Approximation of hyperbolic conservation laws

Tese de doutoramento em Matemática (Análise Matemática), apresentada à Universidade de Lisboa através da Faculdade de Ciências, 2008In a first part, we study the zero diffusion-dispersion limit for a class of nonlinear hyperbolic and multi-dimensional conservation laws regularized in a fashion similar to to the Benjamin-Bona-Mahony-Burgers (BBMB) and Korteweg-deVries-Burgers (KdVB) equations. We establish the strong convergence toward classical entropy solutions by relying DiPerna's theory of entropy measure-valued solutions. Optimal conditions are determined for the balance between diffusion and dispersion coefficients. This allows us to propose criteria for the possible existence or non-existence of nonclassical solutions in the sense investigated by LeFloch. Our analysis distinguishes between several assumptions on the diffusion, the dispersion, and the flux-function and emphasize drastic differences between the BBMB and the KdVB models; distinct convergence behaviors are put in evidence and various energy-type arguments are discussed. In the second part, we study the Riemann problem for nonlinear hyperbolic systems of conservation laws whose flux-function is solely Lipschitz continuous. Typical examples arise in the modelling of multi-phase flows and of elasto-plastic materials. To extend Lax's theory, the main difficulty is to handle possibly discontinuous wave speeds. We revisit certain fundamental notions such as the strict hyperbolicity, the genuine nonlinearity and the entropy inequalities. Our proofs rely on a generalized calculus for Lipschitz continuous mappings and the related Filippov's theory of ordinary differential equations with discontinuous coefficients. We identify here several new features arising in discontinuous solutions of the Riemann problem