Page 522 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 522
NONLOCAL OPERATORS AND RELATED TOPICS (MS-55)
flights to a density function that solves the fractional diffusion equation is not guaranteed when
the jumps follow a bi-modal power-law distribution equal to zero in zero, but, as a matter of
fact, the resulting diffusive process converges to a density function that solves a double-order
fractional diffusion equation. Within this framework, self-similarity is lost. The consequence
of this loss of self-similarity is the emergence of a time-scale for realizing the large-time limit.
Such time-scale results to span from zero to infinity accordingly to the power-law displayed
by the tails of the walker’s density function. Hence, the large-time limit could not be reached
in real systems. The significance of this result is two-fold: i) with regard to the probabilistic
derivation of the fractional diffusion equation and also ii) with regard to recurrence and the
related concept of site fidelity in the framework of Lévy-like motion for wild animals.
A heat equation with memory: large-time behavior
Fernando Quirós, fernando.quiros@uam.es
Universidad Autónoma de Madrid, Spain
We study the large-time behavior in all Lp norms and in different space-time scales of solutions
to the Cauchy problem for a heat equation with a Caputo α-time derivative. The initial data are
assumed to be integrable, and, when required, to be also in Lp. A main difficulty in the analysis
comes from the singularity in space at the origin of the fundamental solution of the equation
when N > 1.
In the characteristic scale |x| tα/2, dictated by the scaling invariance of the equation, so-
lutions behave, when properly scaled to kill their decay, like M times the fundamental solution,
where M is the integral of the initial datum. In compact sets they converge to the newtonian
potential of the initial datum if N ≥ 3, one of the main novelties of the paper, and to a constant
if N = 1, 2, with a logarithmic correction in the decay rate for the critical dimension N = 2.
In intermediate scales, going to infinity more slowly than the characteristic one, solutions ap-
proach a multiple of the fundamental solution of the laplacian if N ≥ 3, and a constant in low
dimensions, again with logarithmic corrections for the critical dimension.
The asymptotic behavior in scales that go to infinity faster than the characteristic one de-
pends strongly on the behavior of the initial datum at infinity. We give results for certain initial
data with specific decays.
Joint work with C. Cortázar and N. Wolanski
On the dynamics of Ginzburg-Landau vortices on a Riemannian Manifold
Antonio Segatti, antonio.segatti@unipv.it
University of Pavia, Italy
We consider a Ginzburg-Landau energy defined for vector fields u on a 2 dimensional closed
Riemannian manifold. The Ginzburg-Landau energy depends on a small parameter ε > 0 that
favors configurations with |u| = 1. It is has been recently proved by R. Jerrard & R. Ignat
that under appropriate hypothesis, as ε → 0, a finite number of point vortices emerges. The
number and the topological charges of the vortices are related to the topology of the manifold.
Moreover, the positions of the vortices is governed by the so called renormalized energy. The
goal of the talk is to show that the vortices move, as in the two dimensional euclidean case,
520
flights to a density function that solves the fractional diffusion equation is not guaranteed when
the jumps follow a bi-modal power-law distribution equal to zero in zero, but, as a matter of
fact, the resulting diffusive process converges to a density function that solves a double-order
fractional diffusion equation. Within this framework, self-similarity is lost. The consequence
of this loss of self-similarity is the emergence of a time-scale for realizing the large-time limit.
Such time-scale results to span from zero to infinity accordingly to the power-law displayed
by the tails of the walker’s density function. Hence, the large-time limit could not be reached
in real systems. The significance of this result is two-fold: i) with regard to the probabilistic
derivation of the fractional diffusion equation and also ii) with regard to recurrence and the
related concept of site fidelity in the framework of Lévy-like motion for wild animals.
A heat equation with memory: large-time behavior
Fernando Quirós, fernando.quiros@uam.es
Universidad Autónoma de Madrid, Spain
We study the large-time behavior in all Lp norms and in different space-time scales of solutions
to the Cauchy problem for a heat equation with a Caputo α-time derivative. The initial data are
assumed to be integrable, and, when required, to be also in Lp. A main difficulty in the analysis
comes from the singularity in space at the origin of the fundamental solution of the equation
when N > 1.
In the characteristic scale |x| tα/2, dictated by the scaling invariance of the equation, so-
lutions behave, when properly scaled to kill their decay, like M times the fundamental solution,
where M is the integral of the initial datum. In compact sets they converge to the newtonian
potential of the initial datum if N ≥ 3, one of the main novelties of the paper, and to a constant
if N = 1, 2, with a logarithmic correction in the decay rate for the critical dimension N = 2.
In intermediate scales, going to infinity more slowly than the characteristic one, solutions ap-
proach a multiple of the fundamental solution of the laplacian if N ≥ 3, and a constant in low
dimensions, again with logarithmic corrections for the critical dimension.
The asymptotic behavior in scales that go to infinity faster than the characteristic one de-
pends strongly on the behavior of the initial datum at infinity. We give results for certain initial
data with specific decays.
Joint work with C. Cortázar and N. Wolanski
On the dynamics of Ginzburg-Landau vortices on a Riemannian Manifold
Antonio Segatti, antonio.segatti@unipv.it
University of Pavia, Italy
We consider a Ginzburg-Landau energy defined for vector fields u on a 2 dimensional closed
Riemannian manifold. The Ginzburg-Landau energy depends on a small parameter ε > 0 that
favors configurations with |u| = 1. It is has been recently proved by R. Jerrard & R. Ignat
that under appropriate hypothesis, as ε → 0, a finite number of point vortices emerges. The
number and the topological charges of the vortices are related to the topology of the manifold.
Moreover, the positions of the vortices is governed by the so called renormalized energy. The
goal of the talk is to show that the vortices move, as in the two dimensional euclidean case,
520
