Time: July 24th, 2024, Wed. 10:00-10:50 am
Location: IMS, RS408
Speaker: Yilun Wu (University of Oklahoma)
Title: Global Bifurcation of Surface Capillary Waves on a 2D Droplet
Abstract: The existence of steady traveling waves bifurcating from a flat surface is a classical problem inwater wave theory. The well-known Stokes waves form a global continuum of periodic steady gravity waves which approach a limiting singular solution with a 120$^\circ$ angle at the top. Recently, Dyachenko et al. obtained numerically a branch of rotating traveling wave solutions bifurcating from a circular droplet in 2D. In this talk, we show a rigorous global bifurcation result constructing a set of such solutions. The obtained solutions are steady surface capillary waves in 2D, and have $m$-fold rotational symmetry. This is joint work with Gary Moon.
Time: July 24th, 2024, Wed. 11:00-11:50 am
Location: IMS, RS408
Speaker: Justin Holmer (Brown University)
Title: A derivation of the Boltzmann equation from quantum many-body dynamics
Abstract:We start by introducing a statistical model for the initial data of an N-body Schrodinger equation, meant to represent a scaled version of an N-particle quantum system with unit-order velocities and interparticle separations. The statistical model yields the expected functional form and scale of the corresponding BBGKY densities. This motivates a general a priori assumption on the Sobolev space norms of the BBGKY densities, which includes quasi-free states. Under this assumption, we prove that the Wigner transformed densities converge to the Boltzmann hierarchy with quadratic collision kernel and quantum scattering cross section. The proof of convergence uses a framework previously applied to the derivation of Bose Einstein condensate from an N-body model, and involves exploiting uniform bounds to obtain compactness and weak convergence. The remaining step is to prove the uniqueness of limits, which is performed using the Hewitt-Savage theorem and an extension of the Klainerman-Machedon board game. Our derivation is optimal with respect to regularity considerations. This is joint work with Xuwen Chen, University of Rochester.
Time: July 24th, 2024, Wed. 2:30-3:20 pm
Location: IMS, RS507
Speaker: Xuwen Chen (University of Rochester)
Title: From Quantum Particles to Compressible Inviscid Fluid
Abstract:We derive the classical compressible Euler equation as the limit of 3D quantum N-particle dynamics as N tends to infinity and Planck's constant tends to zero. We forge together the hierarchy method and the modulated energy method. We establish strong and quantitative convergence up to the 1st blow up time of the limiting Euler equation. During the course of the proof, we prove, as theoretically predicted, that the macroscopic pressure emerges from the space-time averages of microscopic interactions, which are in fact, Strichartz-type bounds andwe have hence found a physical meaning for the Strichartz type bounds. The grand scheme also applies to the Euler-Poisson situation.
Time: July 24th, 2024, Wed. 3:30-4:20 pm
Location: IMS, RS507
Speaker: Kai Yang (Chongqing University)
Title: Stability of the solitary waves and breathers for the L2-subcritical gKdV type equation
Abstract:We discuss the orbital and asymptotic stability of the solitary waves and breathers (when exist) for the L2-subcritical gKdV type equations, which are observed in numerous numerical experiments. Theoretical proof of the asymptotic stability on solitary waves follows the scheme developed by Martel and Merle for the gKdV case, and is extended to the multidimensional cases (Zakharov-Kuznetsov equation) by overcoming the obstacles on 1) the regularity problemfor the limit weak solution from using the regularity boosting technique, and 2) the properly selected modulational subspace which satisfies the resulting linear Liouville problem by intuitively selecting the proper modulational space and verifying the positivity of the Virial operator from numerical computation. This strategy reduces the asymptotic stability problem into showing the positivity of the corresponding Virial operator in a properly selected modulational subspace. Our numerical computation verifies the positivity of the Virial operators in the suitable modulational subspace for all the L2-subcritical ZK equations, which shows the asymptotic stability and agrees with the numerical observations. We also discuss the obstacle in applying this strategy for proving the asymptotic stability on the breathers. This is the joint work with Luiz Farah, Justin Holmer and Svetlana Roudenko.