A vanishing dynamic capillarity limit equation with discontinuous flux

We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the dynamics capillarity equation {partial derivative(t)u(epsilon,delta)+divf(epsilon,delta)(x,u(epsilon,delta)) = epsilon Delta u(epsilon,delta)+delta(epsilon)partial derivative(t)Delta u(epsilon,delta), x is an element of M, t >= 0 u|(t=0)=u(0)(x). Here, f(epsilon,delta) and u(0) are smooth functions while epsilon and delta = delta(epsilon) are fixed constants. Assuming f(epsilon,delta) -> f is an element of L-p(R-d x R; R-d) for some 1 < p < infinity, strongly as epsilon -> 0, we prove that, under an appropriate relationship between epsilon and delta(epsilon) depending on the regularity of the flux f, the sequence of solutions (u(epsilon,delta)) strongly converges in L-loc(1) loc (R+ x R-d) toward a solution to the conservation law partial derivative(t)u + divf(x, u) = 0. The main tools employed in the proof are the Leray-Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second. These results have the potential to generate a stable semigroup of solutions to the underlying scalar conservation laws different from the Kruzhkov entropy solutions concept.