Schedule and book of abstracts

Michele Coti Zelati. Long-time dynamics of stratified fluids

Talks

Paolo Bonicatto: On the strong locality of differential operators.

Abstract: It is well known that, given a Sobolev function vanishing in a measurable set, the gradient must vanish almost everywhere on that set. This property is usually called “locality of the gradient operator”. In the seminar, we will introduce the notion of locality for general linear (first-order) differential operators and we will discuss some sufficient and necessary conditions for locality to hold. We will present several examples and, if time allows, a complete catalogue of differential operators in the 2D setting. This is part of ongoing projects with G. Alberti (Pisa) and G. Del Nin (MPI, Leipzig).

Elia Bruè: Non-Uniqueness and Flexibility in Two-Dimensional Euler Equations

In 1962, Yudovich established the well-posedness of the two-dimensional incompressible Euler equations within the class of solutions with bounded vorticity. Since then, a central unresolved problem has been the question of uniqueness within the broader class of solutions with L^p-vorticities. Recent years have witnessed significant progress in this investigation. In my talk, I aim to provide an overview of these developments and highlight recent results obtained thanks to the convex integration method.

Alexander Cliffe: Shock Reflection and other 2D Riemann Problems in Gas Dynamics (slides)

Abstract: The Riemann problem is the IVP with simple piecewise constant initial data that is invariant under scaling. In 1D, the problem was originally considered by Riemann during the 19th century in the context of gas dynamics, and the general theory was more or less competed by Lax and Glimm in the mid-20th century. In 2D and MD, the situation is much more complicated, and very few analytic results are available. We discuss a shock reflection problem for the Euler equations for potential flow, with initial data that generates four interacting shockwaves. After reformulating the problem as a free boundary problem for a nonlinear PDE of mixed hyperbolic-elliptic type, the problem is solved via a sophisticated iteration procedure. The work is joint work with G-Q Chen (Oxford) et. al. arXiv:2305.15224.

Ricardo Grande Izquierdo: Wave kinetic theory for the forced/dissipated NLS equation (slides)

Abstract: We will present some recent developments in the justification of kinetic equations in the presence of forcing and dissipation. Such settings are of particular physical relevance as they allow the study of cascades: the transfer of energy from large scales to small scales. 

In this talk, we provide the first rigorous justification of such a kinetic equation in the case of a wave system governed by the cubic Schrödinger equation with a stochastic forcing and viscous dissipation. We will describe various regimes depending on the relative strength of the dissipation, the forcing and the nonlinear interactions, which give rise to different kinetic equations. Based on joint work with Zaher Hani.

Giacomo Maria Leccese: Existence and blow-up for non-autonomous conservation laws with viscosity (slides) Click here for the abstract.

Stefano Modena: Non uniqueness and energy. conservation for 2D Euler equations with vorticity in Hardy spaces (slides)

Camilla Nobili: Large time behaviour of the 2D thermally non-diffusive Boussinesq equation

In this talk we consider the two-dimensional Boussinesq equations without thermal diffusion on a bounded domain with Navier-slip boundary conditions and study large-time asymptotics. We prove that, in suitable norms, the solution converges to the hydrostatic equilibrium and show linear stability for the hydrostatic equilibrium when the temperature is an increasing affine function of the height, i.e. the temperature is vertically stably stratified.

Gianluca Orlando: Long-time behaviour of damped adhesive strings (slides)

Abstract: The talk will focus on an evolutionary PDE modelling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. More precisely, the PDE is a damped semilinear wave equation with Neumann boundary conditions. The main feature of the equation is the nonlinear force acting on the body: it is proportional to the displacement for small displacements and it is zero when the displacement is large. The talk will be divided in three parts. In the first part, we will recall some key aspects of the classica theory of maximal monotone operators to deduce the global well-posedness for the model. In the second part, we will focus on the qualitative asymptotic behaviour of solutions for large time, showing that solutions converge to solutions of the steady problem. Finally, in the third part, we will show how to deduce a quantitative estimate on the rate of convergence to the asymptotic limit. The content of the talk is based on a paper in collaboration with G. M. Coclite (Politecnico di Bari), N. De Nitti (EPFL), F. Maddalena (Politecnico di Bari), E. Zuazua (FAU Erlangen-Nuerenberg).

Andrea Pinamonti: Some regularity results for balance laws and applications to the Heisenberg group

Emanuela Radici: Smoothing effect and particle approximation for a nonlocal conservation law

Click here for the abstract .

Massimo Sorella: Spontaneous stochasticity for 2D autonomous flows