Effet de la diffusion saturante sur les équations hyperboliques

Abstract

In this thesis, we study the effect of the saturating diffusion on the hyperbolic equations. We begin by studying the general Cauchy problem of the hyperbolic equation perturbed with the saturating diffusion. We establish the convergence of the corresponding solutions to the entropy weak solution of the hyperbolic conservation law. Then we consider the diffusivedispersive perturbation by adding a linear dispersion. We establish some relevant estimates of the solutions of the new problem. These latter seem not sufficient to prove the convergence in the vanishing regularisation limit. Therefore, we focus on the special solutions depending on one variable, called the travelling wave solutions. The problem reduces to a second-order ordinary differential equation that we can study through a phase plane analysis. Numerical simulations are developed to show the behaviour of the travelling waves. We also determine numerically the minimal value of the diffusion coefficient for which the perturbed problem has monotone travelling waves, and we classify the wave profiles into three different types: monotone, with a finite number of oscillations, and with infinite oscillations. We prove many theoretical results that confirm the numerical observations. Finally, based on the results of the travelling waves, we formulate conjectures about the convergence of the regularised solutions to the entropy weak solution of the hyperbolic conservation law as the diffusion and dispersion parameters tend to zero.