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.



Speaker: Jonas Lührmann (University of Cologne)

Title: Asymptotic stability of the sine-Gordon kink outside symmetry

Abstract: We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction potentials. Prime examples include the \phi^4 model and the sine-Gordon model. These models feature simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a rich class of problems owing to the combination of weak dispersion in one space dimension, low power nonlinearities, and intriguing spectral features of the linearized operators such as threshold resonances or internal modes. We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is general and not specific to the sine-Gordon model. We conclude with a discussion of potential applications in the generic setting (no threshold resonances) and with a discussion of the outstanding challenges posed by internal modes such as in the well-known \phi^4 model. This is joint work with Gong Chen (GeorgiaTech).



Speaker: Davide Carazzato (University of Vienna)

Title: A strong quantitative isoperimetric inequality for a capillarity problem

Abstract: During the seminar, we will introduce a stronger version of the quantitative isoperimetric inequality, originally developed by Fusco and Julin. Building on that, we will arrive to the analogous inequality for a capillarity problem using the so-called selection principle, based on the regularity theory for the perimeter functional. We will also highlight the difficulties that arise when we apply Fusco and Julin's method to our situation. This result was obtained in collaboration with Giulio Pascale and Marco Pozzetta.



Speaker: Eliot Pacherie (CNRS/CYU)

Title: Stability results for the Gross-Pitaevskii equation

Abstract: Gross-Pitaevskii is a dispersive equation that describes the behavior of superfluids and superconductors. Its dispersive estimates are worse than those for Schrödinger, which leads to difficulties when studying stability problems. In this talk, we present recent progress on these questions, in particular concerning the stability of vortices, which are stationary solutions of this equation.



Speaker: Roberta Bianchini (CNR/IAC)

Title: Stratified steady states of the IPM equation

Abstract: After introducing the Incompressible Porous Media (IPM) equation and its stratified steady states, I will present a recent result - obtained in collaboration with Jo, Park, and Wang - on the sharp asymptotic stability of IPM near stable stratified density profiles. This result is sharp in that it resolves the gap between small-data global well-posedness and strong ill-posedness for IPM in the vicinity of stable stratified states, and it exploits the variational structure underlying the dynamics.