the transition layers. A result on the regularity of the generalized motion by mean curvature is also provided in the Appendix.     

Study of the mixed finite volume method for Stokes and Navier‐Stokes equations

We present finite volume schemes for Stokes and Navier‐Stokes equations. These schemes are based on the mixed finite volume introduced in (Droniou and Eymard, Numer Math 105 (2006), 35‐71), and can be applied to any type of grid (without “orthogonality” assumptions as for classical finite volume methods) and in any space dimension. We present numerical results on some irregular grids, and we prove, for both Stokes and Navier‐Stokes equations, the convergence of the scheme toward a solution of the continuous problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Noncoercive convection–diffusion elliptic problems with Neumann boundary conditions

We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation