Regularity properties of generalized solutions

Abstract

The problem of analyzing the smoothness of solutions of transport equations (i.e., their regularity properties) is, probably, one of the most difficult in mathematical transport theory. One of the reasons is that many problems do not have (or “almost do not have”) the smoothness of their solutions in the classical sense. It is possible that at any point where a solution is defined, nevertheless, it has one or another singularity. Therefore the way to determine and “extract” these singularities cannot usually be realized. But another way exists. Since we deal with generalized solutions, then we have to measure the “generalized regularity” in special spaces (a number of those were offered earlier). We try to develop the second method in this chapter. Here we study the regularity properties of solutions by analyzing to which spaces with differential-difference characteristics they belong. It turns out that often the spaces of functions with bounded variation are rather convenient from this point of view, and the results obtained are close to the “exact”.