Mathematical Analysis Seminars @ DISIM

Title: Computing Branched Optimal Transport with an application for tree branches

Abstract: Due to the lack of convexity of the ramified transportation cost, the computation of optimal irrigation patterns runs into severe difficulties. In the current literature, several methods were developed, ranging from a recursive construction with full Steiner topology to Modica-Mortola approximations of 1-rectifiable currents. 

In the present talk, we introduce an alternative approach, based on a mollification of the irrigation cost expressed in Lagrangian variables. Furthermore, we demonstrate how this mollified approach can address more complex problems, such as determining optimal shapes for tree branches.

Title: On leapfrogging vortex rings in ideal fluids

Abstract: We consider the dynamics of several vortex rings in an ideal fluid governed by the 3D incompressible Euler equations. While the self-induced motion of a single vortex ring amounts to a translation along its symmetry axis,  Helmholtz conjectured that two coaxial vortex rings may exhibit the so-called leapfrogging dynamics. Specifically, the vortex rings display a periodic motion in which they repeatedly pass through each other by shrinking and widening respectively due to mutual interactions. The phenomenon is well observed numerically and experimentally. 

We rigorously justify this phenomenon for a general class of initial data corresponding to sharply concentrated coaxial vortex rings on a logarithmic time scale. The key mathematical difficulty consists in dealing with singular interactions of the vortex rings in the suitable regime for the leapfrogging to occur.  Our method is based on new and improved weak and strong localization estimates for the vorticity that allow for the motion law to be derived. 

Joint work with M. Donati (Grenoble), C. Lacave (Chambery) and E. Miot (Grenoble)

Title: Managing uncertainty in control systems: averaged and uniform ensemble optimal control

Abstract: In this talk, we focus on problems related to the simultaneous optimal control of ensembles of dynamical systems (ODEs). These questions arise naturally in several situations in Applied Mathematics, for example when a usual control system (e.g. related to a physical or biomedical model) depends on parameters affected by uncertainty, or when the Cauchy datum is not available with precision due to measurement errors. In this setting, we typically aim at finding a strategy that should be the same for every system of the ensemble, and that minimizes a proper cost.

In these cases, the proposed policy should incorporate the uncertainty that affects the system, and typically we seek one that results in a good performance in the most likely scenarios (averaged optimization), or one that guarantees resilience in the least favorable conjuncture (worst-case optimization).

Title: Large amplitude traveling waves for the non-resistive MHD system

Abstract: The goal of this talk is to discuss the existence of large amplitude traveling waves of the two-dimensional non-resistive Magnetohydrodynamics (MHD) system with a traveling wave external force. More precisely, we assume that the force is a smooth bi-periodic traveling wave propagating in the direction 

ω = (ω_1,ω_2) ∈R^2, with large amplitude of order O(λ^{1+}) and with large velocity speed λω. Then, for most values of ωand for λ≫1 large enough, we construct bi-periodic traveling wave solutions of arbitrarily large amplitude. Due to the presence of small divisors, the proof is based on a nonlinear Nash-Moser scheme adapted to construct nonlinear waves of large amplitude. The main difficulty is that the linearized equation at any approximate solution is an unbounded perturbation of large size of a diagonal operator and hence the problem is not perturbative. The invertibility of the linearized operator is then performed by using tools from micro-local analysis and normal forms together with a sharp analysis of high and low frequency regimes with respect to the large parameter λ. This works offers the first existence result for large amplitude quasi-periodic solutions for a nonlinear PDE in higher space dimensions. This research extends KAM techniques to a complex scenario in fluid mechanics. This is a joint work with G. Ciampa and R. Montalto.

Title: High-order estimates for fully nonlinear second order equations under "weak concavity" assumptions

Abstract: In 1982 L.C. Evans and N.V. Krylov proved interior a priori second derivatives estimates in Hölder spaces for fully nonlinear second  order uniformly elliptic/parabolic equations under the main assumption that the operator is concave or convex in the Hessian variable. L. Caffarelli then provided a W^{2,q} regularity theory of solutions to the same class of PDEs. Since then, a remarkable question in the theory is to determine which hypotheses on the operator in between convexity/concavity and no assumptions ensure that solutions to general second  order fully nonlinear equations are classical. In this direction, N. Nadirashvili and  S. Vladut exhibited counterexamples in dimension higher than or equal  to 5 to show that the sole uniform ellipticity is in general not  enough to reach classical regularity. Despite this progress, the  minimal assumptions guaranteeing classical regularity are unknown, and  the above question has remained largely open.

After a review of the regularity theory for fully nonlinear second order equations, I will discuss some advances on the Evans-Krylov and Calderón-Zygmund theory and show how to prove interior  C^{2,alpha}, C^{1,1} and W^{2,q} regularity for fully nonlinear elliptic and parabolic problems  under the assumption that the operator is quasi-concave/convex. I will also consider interior estimates for functions that are concave/convex or “close to a hyperplane” at  infinity as well as C^{2,alpha} bounds for some special non-concave operators. The approach is based on  linearization arguments and Bernstein methods combined with the Krylov-Safonov theory.  I will conclude with some consequences about the Calderón-Zygmund regularity of solutions to (fully nonlinear) second order Hamilton-Jacobi-Bellman/Isaacs equations with power-growth gradient terms and discuss the relation with a conjecture posed by P.-L. Lions. 

Title: A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations 

Abstract:  In this talk we derive a strain-gradient theory for plasticity as the Γ-limit of discrete dislocation fractional energies, without the introduction of a core-radius. By using the finite horizon fractional gradient introduced by Bellido, Cueto, and Mora-Corral of 2023, we consider a nonlocal model of semi-discrete dislocations, in which the stored elastic energy is computed via the fractional gradient of order 1 − α. As α goes to 0, we show that suitably rescaled energies Γ-converge to the macroscopic strain-gradient model of Garroni, Leoni, and Ponsiglione of 2010. 

This is a joint work with Stefano Almi (Università degli Studi di Napoli "Federico II"), Maicol Caponi (Università degli Studi dell’Aquila), and Manuel Friedrich (FAU Erlangen-Nürnberg).

Title: Diffusion effects  in optimal transport models penalizing congestion

Abstract: We discuss optimal transport problems with density dependent costs, penalizing congestion effects. Those terms enhance some form of dissipation compared to the classical Monge-Kantorovich transport. The underlying diffusive effect is to be observed through the optimality system coupling Hamilton-Jacobi and continuity equations, using different tools which lie at the crossroad between transport and diffusion methods as well as ideas from mean-field games. We discuss properties like L^1-L^\infty regularization, displacement convexity, finite or infinite speed of propagation of the support, and the possible formation of Lipschitz free-boundaries.

Title: Mean-field sparse optimal control of systems with additive white noise

Abstract: We analyze the problem of controlling a multiagent system with additive white noise through parsimonious interventions on a selected subset of the agents (leaders). For such a controlled system with an SDE constraint, we introduce a rigorous limit process toward an infinite dimensional optimal control problem constrained by the coupling of a system of ODEs for the leaders with a McKean--Vlasov type of SDE, governing the dynamics of the prototypical follower. The latter is, under some assumptions on the distribution of the initial data, equivalent with a (nonlinear parabolic) PDE-ODE system. The derivation of the limit mean-field optimal control problem is achieved by linking the mean-field limit of the governing equations together with the Gamma-limit of the cost functionals for the finite-dimensional problems.

This is a joint research project with Francesca Anceschi (Ancona), Giacomo Ascione (SSM Napoli) and Francesco Solombrino (Napoli Federico II).

Title: Slow dynamics in reaction-diffusion equations

Abstract: When I started my PhD at the University of L’Aquila, the main topic of my research activity concerned the analysis of a particular phenomenon, known in literature as metastability, that some one-dimensional reaction-diffusion (R-D) equations exhibit. The aim of this talk is to briefly review the main results obtained in the past years and, in particular, to focus on the case of a R-D equation with a Perona-Malik type diffusion.