Page 521 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 521
NONLOCAL OPERATORS AND RELATED TOPICS (MS-55)
Uniqueness issues for nonlinear diffusions
Gabriele Grillo, gabriele.grillo@polimi.it
Politecnico di Milano, Italy
I discuss uniqueness, and nonuniqueness, of solutions to classes of nonlinear diffusions in dif-
ferent contexts, that include slow and fast diffusions on general manifolds and the fractional
porous medium equation on the hyperbolic space.
Blow-up analysis of conformal metrics with prescribed curvatures on the
disk
María Medina, maria.medina@uam.es
Universidad Autónoma de Madrid, Spain
We will establish necessary conditions on the blow-up points of conformal metrics of the disk
with prescribed Gaussian and geodesic curvatures, where a non local restriction will appear.
Conversely, given a point satisfying these conditions, we will construct an explicit family of
approximating solutions that explode at such a point. These results are contained in several
works in collaboration with A. Jevnikar, R. López-Soriano and D. Ruiz, and with L. Battaglia
and A. Pistoia.
Uniqueness of very weak solutions for a fractional nonlinear diffusion
Matteo Muratori, matteo.muratori@polimi.it
Politecnico di Milano, Italy
In this talk I will present some recent results, obtained in collaboration with G. Grillo and F.
Punzo, dealing with existence and uniqueness of (distributional) bounded solutions to a frac-
tional parabolic equation of porous medium type, posed in the whole Euclidean space. Exis-
tence is established by means of a classical approximation scheme, while uniqueness is much
more involved and requires a careful analysis of a suitable dual problem.
Should I stay or should I go? Zero-size jumps in random walks for Lévy
flights
Gianni Pagnini, gpagnini@bcamath.org
BCAM - Basque Center for Applied Mathematics, Spain
Motivated by the fact that, in the literature dedicated to random walks for anomalous diffusion,
it is disregarded if the walker does not move in the majority of the iterations because the most
frequent jump-size is zero (i.e., the jump-size distribution is unimodal with mode located in
zero) or, in opposition, if the walker always moves because the jumps with zero-size never
occur (i.e., the jump-size distribution is bi-modal and equal to zero in zero), we provide an
example in which indeed the shape of the jump-distribution plays a role. In particular, we show
that the convergence of Markovian continuous-time random walk (CTRW) models for Lévy
519
Uniqueness issues for nonlinear diffusions
Gabriele Grillo, gabriele.grillo@polimi.it
Politecnico di Milano, Italy
I discuss uniqueness, and nonuniqueness, of solutions to classes of nonlinear diffusions in dif-
ferent contexts, that include slow and fast diffusions on general manifolds and the fractional
porous medium equation on the hyperbolic space.
Blow-up analysis of conformal metrics with prescribed curvatures on the
disk
María Medina, maria.medina@uam.es
Universidad Autónoma de Madrid, Spain
We will establish necessary conditions on the blow-up points of conformal metrics of the disk
with prescribed Gaussian and geodesic curvatures, where a non local restriction will appear.
Conversely, given a point satisfying these conditions, we will construct an explicit family of
approximating solutions that explode at such a point. These results are contained in several
works in collaboration with A. Jevnikar, R. López-Soriano and D. Ruiz, and with L. Battaglia
and A. Pistoia.
Uniqueness of very weak solutions for a fractional nonlinear diffusion
Matteo Muratori, matteo.muratori@polimi.it
Politecnico di Milano, Italy
In this talk I will present some recent results, obtained in collaboration with G. Grillo and F.
Punzo, dealing with existence and uniqueness of (distributional) bounded solutions to a frac-
tional parabolic equation of porous medium type, posed in the whole Euclidean space. Exis-
tence is established by means of a classical approximation scheme, while uniqueness is much
more involved and requires a careful analysis of a suitable dual problem.
Should I stay or should I go? Zero-size jumps in random walks for Lévy
flights
Gianni Pagnini, gpagnini@bcamath.org
BCAM - Basque Center for Applied Mathematics, Spain
Motivated by the fact that, in the literature dedicated to random walks for anomalous diffusion,
it is disregarded if the walker does not move in the majority of the iterations because the most
frequent jump-size is zero (i.e., the jump-size distribution is unimodal with mode located in
zero) or, in opposition, if the walker always moves because the jumps with zero-size never
occur (i.e., the jump-size distribution is bi-modal and equal to zero in zero), we provide an
example in which indeed the shape of the jump-distribution plays a role. In particular, we show
that the convergence of Markovian continuous-time random walk (CTRW) models for Lévy
519
