Regularity of weak solutions of the nonlinear Fokker-Planck equation

We study regularity properties of weak solutions of the degenerate parabolic equation u_t + f(u)_x = K(u)_xx, where Q(u) := K′(u) > 0 for all u ≠ 0 and Q(0) = O (e.g., the porous media equation, K(u) = \u\^m-1 u, m > 1). We show that whenever the solution u is nonnegative, Q(u(·, t)) is uniformly Lipschitz continuous and K(u(·, t)) is C¹-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.