Page 231 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 231
ARIATIONAL AND EVOLUTIONARY MODELS INVOLVING LOCAL/NONLOCAL
INTERACTIONS (MS-58)
The antiferromagnetic XY model
Marco Cicalese, cicalese@ma.tum.de
Technical University of Munich, Germany
We introduce the antiferromagnetic XY system on the triangular lattice, a spin model driven by
an energy functional that favours anti-alignment on each pair of interacting spins. We start re-
calling the main results concerning the variational discrete-to-continuum analysis at the surface
scaling at which chirality transitions take place. Then, we focus on the vortex scaling: we intro-
duce a notion of discrete vorticity and explain how to gain compactness for this order parameter
and how to prove a Gamma-limit result of the energy functionals as the lattice spacing goes to
zero.
Joint work with A. Bach, L. Kreutz and G. Orlando.
Deterministic particle approximation of aggregation-diffusion equations
on unbounded domains
Sara Daneri, sara.daneri@gssi.it
Gran Sasso Science Institute, Italy
Coauthors: Eris Runa, Emanuela Radici
We consider a one-dimensional aggregation-diffusion equation, which is the gradient flow in
the Wasserstein space of a functional with competing attractive-repulsive interactions. We prove
that the fully deterministic particle approximations with piecewise constant densities introduced
by Di Francesco and Rosini starting from general bounded initial densities converge strongly
in L1 to bounded weak solutions of the PDE. In particular, the result is achieved in unbounded
domains and for arbitrary nonnegative bounded initial densities, thus extending the results by
Gosse-Toscani and Matthes-Osberger (in which a no-vacuum condition is required) and giving
an alternative approach to the one proposed by Carrillo-Craig-Patacchini in the one-dimensional
case, including also subquadratic and superquadratic diffusions.
Deterministic many-particle limit for a system of interaction equations
driven by Newtonian potentials
Marco Di Francesco, marco.difrancesco@univaq.it
University of L’Aquila, Italy
We consider a discrete particle system of two species coupled through nonlocal interactions
driven by the one-dimensional Newtonian potential, with repulsive self-interaction and attrac-
tive cross-interaction. After providing a suitable existence theory in a finite-dimensional frame-
work, we explore the behaviour of the particle system in case of collisions and analyse the be-
haviour of the solutions with initial data featuring particle clusters. Subsequently, we prove that
the empirical measure associated to the particle system converges to the unique 2-Wasserstein
gradient flow solution of a system of two partial differential equations (PDEs) with nonlocal
interaction terms in a proper measure sense. The latter result uses uniform estimates of the
Lm-norms of a piecewise constant reconstruction of the density using the particle trajectories.
The results are a joint work with A. Esposito and M. Schmidtchen.
229
INTERACTIONS (MS-58)
The antiferromagnetic XY model
Marco Cicalese, cicalese@ma.tum.de
Technical University of Munich, Germany
We introduce the antiferromagnetic XY system on the triangular lattice, a spin model driven by
an energy functional that favours anti-alignment on each pair of interacting spins. We start re-
calling the main results concerning the variational discrete-to-continuum analysis at the surface
scaling at which chirality transitions take place. Then, we focus on the vortex scaling: we intro-
duce a notion of discrete vorticity and explain how to gain compactness for this order parameter
and how to prove a Gamma-limit result of the energy functionals as the lattice spacing goes to
zero.
Joint work with A. Bach, L. Kreutz and G. Orlando.
Deterministic particle approximation of aggregation-diffusion equations
on unbounded domains
Sara Daneri, sara.daneri@gssi.it
Gran Sasso Science Institute, Italy
Coauthors: Eris Runa, Emanuela Radici
We consider a one-dimensional aggregation-diffusion equation, which is the gradient flow in
the Wasserstein space of a functional with competing attractive-repulsive interactions. We prove
that the fully deterministic particle approximations with piecewise constant densities introduced
by Di Francesco and Rosini starting from general bounded initial densities converge strongly
in L1 to bounded weak solutions of the PDE. In particular, the result is achieved in unbounded
domains and for arbitrary nonnegative bounded initial densities, thus extending the results by
Gosse-Toscani and Matthes-Osberger (in which a no-vacuum condition is required) and giving
an alternative approach to the one proposed by Carrillo-Craig-Patacchini in the one-dimensional
case, including also subquadratic and superquadratic diffusions.
Deterministic many-particle limit for a system of interaction equations
driven by Newtonian potentials
Marco Di Francesco, marco.difrancesco@univaq.it
University of L’Aquila, Italy
We consider a discrete particle system of two species coupled through nonlocal interactions
driven by the one-dimensional Newtonian potential, with repulsive self-interaction and attrac-
tive cross-interaction. After providing a suitable existence theory in a finite-dimensional frame-
work, we explore the behaviour of the particle system in case of collisions and analyse the be-
haviour of the solutions with initial data featuring particle clusters. Subsequently, we prove that
the empirical measure associated to the particle system converges to the unique 2-Wasserstein
gradient flow solution of a system of two partial differential equations (PDEs) with nonlocal
interaction terms in a proper measure sense. The latter result uses uniform estimates of the
Lm-norms of a piecewise constant reconstruction of the density using the particle trajectories.
The results are a joint work with A. Esposito and M. Schmidtchen.
229
