On uniqueness techniques for degenerate convection-diffusion problems

We survey recent developments and give some new results concerning uniqueness of weak and renormalized solutions for degenerate parabolic problems of the form ut − div (a0(∇w) + F (w)) = f , u ∈ β(w) for a maximal monotone graph β, a Leray-Lions type nonlinearity a0, a continuous convection flux F , and an initial condition u|t=0 = u0. The main difficulty lies in taking boundary conditions into account. Here we consider Dirichlet or Neumann boundary conditions or the case of the problem in the whole space. We avoid the degeneracy that could make the problem hyperbolic in some regions; yet our starting point is the notion of entropy solution, notion that underlies the theory of general hyperbolic-parabolic-elliptic problems. Thus, we focus on techniques that are compatible with hyperbolic degeneracy, but here they serve to treat only the “parabolic-elliptic aspects”. We revisit the derivation of entropy inequalities inside the domain and up to the boundary; technique of “going to the boundary” in the Kato inequality for comparison of two solutions; uniqueness for renormalized solutions obtained via reduction to weak solutions. On several occasions, the results are achieved thanks to the notion of integral solution coming from the nonlinear semigroup theory.