Systèmes hyperboliques à caractéristiques de multiplicité variable
Le but de l’exposé est de présenter les résultats obtenus par S. Bianchini et A. Bressan sur le problème de Cauchy pour des perturbations visqueuses de systèmes strictement hyperboliques en une dimension d’espace. Ils ont en particulier montré l’existence globale (), l’unicité et la stabilité des solutions et justifié la convergence quand tend vers zéro pour des données initiales à petite variation totale. Leur analyse montre aussi que les solutions du système hyperbolique ainsi obtenues...
We construct a Roe-type numerical scheme for approximating the solutions of a drift-flux two-phase flow model. The model incorporates a set of highly complex closure laws, and the fluxes are generally not algebraic functions of the conserved variables. Hence, the classical approach of constructing a Roe solver by means of parameter vectors is unfeasible. Alternative approaches for analytically constructing the Roe solver are discussed, and a formulation of the Roe solver valid for general closure...
Consider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.
We discuss the local existence and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones and by use of the Tanabe-Sobolevski’s method and the Banach scale method we construct a semi-group which gives a representation of the solution to the Cauchy problem.