Long Time Behavior and Singularity Formation in PDEs: Part V

Monday, May 30, 2024

1.15 pm: Opening remarks by Sehamuddin Galadari, Senior Vice Provost of Research; Managing  Director, Research Institute; Professor of Biology, New York University, Abu Dhabi 

1.20 - 2.20 pm: Sylvia Serfaty — Mean-field limits for singular flows 

  • Abstract: We discuss the derivation of PDEs as limits as N tends to infinity of the dynamics of N  points for a certain class of Riesz-type singular pair interactions. The method is based on  studying the time evolution of a certain "modulated energy" and on proving a functional  inequality relating certain "commutators" to the modulated energy. When additive noise is  added, in dimension at least 3 a uniform in time convergence can even be obtained. Based on  joint works with Hung Nguyen, Matthew Rosenzweig. 

2.30 – 3.30 pm: Michele Coti Zelati — Exponential mixing for random flows 

  • Abstract: We consider random dynamical systems driven by noise that is absolutely continuous  with respect to the Lebesgue measure, and exhibit sufficient conditions that imply exponential  mixing. As a corollary, we show that the so-called Pierrehumbert model, consisting of  alternating shear flows with randomized phases, is exponentially mixing. 

5.30 – 6.30 pm: Avraham Soffer — The Asymptotic States of Nonlinear Dispersive Equations with Large Initial Data and  General Interactions 

  • Abstract: I will describe a new approach to scattering theory, which allows the analysis of  interaction terms which are linear and space-time dependent, and nonlinear terms as well. This  is based on deriving (exterior) propagation estimates for such equations, which micro-localize  the asymptotic states as time goes to infinity. In particular, the free part of the solution  concentrates on the propagation set (x=vt), and the localized leftover is characterized in the  phase-space as well. The NLS with radial data in three dimensions is considered, and it is shown  that besides the free asymptotic wave, in general, the localized part is smooth, and is localized  in the region where |x|^2 is less than t. Furthermore, the localized part has a massive core and  possibly a halo which may be a self-similar solution. This work is joint with Baoping Liu. This is then followed by new results on the non-radial case and Klein-Gordon equations (Joint works  with Xiaoxu Wu).

Tuesday, May 31

1.15 – 2.15 pm: Matthew R. I. Schrecker — Self-similar gravitational collapse for Newtonian stars 

  • Abstract: The Euler-Poisson equations give the classical model of a self-gravitating star under  Newtonian gravity. It is widely expected that, in certain regimes, initially smooth initial data may  give rise to blow-up solutions, corresponding to the collapse of a star under its own gravity. In  this talk, I will present recent work with Yan Guo, Mahir Hadzic and Juhi Jang that demonstrates  the existence of smooth, radially symmetric, self-similar blow-up solutions for this problem. At  the heart of the analysis is the presence of a sonic point, a singularity in the self-similar model  that poses serious analytical challenges in the search for a smooth solution. 

2.30 – 3.30 pm: Gregory Seregin — Long time behaviour and local regularity for solutions to the Navier-Stokes equations

  • Abstract: We shall discuss the problem of local regularity for solutions to the Navier-Stokes  equations under certain scale-invariant conditions. Using zooming and duality, we reduce it to  the problem of long time behaviour of a certain Stokes equations with a drift. This is in part a  joint work with M. Schonbek. 

5.30– 6.30 pm: Zhiyuan Zhang — Stability of solitary waves of the NLS equation 

  • Abstract: We consider the asymptotic stability of the solitary waves of 1D NLS equations, under  the assumption that the linearized operator is generic (no endpoint resonance) and has no  internal modes. Moreover, we also consider the 1D nonlinear Klein-Gordon equation with a  potential, and give a result on small data existence. The method of analysis is based on the  distorted Fourier transform. This is joint work with P. Germain and F. Pusateri. 

Wednesday, June 1

1.15 – 2.15 pm: Klaus Widmayer — On the stability of a point charge for the Vlasov-Poisson system 

  • Abstract: A Dirac mass is a particularly simple yet relevant equilibrium of the (repulsive) Vlasov Poisson equations. This talk addresses the question of its stability in the repulsive setting: we  capture the precise asymptotic dynamics of solutions which start as small, smooth and suitably  localized perturbations of a point charge. Our analysis builds on the Hamiltonian/symplectic  structure of the equations, and makes use of an exact integration of the linearized equation  through angle-action coordinates. This allows us to obtain optimal decay estimates and reveals a  modified scattering dynamic. This is joint work with Jiaqi Yang (ICERM) and Benoit Pausader  (Brown University).

2.30 – 3.30 pm: Jaemin Park — Existence of non-radial stationary solutions to the 2D Euler equation 

  • Abstract: In this talk, we study stationary solutions to the 2D incompressible Euler equations in  the whole plane. It is well-known that any radial vorticity is stationary. For compactly supported  vorticity, it is more difficult to see whether a stationary solution has to be radial. In the case  where the vorticity is non-negative, it has been shown that any stationary solution has be radial.  By allowing the vorticity to change the sign, we prove that there exist non-radial stationary  patch-type solutions. We construct patch-type solutions whose kinetic energy is infinite or finite.  For the finite energy case, it turns out that a construction of a stationary solution with  compactly supported velocity is possible. 

5.30 – 6.30 pm: Robert Strain — On the 2D fully nonlinear Peskin problem 

  • Abstract: The Peskin problem models the dynamics of a closed elastic string immersed in an  incompressible 2D stokes fluid. This set of equations was proposed as a simplified model to  study blood flow through heart valves. The immersed boundary formulation of this problem has  proven very useful in particular giving rise to the immersed boundary method in numerical  analysis. In a joint work with Stephen Cameron, we consider the general case of a fully non linear tension law. We prove local wellposedness for arbitrary initial data in the scaling critical  Besov space \dot{B}^{3/2}_{2,1}, and the high order smoothing effects for the solution. 

Thursday, January 2

1.15 – 2.15 pm: Tak Kwong Wong — Regularity structure, global-in-time existence and uniqueness of energy conservative  solutions to the Hunter-Saxton equation 

  • Abstract: The Hunter-Saxton equation is an integrable equation in one spatial dimension, and  can be used to study the nonlinear instability in the director field of a nematic liquid. In this talk,  we will discuss the regularity structure, global-in-time existence and uniqueness of energy  conservative solutions to the Hunter-Saxton equation. In particular, singularities for the energy  measure may only appear at at most countably many times, and are completely determined by  the absolutely continuous part of initial energy measure. The temporal and spatial locations of  singularities are explicitly determined by the initial data as well. The analysis is based on using  the method of characteristics in a generalized framework that consists of the evolutions of  solution to the Hunter-Saxton equation and the energy measure. This is a joint work with Yu Gao  and Hao Liu. 

2.30 – 3.30 pm: Christophe Prange — Concentration and quantitative regularity for the Navier-Stokes equations

  • Abstract: In this talk I will show concentration phenomena near potential singularities of the  three-dimensional Navier-Stokes equations. I will also investigate the connection between concentration estimates and quantitative regularity. This is joint work with Tobias Barker  (University of Bath).