Workshop on nonlinear dispersive equations

Sung-Jin Oh     Long-term behavior of nonlinear waves

Abstract: In this series of lectures, I will discuss classical and recent physical-space techniques for determining the precise long-term behavior of solutions to various linear and nonlinear wave equations on (possibly dynamic) asymptotically flat backgrounds.

Frank Merle     On the implosion of a three dimensional compressible fluid

Abstract: We consider the compressible three dimensional Navier Stokes and Euler equations. In a suitable regime of barotropic laws,  we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity.

Two essential steps of the analysis are the existence of $\mathcal C^\infty$ smooth self-similar solutions to the compressible Euler equations for quantized values of the speed and the derivation of spectral gap estimates for the associated linearized flow.

All blow up dynamics obtained for the Navier-Stokes problem are of type II (non self-similar).

This talk is based on joint works with Pierre Raphaël, Igor Rodnianski, and Jeremie Szeftel.

Beomjong Kwak     Strichartz estimates and global well-posedness of the cubic NLS on 2D turus

Abstract: In this talk, we present an optimal $L^4$-Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus $\mathbb{T}^2$. We first recall the previously known results and counterexamples on the Strichartz estimates on the torus. Then we present our new Strichartz estimate, which has an optimal amount of loss, and the small-data global well-posedness of (mass-critical) the cubic NLS in $H^s,s>0$ as its consequence. An intuition for the relation between them is then provided.

Our Strichartz estimate is based on a combinatorial proof. We introduce our key proposition, the Szemerédi-Trotter theorem, and explain the idea of the proof. This is a joint work with S. Herr.

Taegyu Kim     Soliton resolution for Calogero—Moser derivative NLS 

Abstract: We consider Calogero–Moser derivative NLS (CM-DNLS) equation which can be seen as a continuum version of completely integrable Calogero-Moser many-body systems in classical mechanics. Soliton resolution refers to the phenomenon where solutions asymptotically decompose into a sum of solitons and a dispersive radiation term as time progresses. Our work proves soliton resolution for both finite-time blow-up and global solutions without radial symmetry or size constraints. Although the equation exhibits integrability, our proof does not depend on this property, potentially providing insights applicable to other non-integrable models. This work is based on the joint work with Soonsik Kwon (KAIST).

Satoshi Masaki     On the asymptotic behavior of cubic NLS systems without weak null gauge condition

Abstract: In this talk, we discuss cubic NLS systems in one dimension (1D). It is well-known that cubic nonlinearity is critical in 1D, where nonlinear effects significantly influence the asymptotic behavior of small solutions (e.g., modified scattering). In the case of systems, these nonlinear effects exhibit a wide variety of phenomena.

Previous studies in this field often assume the weak null gauge condition, a structural condition that ensures the existence of an effective conserved quantity. In this work, we present an example of a system that does not satisfy this condition. Despite this, we succeed in determining the asymptotic behavior of small solutions. The key lies in identifying a new type of conserved quantity for the corresponding ODE system.

Jinmyoung Seok     From Lane-Emden Stars to Binary Stars: Stability in the gravitational Euler–Poisson System

Abstract: In astrophysical fluid dynamics, stars are modeled as isolated fluid masses governed by self-gravity. A fundamental hydrodynamic framework for describing the dynamics of Newtonian stars is provided by the gravitational Euler–Poisson (EP) system. In this talk, I will explore stability issues for both one-body and multi-body solutions of the EP system, which correspond to individual stars and star systems, respectively. In the first half of the talk, I will review key results on the stability of one-body solutions, specifically the so-called Lane-Emden stars. In the second half, I will present my results on the stability and instability of two-body rotating solutions of the EP system, which serve as models for binary star systems.