Abstract or Additional Information
In this talk we will discuss how the tools from the theory of kinetic equations can be applied to the problem of constructing solutions (both classical and weak) of hyperbolic systems of conservation laws, and in particular the equations of gas dynamics. We will start with the basic properties of hyperbolic systems of PDEs and recall the classical results on the existence of solutions to the Cauchy problem for linear and quasilinear hyperbolic systems. Then, we will turn to the kinetic formalism in which solutions are represented as moments of a kinetic function determined by the Gibbs minimization property. The kinetic formalism can be expressed as a differential inclusion for the dual kinetic function. Based on that, we will describe a transport-projection splitting algorithm for classical solutions of the Euler equations in multi-dimensions. As another application of the kinetic equations, we will introduce the class of approximate weak solutions of the isentropic gas dynamics and scalar conservation laws. The solutions of this type can identified with measure-valued solutions of small variation, or, alternatively, with weak solutions to a system of conservation laws, in which the fluxes are close the the fluxes in the original equations.