Applications of Generalized Functions to Shocks and Discrete Models

Abstract

Generalized functions, also called distributions, have proved useful in many problems related to physics even before their rigorous and general definition. Generalized functions are also a powerful tool to formulate in a unified manner fundamental physical laws. We shall give here two general examples of such an application of generalized functions. The first is to the writing of the equations satisfied by shock waves : we establish in particular the algebraic conditions satisfied across a gravitational shock wave and the propagation of the corresponding discontinuities. In the second example we use a representation of the distribution function of kinetic theory by a sum of discrete measures to find the boltzmann equation for a model with a finite number of velocities, together with propagation equations satisfied by these velocities.