Kinetic Formulation for a Parabolic Conservation Law. Application to Homogenization

We derive a kinetic formulation for the parabolic scalar conservation law ∂tu + divyA(y, u) − ∆yu = 0. This allows us to define a weaker notion of solutions in L1, which is enough to recover the L1 contraction principle. We also apply this kinetic formulation to a homogenization problem studied in a previous paper; namely, we prove that the kinetic solution uε of ∂tu+divxA (x/ε, u)−ε∆xu = 0 behaves in Lloc as v (x/ε, ū(t, x)), where v is the solution of a cell problem and ū the solution of the homogenized problem.