# Functional Analysis

## Nonlocal diffusion equations

Summer 2012

### Seminar

Mondays 14-16, Zeichensaal, Wegelerstr. 10

Dr. Y. Seki, Prof. J. J. L. Velázquez

### Office hours

By appointment.

### Synopsis

The goal of the seminar is to study some recent results for equations containing nonlinear diffusion operators. These operators, which can often be considered as a nonlocal generalization of the Laplace operator, appear naturally in the study of many kinetic equations (gas dynamics, coagulation models, etc.) as well as in Mathematical Biology and many problems involving stochastic processes.

The mathematical study of such equations using PDE methods is relatively recent. The theory that is arising shows that many features of the equations are similar to those of classical elliptic and parabolic problems. There are, however, also examples which indicate that the presence of nonlocal operators can produce effect that do not appear in the corresponding PDE counterpart of the problem.

One recent progess is the formulation of some problems involving nonlocal operators using boundary values of standard PDEs which are solved in spaces with higher dimensionality, see Ref. 3 below. In the seminar some of the relevant papers in this area will be studied. Another goal of the seminar is to clarify in which cases a similar behaviour of nonlocal equations and PDEs can be expected and in which cases both differ.

### Organization

Interested students please contact Yukihiro Seki. There will be a meeting in March in order to discuss details of the organization.

Participating students are expected to give a talk explaining the main results and methods of at least (part of) one of the papers.

### Literature

- X. Cabré and J.-M. Roquejoffre. Propagation de fronts dans les e'quations de Fisher-KKP avec diffusion fractionnaire. (French. English, French summary) [Front propagation in Fisher-KKP equations with fractional diffusion]
*C. R. Math. Acad. Sci. Paris*347 (2009), no. 23-24, 1361–1366. - L. Caffarelli, S. Salsa and L. Silvestre. Regularity estimates for the solution and free boundary of the obstacle problem for the fractional Laplacian.
*Invent. Math.*171 (2008), no. 2, 425–461. - L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian.
*Comm. Partial DIfferential Equations*32 (2007), no. 7-9, 1245–1260. - L. Caffarelli and L. Silvestre. Regularity results for nonlocal equations by approximation.
*Arch. Ration. Mech. Anal.*200 (2011), no. 1, 59–88. - J. Garnier. Accelerating solutions in integro-differential equations.
*SIAM J. Math. Anal.*43 (2011), no. 4, 1955–1974. - L. Silvestre. Hölder estimates for solutions of integro-differential equations like the fractional Laplace.
*Indiana Univ. Math. J.*55 (2006), no. 3, 1155–1174. - L. Silvestre. The regularity of the obstacle problem for a fractional power of the Laplace operator.
*Comm. Pure Appl. Math.*60 (2007), 67–112.