Petra Wittbold

Existence of renormalized solutions of degenerate elliptic-parabolic problems

We consider a general class of degenerate elliptic-parabolic problems associated with the equation b (υ) t = div a (υ, Dυ ) + f . Existence of renormalized solutions is established for general L 1 data. Uniqueness of renormalized solutions has already been shown in a previous work.

ON A STOCHASTIC FIRST-ORDER HYPERBOLIC EQUATION IN A BOUNDED DOMAIN

In this paper, we are interested in the stochastic perturbation of a first-order hyperbolic equation of nonlinear type. In order to illustrate our purposes, we have chosen a scalar conservation law in a bounded domain with homogeneous Dirichlet condition on the boundary. Using the concept of measure-valued solutions and Kruzhkovs entropy formulation, a result of existence and uniqueness of the entropy solution is given.

The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation

Abstract In this paper, we are interested in the Dirichlet boundary value problem for a multi-dimensional nonlinear conservation law with a multiplicative stochastic perturbation. Using the concept of measure-valued solutions and Kruzhkovʼs semi-entropy formulations, a result of existence and uniqueness of the entropy solution is proved.

A degenerate parabolic-hyperbolic Cauchy problem with a stochastic force

In this paper we are interested in the Cauchy problem for a nonlinear degenerate parabolic-hyperbolic problem with multiplicative stochastic forcing. Using an adapted entropy formulation a result of existence and uniqueness of a solution is proved.

Nonlinear Diffusion with Absorption

We consider the nonlinear diffusion-absorption problem in L1(Ω) u - div a(⋅, Du) + ∂j (⋅, u) ∋ f on Ω, u = 0 on ∂Ω where Ω is an arbitrary open set in ℝN. Conditions for the existence of a strong solution are presented and the existence of a natural generalized solution in the general case is established. We study the dependence of the generalized solution on the absorption term ∂j and characterize generalized solutions which are essentially bounded.