Regularizing effects for multidimensional scalar conservation laws
It has been known for a long time that multidimensional scalar conservation laws possess a unique weak entropy solution in \(L^1\cap L^{\infty}\) which however need not be of bounded variation. The present paper deals with the important question whether the nonlinearity of the flux acts as a regularizing effect. For a flux which in each direction is either strictly convex or possesses at most one nondegenerate inflection point, the author is able to derive estimates for some quantity \(\Theta\) that is obtained via a scattering procedure. These estimates are sufficient to derive a number of interesting conclusions, e.g. BV- and Sobolev estimates for some averaged quantities or decay estimates for spatially periodic solutions.
BV- and Sobolev estimates for some averaged quantities
decay estimates for spatially periodic solutions
scientific article; zbMATH DE number 1546941
scientific article