A note on strong solutions to the variational kinetic equation for scalar conservation laws

摘要

We consider a variational kinetic formulation for weak, entropy solutions of scalar conservation laws due to Brenier. The solutions in this formulation are represented by a kinetic density function Y that solves a differential inclusion ?_tY ? —A(Y) = —?_vf? ▽_xY — ?I_K(Y)， where I_K is the indicator function of a closed， convex cone K. Under a certain "non-degeneracy" condition we determine a maximal monotone extension of A and use it to prove the existence of strong and weak solutions of the differential inclusion for a general, possibly degenerate, flux ?_υf(v). Furthermore， we discuss several properties of strong solutions.