Regularity in Sobolev spaces for the fast diffusion and the porous medium equation

Abstract The degenerate parabolic equation u t = Δ ( | u | m − 1 u ) , m > 0 is considered in a cylinder Ω × ( 0 , T ) under homogeneous Dirichlet boundary values. The regularity of weak solutions for the fast diffusion equation ( m 1 ) and the porous medium equation ( m > 1 ) are investigated. Regularity of u and | u | m − 1 u in weighted Sobolev spaces and in fractional order Nikolskii spaces are proved.

[1]  Avner Friedman,et al.  Regularity of the free boundary of a gas flow in an n-dimensional porous medium. , 1980 .

[2]  J. Vázquez,et al.  Lipschitz Continuity o f Solutions and Interfaces of the N-Dimensional Porous Medium Equation , 1985 .

[3]  J. Vázquez,et al.  Regularity of solutions and interfaces of a generalized porous medium equation inRN , 1991 .

[4]  J. Vázquez.  An Introduction to the Mathematical Theory of the Porous Medium Equation , 1992 .

[5]  L. Caffarelli,et al.  C1,α regularity of the free boundary for the N-dimensional porous media equation , 1990 .

[6]  Carsten Ebmeyer.  A Non-Degeneracy Property for a Class of Degenerate Parabolic Equations , 1996 .

[7]  Michel C. Delfour,et al.  Shape optimization and free boundaries , 1992 .

[8]  P. Daskalopoulos,et al.  Regularity of the free boundary for the porous medium equation , 1998 .

[9]  J. M. Urbano,et al.  Regularity in Sobolev spaces for doubly nonlinear parabolic equations , 2003 .

[10]  Felix Otto,et al.  L1-Contraction and Uniqueness for Quasilinear Elliptic–Parabolic Equations , 1996 .

[11]  Sigurd B. Angenent,et al.  The focusing problem for the radially symmetric porous medium equation , 1995 .

[12]  Stephan Luckhaus,et al.  Quasilinear elliptic-parabolic differential equations , 1983 .

[13]  E. Giusti.  Direct methods in the calculus of variations , 2003 .

[14]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .

[15]  Paul E. Sacks,et al.  The initial and boundary value problem for a class of degenerate parabolic equations , 1983 .

[16]  J. M. Urbano,et al.  The smoothing property for a class of doubly nonlinear parabolic equations , 2005 .

[17]  James G. Berryman,et al.  Stability of the separable solution for fast diffusion , 1980 .

[18]  Ya-Zhe Chen,et al.  Boundary estimates for solutions of nonlinear degenerate parabolic systems. , 1989 .