Tuesday, May 30 |
07:00 - 09:00 |
Breakfast (Restaurant - Hotel Granada Center) |
09:00 - 09:45 |
Giuseppe Savare: A Lagrangian approach to dissipative evolutions of probability measures. ↓ While the case of Wasserstein gradient flows generated by displacement-convex functionals is well understood, in part due to the possibility of constructing the evolution using the JKO/Minimizing Movement method, the general picture of dissipative evolutions (i.e., when the driving probability vector field is not generated by the Wasserstein gradient of some functional) appears more complicated.
The theory seems to present two very different situations, depending on whether the probability vector field is dissipative along any coupling of measures (which we will call totally dissipative) or only along the optimal ones, that minimize the Wasserstein distance.
In this talk we will focus on the first case and show that it is possible to construct a fully satisfying Lagrangian theory based on classical results in Hilbert spaces; this leads to new interesting results also in the case of gradient flows. Perhaps more surprisingly, we will also show that a dissipative probability vector field defined everywhere and continuous is always totally dissipative. As a consequence we can see that in the Wasserstein setting it is impossible to approximate a dissipative (but not totally dissipative) probability vector field by continuous ones. This fact clarifies why the case of non-totally dissipative probability vector fields needs to be investigated on an ad hoc basis.
(In collaboration with Giulia Cavagnari and Giacomo Sodini) (Main Meeting Room - Calle Rector López Argüeta) |
09:45 - 10:30 |
Alexander Mielke: EDP-convergence for gradient systems and Non-Equilibrium Steady States ↓ The convergence of gradients systems (X,Eε,Rε) in the sense of the “Energy-Dissipation Principle”
is discussed. Generalized gradient systems on a Banach space X are given by a driving
functional Eε (e.g. free energy or relative entropy) and a dissipation potential Rε.
In analogy to Γ convergence for functionals, one defines EDP-convergence by looking at functionals
only, avoiding the usage of the solutions of the gradient-flow equation; however, convergence
of solutions is implied by EDP-convergence. The major feature of EDP convergence is that the
effective dissipation Reff may depend on the microscopic information in the energy encoded in
Eε−Eeff, and quadratic dissipation potentials Rε may converge to non-quadratic Reff.
The theory is applied to slow-fast gradient systems which arise in chemical reaction-diffusion systems
where the fast subsystem moves along a family of Non-Equilibrium Steady States determining
the effective slow dynamics. For a scalar diffusion equation with a thin membrane region with low
mobility we show that the arising transmission is generated by a nonlinear kinetic relation.
The research is partially joint work with Th. Frenzel, M. Peletier, and A. Stephan. (Main Meeting Room - Calle Rector López Argüeta) |
10:30 - 11:00 |
Coffee Break (Main Meeting Room - Calle Rector López Argüeta) |
11:15 - 11:45 |
David Gomez Castro: Newtonian vortex equations with non-linear mobility ↓ In this talk we will consider conservation equations of the form
{ut=div(uα∇v)−Δv=u
This system can equivalently be written as one equation ut=div(uα∇W∗u) where W is the Newtonian potential. For linear mobility, α=1, the equation and some variations have been proposed as a model for superconductivity or superfluidity. In that case the theory leads to uniqueness of bounded weak solutions having the property of compact space support, and in particular there is a special solution in the form of a disk vortex showing a discontinuous leading front.
The aim of the talk is to discuss the cases α∈(0,1) and α>1 which, as for the Porous Medium Equation, exhibit very different behaviours. First, we discuss self-similar solutions. Then, we restrict the analysis to radial solutions and construct solutions by the method of characteristics. We introduce the mass function, which solves an unusual variation of Burger's equation, and plays an important role in the analysis. We show well-posedness in the sense of viscosity solutions and construct a convergent finite-difference numerical schemes.
For sublinear mobility 0<α<1 nonnegative solutions recover positivity everywhere, and moreover display a fat tail at infinity. The model acts in many ways as a regularization of the previous one.
For superlinear mobility α>1 we show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case α=1. We show a waiting time phenomena.
The talk presents joint work with José A. Carrillo (U. Oxford) and Juan Luis Vázquez (U. Autónoma de Madrid). (Main Meeting Room - Calle Rector López Argüeta) |
11:45 - 12:15 |
Emanuela Radici: Stability of quasi-entropy solutions for nonlocal scalar conservation laws ↓ In this talk we consider the stability of entropy solutions for nonlinear scalar conservation laws with
respect to perturbations of the initial datum, the space-time dependent flux and the entropy inequalities.
Such general stability theorem has several primary benefits, among which the possibility to study
conditional existence and uniqueness of entropy solutions for evolutions whose flux depends non-locally
on the solution itself. In addition, the relaxation of the entropy inequality allows to treat approximate
solutions arising from various numerical schemes and to derive their corresponding rates of convergence.
In particular, we discuss the case of a recent deterministic particle method introduced to study nonlocal
transport models with congestion and we obtain for the first time the (sharp) convergence rate. This is
a joint work with E. Marconi and F. Stra. (Main Meeting Room - Calle Rector López Argüeta) |
12:15 - 13:00 |
Adriana Garroni: Homogenisation of non local phase transition models and application to crystal plasticity ↓ I will present a recent result in collaboration with Sergio Conti and Stefan Muller. We analyse a nonlocal phase transition model in dimension two known, in the literature of crystal defects (dislocations), as the Nabarro Peierls model. The phase field represents the slip of a crystal (i.e. a lattice) along a given slip plane, the multi-well potential penalises slips which are not lattice preserving, while the singular perturbation with the regularised effect is a non local fractional norm representing the elastic distortion due to incompatibilities of the lattice. Under an appropriate scaling we study the Gamma limit of this energy which results in a macroscopic model for elasto-plastic deformations. (Main Meeting Room - Calle Rector López Argüeta) |
13:00 - 13:15 |
Group Photo (Main Meeting Room - Calle Rector López Argüeta) |
13:15 - 14:45 |
Lunch (Restaurant - Hotel Granada Center) |
14:45 - 15:30 |
Simone Di Marino: On curvature and Five Gradients Inequality on Manifolds ↓ Introduced almost ten years ago, the five gradients inequality has been used to provide estimates on sobolev norms of minimizers involving the Wasserstein distance. In conjunction with the JKO scheme, this inequality can grant compactness for the minimizing movement scheme. We investigate the geometric and functional meaning of the five gradients inequality in two generalizations. In the setting of Lie groups the proof naturally suggest that it is a second order optimality condition for the Kantorovich potentials, while in general compact Riemannian manifolds the curvature plays a role. This is a joint work with Simone Murro and Emanuela Radici. (Main Meeting Room - Calle Rector López Argüeta) |
15:30 - 16:00 |
Stephan Wojtowytsch: Convergence to an invariant distribution for stochastic gradient descent ↓ Stochastic gradient descent is one of the most common optimization algorithms in deep learning. In these applications, the noise intensity generally scales with the objective function to be minimized and the covariance matrix of the gradient estimators has low rank. We discuss impacts of low rank and noise degeneracy in separate models. In a toy model capturing only noise degeneracy, we identify the invariant distribution and discuss convergence in a continuous time model both for `ML type noise' and classical homogeneous and isotropic noise. Our methods are associated to fast diffusion equations and related Poincaré inequalities. (Main Meeting Room - Calle Rector López Argüeta) |
16:00 - 16:30 |
Coffee Break (Main Meeting Room - Calle Rector López Argüeta) |
16:30 - 17:00 |
Gissell Estrada-Rodriguez: Diffusion and superdiffusion in complex domains: Introduction of a networks of subdomains ↓ In this talk I will introduce the concept of metaplexes and the dynamical systems on them. A metaplex combines the internal structure of the
entities of a complex system with the discrete interconnectivity of these
entities in a global topology. We focus here on the study of diffusive processes on metaplexes, both model and real-world examples. We provide
theoretical and computational evidence pointing out the role of the endoand exo-structure of the metaplexes in their global dynamics, including the
role played by the size of the nodes, the location, strength and range of
the coupling between nodes. We show that the internal structure of brain
regions (corresponding to the nodes of the network) in the macaque visual
cortex metaplex dominates almost completely the global dynamics. On the
other hand, in the linear metaplex chain and the landscape metaplex the
diffusion dynamics display a combination of the endo- and exo-dynamics,
which we explain analytically and in numerical results. Metaplexes are
expected to facilitate the understanding of complex systems in an integrative way, which combine dynamical processes inside the nodes and between
them. (Main Meeting Room - Calle Rector López Argüeta) |
17:00 - 17:30 |
Nikita Simonov: Stability in Gagliardo-Nirenberg-Sobolev inequalities ↓ In some functional inequalities, best constants and minimizers are known. The next question is stability: suppose that a function "almost attains the equality", in which sense it is close to one of the minimizers? In this lecture, I will address a recent result on the quantitative stability of a subfamily of Gagliardo-Nirengerg-Sobolev inequalities. The approach is based on the entropy method for the fast diffusion equation and allows us to obtain completely constructive estimates. The results are based on joint work with M. Bonforte, J. Dolbeault, and B. Nazaret. (Front Desk - Hotel Granada Center) |
17:30 - 18:00 |
Alexandre Rege: Propagation of velocity moments for the magnetized Vlasov–Poisson system ↓ The evolution of a cloud of charged particles subjected to an external magnetic field can
be described by the magnetized Vlasov–Poisson system. We study the existence of classical solutions to this system by using the velocity moment method introduced by Lions and
Perthame. In the case of a constant external magnetic field, one can obtain an explicit representation formula for the charge density which allows us to show propagation of velocity
moments directly using an Eulerian approach. Conversely, for a time-dependent and position independent magnetic field this method breaks down and we need to use a Lagrangian
approach, where the characteristic flow of the system is carefully studied, in order to show
our propagation result. A common interesting feature in both works is the use of a recurrence argument depending on the cyclotron frequency. If time permits, we will also discuss
uniqueness results for our system. (Front Desk - Hotel Granada Center) |
19:00 - 21:00 |
Dinner (Restaurant - Hotel Granada Center) |