Regularity of solutions to the perturbed conservation laws

Abstract

A kind of regularity for the mild solution of perturbed conservation laws is proposed. This regularity is described in term of variations measured in the L¹-norm. A dissipativity condition from the semigroup approach is used to show that the mild solution stays within a class of bounded variation in this sense of regularity. This shows that this class of functions is an invariant of the semigroup. The same analysis carries over to the periodic problem. The class of boundedL¹-variation functions used here can be normed to give a Banach space structure. It also has an analogue with the space of Lipschitz functions

Keywords:

• Conservation laws
• perturbed conservation law
• balance law
• regularity of solutions
• m-dissipative operato operator
• L^P-variation.

AMS:

• 35L65
• 47H20