Regularity of weak solutions of the nonlinear fokker-planck equation

Abstract

Abstract. We study regularity properties of weak solutions of the degenerate parabolic equation. We show that whenever the solution u is nonnegative, Q(u (·, t)) is uniformly Lipschitz continuous and K(u(·, 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(·, 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.

Velocity averaging, kinetic formulations, and regularizing effects in quasi‐linear PDEs

“…The velocity averaging yields W s,1 -regularity of order 2/(n + 2) and, as in the hyperbolic case, it does not recover the optimal Hölder continuity in this case, e.g. [DiB93,Ta96]. In fact, the kinetic arguments do not yield continuity.…”

Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws

“…In particular, in [15] the concept of bounded-flux solutions was put forward; it is clear that the L ∞ bound on F[u] investigated in [15] implies equi-integrability of F[u] (•, x) x<0 , so that bounded-flux solutions are in particular weakly trace-regular. Some one-dimensional regularity results for the flux F[u] can also be found in [35,36,20]. The techniques of these works are limited to the onedimensional situation, and the justification of weak trace-regularity in the general multi-dimensional case requires new ideas.…”