Antonio Hidalgo-Torné, Max Planck Institute for Mathematics in the Sciences, Leipzig, Global well-posedness of 3D Navier-Stokes with helical vortex filament data, Wednesday, October 16, 2024, time 15:00, Aula Seminari - III Piano
Abstract:Abstract:
We address the global-in-time existence, uniqueness, and regularity of solutions to the Cauchy problem for the three-dimensional Navier-Stokes equation with the initial vorticity concentrated on a helix. More specifically, we establish a local-in-time well-posedness result for vortex filaments that are periodic in one spatial direction, following the approach of Bedrossian, Germain and Harrop-Griffiths for closed filaments. Then, we use local energy weak solutions and helical estimates to extend the solution uniquely and globally in time in the helical case.

Filippo Gazzola, Politecnico di Milano, New tools for detecting the epochs of irregularity of Leray-Hopf solutions to some 3D Navier-Stokes equations, Friday, October 04, 2024, time 11:00, Aula Seminari III Piano
Abstract:Abstract:
We study global Leray-Hopf solutions to Cauchy problems for the 3D Navier-Stokes equations in a cube under Navier boundary conditions. With a suitable reflection procedure, these solutions become space-periodic over the whole space R^3. Since the pioneering work by Jean Leray, it is known that solutions exist for any initial data with finite energy but it is not known whether their enstrophy may blow up in finite time in the so-called epochs of irregularity. Our simplified geometric and functional-analytic framework enables us to determine explicit bounds both for the epochs of irregularity and for the enstrophy. By using these information we bring strong evidence that the enstrophy blow-up may indeed occur in finite time due to the energy equipartition among the Fourier components of the solution to a finite-dimensional Galerkin approximation of the problem. This is a joint work with Gianni Arioli and Alessio Falocchi.

Nicolas Zadeh, Université Libre de Bruxelles, Belgium, Description and numerical study of a kinetic Fokker-Planck equation in neuroscience, Monday, July 01, 2024, time 15:15, Aula seminari MOX, VI piano
Abstract:Abstract:
The well-known Integrate and fire model in a partial differential equation form has been at the center of many mathematical developments since Brunel's work and the seminal paper by Caceres, Carrillo and Perthame in 2011. Its descriptive shortcomings such as the inability to see sub-threshold oscillations or to obtain resonances gave birth to the resonate and fire model (Izhikevich, 2001), phenomenologically complex enough but computationally not costly. However, there has been no deep mathematical study of it yet. In this work, we first establish a PDE corresponding to the mean-field limit of a population of resonate and fire neurons. The obtained formulation corresponds to a non-linear kinetic Fokker-Planck equation, with a non-local linearity and a measure source term, studied on a half plane. Even though the obtained operator has properties of hypoellipticity, the theoretical study is tedious, encouraging the pursuit of a numerical study to obtain information about the behaviour of the solutions. We thereby will describe the positivity and mass preserving finite differences scheme of experimental order one we developed, which allows us to observe all the properties we were expecting from the original single neuron model, and even giving birth to some conjectures which we shall detail.

Francesca Anceschi, Università Politecnica delle Marche, Well-posedness results for Kolmogorov equations with applications to mean-field control problems for multi-agent systems, Thursday, May 16, 2024, time 15:00, Aula Seminari - III Piano
Abstract:Abstract:
In this talk, based on a joint project with G. Ascione, D. Castorina and F. Solombrino, we discuss some well-posedness results for Kolmogorov-Fokker-Planck equations with measurable coefficients in time and locally Holder continuous coefficients in space with possibly unbounded drift terms and their application to the study of particle systems of the second order whose prototypical agent is driven by a McKean-Vlasov SDE, or by a Vlasov-Fokker-Planck PDE.

Eduardo Munoz Hernandez, Universidad Complutense de Madrid, Existence and multiplicity of coexistence states in a heterogeneous predator-prey model with saturation, Tuesday, May 14, 2024, time 11:30, Aula Seminari - III Piano
Abstract:Abstract:
In this talk we analyze a predator-prey model coming from the heterogeneous counterparts of the classical Lotka-Volterra and Holling-Tanner models by considering a non-negative saturation term that can vanish inside the domain. By the use of the maximum principle and bifurcation theory the regions of coexistence can be exactly determined. Furthermore, by varying the amplitude of the saturation different multiplicity results will be obtained. This is a joint work with Julián López-Gómez.

Alessio Falocchi, Politecnico di Milano, On the long-time behaviour of solutions to unforced evolution Navier-Stokes equations under Navier boundary conditions, Thursday, May 09, 2024, time 16:30, Aula Seminari - III Piano
Abstract:Abstract:
We study the asymptotic behaviour of the solutions to Navier-Stokes unforced equations under Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest that we call sectors. The main motivations come from the celebrated results by Foias-Saut related to the long time behaviour of the solutions to Navier-Stokes equations under Dirichlet conditions. Here the choice of the boundary conditions requires carefully considering the geometry of the domain, due to the possible lack of the Poincaré inequality in presence of axial symmetries. In non-axially symmetric domains we show the validity of the Foias-Saut result about the limit at infinity of the Dirichlet quotient, in axially symmetric domains we provide two invariants of the flow which completely characterize the motion and we prove that the Foias-Saut result holds for initial data belonging to one of the invariants. This is a joint work with Prof. Elvise Berchio (Politecnico di Torino, Italy) and Clara Patriarca (Université libre de Bruxelles, Belgium).