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 $u \neq 0$ and $Q(0)=0$ (e.g., the porous media equation, $K(u)=|u|^{m-1}u$, $m >1$). We show that whenever the solution $u$ is nonnegative, $Q(u{(\cdot,t)})$ is uniformly Lipschitz continuous and $K(u{(\cdot,t)})$ is $C^1$-smooth and note that these global regularity results are optimal. Weak solutions with changing sign are proved to possess a weaker regularity – $K(u{(\cdot,t)})$, rather than $Q(u{(\cdot,t)})$, is uniformly Lipschitz continuous. This regularity is also optimal, as demonstrated by an example due to Barenblatt and Zeldovich.

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Published 1 January 1996