Kinetic formulation for a parabolic conservation law. Application to homogenization

We derive a kinetic formulation for the parabolic scalar conservation law partial derivative(t)u + div(y)A(y,u) - Delta(y)u = 0. This allows us to de. ne a weaker notion of solutions in L-1, which is enough to recover the L-1 contraction principle. We also apply this kinetic formulation to a homogenization problem studied in a previous paper; namely, we prove that the kinetic solution of partial derivative(t)u(epsilon) + div(x)A(x/epsilon, u(epsilon)) - epsilon Delta(x)u(epsilon) = 0 behaves in L-loc(1) as v (x/epsilon,(u) over bar( t, x)), where v is the solution of a cell problem and (u) over bar the solution of the homogenized problem.