Guy Vallet

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.

Some Stratigraphic Control Problems

In this paper, we are interested in lithology diffusion models applied in the field of stratigraphic basin simulations for large-scale depositional transport processes of sediments. Such models describe erosion-sedimentation processes and take into account limited weathering via nonstandard unilateral problems. Various theoretical results, illustrations, and numerical solutions are presented for the monolithologic column case. A new conservation law involved in modeling is formulated, and mathematical tools for solving the problem are described.

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.

Functional inequalities and theorems of the alternative involving composite functions

We propose variants of non-asymptotic dual transcriptions for the functional inequality of the form

ON THE EQUATION OF BARENBLATT–SOBOLEV

f + g + k\circ H \ge h

Continuous dependence estimate for a degenerate parabolic–hyperbolic equation with Lévy noise

f+g+k∘H≥h. The main tool we used consists in purely algebraic formulas on the epigraph of the Legendre-Fenchel transform of the function