TY - JOUR

T1 - Regularity of weak solutions of the nonlinear Fokker-Planck equation

AU - Tassa, Tamir

PY - 1996

Y1 - 1996

N2 - 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 ≠ 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.

AB - 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 ≠ 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.

UR - http://www.scopus.com/inward/record.url?scp=0030305159&partnerID=8YFLogxK

U2 - 10.4310/MRL.1996.v3.n4.a6

DO - 10.4310/MRL.1996.v3.n4.a6

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AN - SCOPUS:0030305159

SN - 1073-2780

VL - 3

SP - 475

EP - 490

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 4

ER -