Variational properties of the kinetic solutions of scalar conservation laws

We consider the variational kinetic formulation of the Cauchy problem for a scalar conservation law due to Brenier[] and Panov[]. The solutions in this formulation are represented by a kinetic density function Y that solves a certain differential inclusion ∂tY ∈ A(Y ), in a suitable Hilbert space. In this paper we establish a sufficient “nondegeneracy” condition under which the operator A is the maximal monotone operator. When this condition is satisfied, the theory of maximal monotone operators asserts that the solutions of the above differential inclusion are “slow” solutions, i.e., the solutions for which ∂tY is the minimal norm element in the values of A(Y ). When the non-degeneracy condition doesn’t hold, the differential inclusion still has a unique solution, as was proved in Brenier[]. We show that this solution is also a slow solution. We give an example of a kinetic density Y containing a traveling wave discontinuity wave and show that Y is the a slow solution when the traveling wave moves with the classical shock speed. 0.