Danica Basaric, Politecnico di Milano, On well-posedness of systems arising from fluid dynamics, Friday, November 15, 2024, time 10:00, Aula Seminari - III Piano
Abstract:Abstract:
Well-posedness of systems describing the motion of compressible fluids in the class of strong and weak solutions represents one of the most challenging problems of the modern theory of PDEs. In the first part of the talk, we are going to define and construct a larger class of solutions, the measure-valued and dissipative ones, which unable us to handle the problem of existence for large times and large initial data; we will also discuss the possible advantages of considering this weaker notion of solution when solving some related problems arising from fluid dynamics. In the second part of the talk, we are going to show that, by performing a suitable selection, it is possible to select one "good" solution satisfying the semigroup property, even in the context when the system lacks uniqueness. After showing this procedure in an abstract setting, we will apply it to specific systems, such as the compressible Euler and Navier-Stokes ones.

Riccardo Molinarolo, Università degli Studi del Piemonte Orientale "A. Avogadro", A general integral identity with applications to a reverse Serrin problem, Wednesday, October 30, 2024, time 15:00, Aula Seminari - III Piano
Abstract:Abstract:
The talk aims to present a new general differential identity and an associated integral identity, which entails a pair of solutions of the Poisson equation with constant source term. This generalizes a formula that R. Magnanini and G. Poggesi previously proved and used to obtain quantitative estimates of spherical symmetry for the Serrin overdetermined boundary value problem. As a first application of this new general differential identity, we prove a quantitative symmetry result for the ``reverse Serrin problem'', which we will introduce. In passing, we obtain a rigidity result for solutions of the aforementioned Poisson equation subject to a constant Neumann condition. This is a joint work with R. Magnanini and G. Poggesi.

Kazuhiro Ishige, The University of Tokyo, Preservation of concavity properties by the Dirichlet heat flow, Wednesday, October 23, 2024, time 16:00, Aula Seminari - III Piano
Abstract:Abstract:
We characterize concavity properties preserved by the Dirichlet heat flow in convex domains of the Euclidean space. (This is a joint work with Paolo Salani and Asuka Takatsu.) Next, we show that no concavity properties are preserved by the Dirichlet heat flow in a totally convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the domain. (This is a joint work with Asuka Takatsu and Haruto Tokunaga.)

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.