Maximal dissipation in equations of evolution (Q649771)

This paper is concerned with an evolution equation or a hyperbolic system with which an energy functional is associated that is conserved by classical solutions but not necessarily by weak solutions; in hyperbolic conservation laws, the role of energy is played by entropy. For uniqueness of a weak solution certain restrictions are imposed; these restricted weak solutions are termed dissipative as their energy/entropy is non-increasing in time. This is verified by establishing that maximal energy dissipates at the highest possible rate; proofs are presented in the context of scalar conservation laws and some special \(2 \times 2\) systems.