Seminario doble de ecuaciones en derivadas parciales 

Fecha y hora: viernes 4 de marzo, de 12:30 a 14.

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Jorgen Endal (Universidad Autónoma de Madrid)

Título: Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities.

Abstract: We will consider the Cauchy problem for generalized porous medium type equations $partial_t u + (-L)[u^m]=0$, with $m>1$, and investigate when integrable data produce bounded solutions. The diffusion operator $L$ belongs to a general class of local and nonlocal operators, and we will see how different assumptions on the Green function of the operator are related to smoothing effects. On one hand, we show that if the linear case ($m=1$) enjoys smoothing effects, also the nonlinear will. On the other hand, we provide an example in which only the nonlinear diffusion enjoys smoothing effects. This is due to the convex nonlinearity. A comparison between the approach based on the Moser iteration and the approach through Green functions is commented on.

Following the ideas of Nash, we also explore the connections between smoothing effects, the validity of Gagliardo-Nirenberg-Sobolev (and Nash) inequalities, heat kernel estimates, and Green function estimates.

This is a work in progress with M. Bonforte.

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Xavier Ros-Oton (Universitat de Barcelona)

Título: The singular set in the Stefan problem.

Abstract: The Stefan problem, dating back to the XIXth century, is probably the most classical and important free boundary problem. The regularity of free boundaries in the Stefan problem was developed in the groundbreaking paper (Caffarelli, Acta Math. 1977). The main result therein establishes that the free boundary is $C^infty$ in space and time, outside a certain set of singular points.

The fine understanding of singularities is of central importance in a number of areas related to nonlinear PDEs and Geometric Analysis. In particular, a major question in such a context is to establish estimates for the size of the singular set. The goal of this talk is to present some new results in this direction for the Stefan problem. This is a joint work with A. Figalli and J. Serra.