Abstract

I will give an overview on two seminal papers of Klainerman and Majda, “Singular limits of quasi-linear hyperbolic systems with large parameters and the incompressible limit of compressible fluids”. Comm. Pure Appl. Math. 34 (1981) and “Compressible and incompressible fluids”. Comm. Pure Appl. Math. 35 (1982).

In the first paper, the authors treat a class of systems of conservation laws which depend on a possibly singular parameter. They establish conditions on the flux functions and the initial data under which a uniform a-priori estimate holds for the solutions. Using this, they prove convergence for several systems as the parameter becomes singular. Note that this includes a change of type in the pde. In the second paper, the authors focus on isentropic gas dynamics, and rigorously prove the existence of an asymptotic expansions. Such expansions are the basis of several recent low-Mach number schemes, which are relevant to our RTG.