Departamento de Matemáticas UAM

Difusión no lineal en Madrid

Seminario conjunto de las universidades: UAM, UC3M, UCM, UPM y URJC

Recent results on nonlinear aggregation-diusion equations:
radial symmetry and long time asymptotics

Prof. Bruno Volzone

Universita degli Studi di Napoli Parthenope"

Viernes 9 de febrero de 2018, 12:30

Seminario del Departamento de Matematicas, aula C-17-520, UAM

Resumen
One of the archetypical aggregation-diusion models is the so-called classical parabolic-elliptic
Patlak-Keller-Segel (PKS for short) model. This model was classically introduced as the simplest
description for chemotatic bacteria movement in which linear diusion tendency to spread ghts
the attraction due to the logarithmic kernel interaction in two dimensions. For this model there
is a well-dened critical mass. In fact, here a clear dichotomy arises: if the total mass of the
system is less than the critical mass, then the long time asymptotics are described by a selfsimilar
solution, while for a mass larger than the critical one, there is nite time blow-up. In
this talk we will show some recent results concerning the symmetry of the stationary states for
a nonlinear variant of the PKS model, of the form
(1) @t =
m + r
(r(W  ));
being W 2 C1(Rd n f0g), d  2, a suitable aggregation kernel, in the assumptions of dominated
diusion, i.e. when m > 2?2=d. In particular, if W represents the classical logarithmic kernel in
the bidimensional case, we will show that there exists a unique stationary state for the model (1)
and it coincides, according to one of the main results in the work [1], with the global minimizer of
the free energy functional associated to (1). In the case d = 2 we will also show how such steady
state coincides with the aymptotic prole of (1). Finally, we will also discuss some recent results
concerning the model (1) with a Riesz potential aggregation, namely when W(x) = cd;sjxj2s?d
for s 2 (0; d=2), again in the diusion dominated regime, i.e. for m > 2 ? (2s)=d. In particular,
all stationary states of the model are shown to be radially symmetric decreasing and that global
minimizers of the associated free energy are compactly supported, uniformly bounded, Holder
regular, and smooth inside their support. These results are objects of the joint works [2], [3].

Referencias
[1] J. A. Carrillo, D. Castorina, B. Volzone, Ground States for Diusion Dominated Free Ener-
gies with Logarithmic Interaction, SIAM J. Math. Anal. 47 (2015), no. 1, 1{25.
[2] J. A. Carrillo, S. Hittmeir, B. Volzone, Y. Yao, Nonlinear Aggregation-Diusion Equations:
Radial Symmetry and Long Time Asymptotics,, arXiv:1603.07767.
[3] J. A. Carrillo, F. Hoffmann, E. Mainini, B. Volzone, Ground States in the Diusion-
Dominated Regime,, arXiv:1705.03519.

Organizado por los proyectos: MTM2014-52240-P, MTM2014-53037-P y Fundacion BBVA para
Investigadores y Creadores Culturales 2016.
Comite organizador: Matteo Bonforte, Mar Gonzalez, Arturo de Pablo y Fernando Quiros.