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Regularity of solutions of the fractional porous medium flow

Luis Caffarelli, Fernando Soria, Juan Luis Vázquez (2013)

Journal of the European Mathematical Society

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is u t = ∇ · ( u ∇ ( - Δ ) - s u ) , 0 < s < 1 . The problem is posed in { x ∈ ℝ n , t ∈ ℝ } with nonnegative initial data u ( x , 0 ) that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and C α regularity of such weak solutions. Finally, we extend the existence...

Regularity of stable solutions of p -Laplace equations through geometric Sobolev type inequalities

Daniele Castorina, Manel Sanchón (2015)

Journal of the European Mathematical Society

We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of – Δ p u = g ( u ) in a smooth bounded domain Ω ⊂ ℝ n . In particular, we obtain new L r and W 1 , r bounds for the extremal solution u ☆ when the domain is strictly convex. More precisely, we prove that u ☆ ∈ L ∞ ( Ω ) if n ≤ p + 2 and u ☆ ∈ L n p n - p - 2 ( Ω ) ∩ W 0 1 , p ( Ω ) if n > p + 2 .

Regularity problem for one class of nonlinear parabolic systems with non-smooth in time principal matrices

Arina A. Arkhipova, Jana Stará (2019)

Commentationes Mathematicae Universitatis Carolinae

Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called A ( t ) -caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems.

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