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.     

Finite volume schemes for the approximation via characteristics of linear convection equations with irregular data

We consider the approximation by multidimensional finite volume schemes of the transport of an initial measure by a Lipschitz flow. We first consider a scheme defined via characteristics, and we prove the convergence to the continuous solution, as the time step and the ratio of the space step to the time step tend to zero. We then consider a second finite volume scheme, obtained from the first one by addition of some uniform numerical viscosity. We prove that this scheme converges to the continuous solution, as the space step tends to zero, whereas the ratio of the space step to the time step remains bounded by below and by above, and under assumption of uniform regularity of the mesh. This is obtained via an improved discrete Sobolev inequality and a sharp weak BV estimate, under some additional assumptions on the transport flow. Examples show the optimality of these assumptions.     

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...     

Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

We study approximations by conforming methods of the solution to the variational inequality \({\langle \partial_t u,v-u\rangle + \psi(v) - \psi(u) \ge \langle f,v-u\rangle}\) , which arises in the context of inviscid incompressible Bingham type fluid flows and of the total variation flow problem. In the general context of a convex lower semi-continuous functional \({\psi}\) on a Hilbert space, we prove the convergence of time implicit space conforming approximations, without viscosity and for nonsmooth data. Then, we introduce a general class of total variation functionals \({\psi}\) , for which we can apply the regularization method. We consider the time implicit regularized, linearized or not, algorithms and prove their convergence for general total variation functionals. A comparison with an analytical solution concludes this study.     

Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data

We consider the nonlinear heat equation (with Leray-Lions operators) on an open bounded subset of R^N with Dirichlet homogeneous boundary conditions. The initial condition is in L¹ and the right hand side is a smooth measure. We extend a previous notion of entropy solutions and prove that they coincide with the renormalized solutions.