A.A. 2024/25

Abstract

We investigate a passive vector field which is transported and stretched by a divergence-free Gaussian velocity field, delta-correlated in time and poorly correlated in space (spatially nonsmooth). Although the advection of a scalar field (Kraichnan's passive scalar model) is known to enjoy regularizing properties, the potentially competing stretching term in vector advection may induce singularity formation. We establish that the regularization effect is actually retained in certain regimes. While this is true in any dimension d ≥ 3, it notably implies a regularization result for linearized 3D Euler equations with stochastic modeling of turbulent velocities, and for the induction equation in magnetohydrodynamic turbulence.

The presentation is based on a joint work with Francesco Grotto and Mario Maurelli.

Abstract 

Introduced by Jordan, Kinderlehrer, and Otto, the JKO scheme is a time discretization method based on optimal transport theory, designed for several parabolic (and possibly degenerate) equations. It can be interpreted as an implicit Newton-type scheme in the Wasserstein space, where the usual Euclidean distance is replaced by its optimal transport counterpart. While the convergence of the flow under various levels of generality is well-understood in the weak setting, one can hope to recover information that is known to hold for the continuous-time equation in the case of certain specific flows. In this talk, I will present second-order estimates of the Li-Yau and Hamilton matrix forms, which hold for the JKO scheme associated with the Fokker-Planck equation, either on the torus or on the whole space, and I will discuss some applications of these estimates. If time permits, I will also explain how to modify these estimates for the porous medium equation.

This is based on joint work with F. Santambrogio, which will be published soon.

Abstract

We deal with the problem of characterizing rotationally symmetric solutions to

\Delta u = n \text{ in } \Omega

u = 0 \text{ in } \partial\Omega

when considering ring-shaped domains in the n-dimensional Euclidean space \mathbb{R}^n (n ≥3). Taking advantage of a suitable conformal reformulation of the problem, we obtain some gradient estimates that will be fundamental to obtaining our main comparison result. In particular, if we assume that the set of maximum points of the solutions has positive ℋ n−1-measure, we are able to characterize ring-shaped model solutions to the corresponding partially overdetermined problem. Our analysis also provides sufficient conditions for solutions to the problem to be rotationally symmetric.

This talk is based on a joint work with V.Agostiniani, S.Borghini and L.Mazzieri.

Abstract

We investigate the random bipartite optimal matching problem on a flat torus in two-dimensions, considering general strictly convex power costs of the distance. We extend the successful ansatz first introduced by Caracciolo et al. for the quadratic case, involving a linear Poisson equation, to a non-linear equation of $q$-Poisson type, allowing for a more comprehensive analysis of the optimal transport cost. Our results establish new asymptotic connections between the energy of the solution to the PDE and the optimal transport cost, providing insights on their asymptotic behavior. This is based on a joint work with L. Ambrosio and D. Trevisan. 

Abstract

The problem of prescribing the mean curvature of spacelike hypersurfaces in a Lorentzian manifold is both physically relevant —arising, for instance, in General Relativity and nonlinear electrodynamics— and mathematically challenging. After a brief overview of the state of the art, I will present a recent existence and regularity result for 2-surfaces whose mean curvature is a given Radon measure. This work is based on an ongoing collaboration with Luciano Mari. 

Abstract

In this seminar, I will talk about relations between curvature flows and foliations of asymptotically flat manifolds. Firstly, I will extend to the asymptotically flat setting a classical result by Huisken-Yau which allows to construct a constant mean curvature foliation of the outer part of the manifold by an alternative approach to the ones by Nerz and Eichmair-Koerber. This is a joint work with C. Sinestrari. Secondly, I will provide a generalization and an alternative construction of the constant spacetime-mean curvature foliation by Cederbaum-Sakovich, which has applications in the definition of center of mass of an isolated system in General Relativity. 

Abstract

Uniqueness for weak solutions of the continuity equation on the torus or in the Euclidean space goes back to the theory of Di Perna and Lions for Sobolev vector fields and of Ambrosio in the BV case. In presence of a boundary, the vector field is required to be tangent to it: the usual weak formulation of this condition is not strong enough to ensure uniqueness, while a different notion of "normal trace" suffices. Here we want to compare the two notions of normal trace and explain why one of them is more suitable in our case.