Past Seminars: (see also GSSI Academic Calendar)



Speaker: Théophile Dolmaire (University of L'Aquila)

Title: An Alexander’s theorem for inelastic hard spheres

Abstract: When studying interacting particle systems, the very first step before any qualitative analysis is to establish the well-posedness of the dynamics of the system. In the case of hard spheres, whose trajectories are piecewise affine, the singularities arising at collision times prevent the direct use of Cauchy-Lipschitz-type of arguments. This issue was addressed by Alexander (1975) in the elastic case, where the kinetic energy is conserved during the collisions. For dissipative systems, the question remains largely open, due to the possibility that infinitely many collisions take place in finite time, a phenomenon known as inelastic collapse. We will discuss the case of a particular class of inelastic hard sphere systems, in which a fixed amount of kinetic energy is lost in each sufficiently energetic collision. The results were obtained in collaboration with Juan J. L. Velázquez (Universität Bonn), and may be found in the preprint arXiv:2403.02162v2.



Speaker: Björn Gebhard (University of Münster)

Title: The Rayleigh-Taylor instability with local energy dissipation

Abstract: We consider the inhomgeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity. Initially the fluids are supposed to be at rest and separated by a flat horizontal interface with the heavier fluid being on top of the lighter one. Due to gravity this configuration is unstable, the two fluids begin to mix in a more and more turbulent way. This is one of the most classical instances of the Rayleigh-Taylor instability. In the talk we will see how weak locally dissipative solutions to the Euler equations reflecting a turbulent mixing of the two fluids in a quadratically growing zone can be constructed. If time allows, we will discuss an arising selection problem for the averaged motion of solutions. The core of the talk is based on a joint work with József Kolumbán.



Speaker: Serena Federico (University of Bologna)

Title: Smoothing estimates for third order equations with variable coefficients

Abstract: In this talk, we establish local smoothing estimates for a class of third-order pseudo-differential operators with variable coefficients, which, in particular, includes operators of KdV type. After introducing the main structural properties of these operators, we will outline the key ingredients of the proof, namely Doi’s lemma and suitable energy estimates. We will then present a local well-posedness result that follows from the smoothing effect. This talk is based on joint work with D. Tramontana.



Speaker: Vikram Giri (ETH Zurich)

Title: Non-conservation of (generalized) helicity in the incompressible Euler equations

Abstract: We begin with a discussion of the helicity which is a conserved integral quantity for the Euler equations. We then discuss Nash iteration schemes for the Euler equations and discuss the difficulties associated with defining helicity for such low-regularity, weak solutions. We'll then present forthcoming work that generalizes the helicity and constructs solutions of the Euler equations with a prescribed helicity profile, demonstrating its non-conservation. Based on joint work with Hyunju Kwon and Matthew Novack.



Speaker: Matteo Nesi (University of Basel)

Title: Uniqueness for the continuity equation on bounded domains

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 to 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.