Regularity of Weak Solutions of the Nonlinear Fokker-planck Equation1

We study regularity properties of weak solutions of the degenerate parabolic equation ut + f(u)x = K(u)xx, where Q(u) := K ′(u) > 0 for all u 6= 0 and Q(0) = 0 (e.g., the porous media equation, K(u) = |u|m−1u, m > 1). We show that whenever the solution u is nonnegative, Q(u(·, t)) is uniformly Lipschitz continuous and K(u(·, t)) is C-smooth and note that these global regularity results are optimal. Weak solutions with changing sign are proved to possess a weaker regularity – K(u(·, t)), rather than Q(u(·, t)), is uniformly Lipschitz continuous. This regularity is also optimal, as demonstrated by an example due to Barenblatt and Zeldovich.