UNIST Webinar on Nonlinear Wave Motions

Title :  The incompressible Navier-Stokes limit from the lattice BGK Boltzmann equation

Abstract: The numerical implementation of the lattice Boltzmann equation is a successful tool for simulating fluid flows. In this talk, we will present the rigorous derivation of the incompressible Navier-Stokes equations as the hydrodynamic limit of a velocity discretized Boltzmann equation with simplified BGK collision operator. We will see that as long as the lattice structure has some certain symmetry, the Boltzmann equation fits well in the context of studying the hydrodynamic limit even its velocity variable is discretized. This is a joint work with Xin Hu (The University of Tokyo), Pritpal Matharu (KTH Royal Institute of Technology), Bartosz Protas (McMaster University) and Tsuyoshi Yoneda (Hitotsubashi University).

Title :  Spectral Instability of Electronic Euler-Poisson System

Abstract:  In this talk, we are concerned with the spectral instability of periodic waves for the electronic Euler-Poisson system, a hydrodynamical model for understanding electron dynamics in plasmas. It is found  that even small-amplitude periodic traveling waves in this system exhibit spectral instability, which is neither modulation nor co-periodic, necessitating an usual spectral analysis approach. We will provide an explanation of fundamental tools employed to investigate the stability of periodic waves, including Bloch-Floquet theory, Kato's perturbation theory, while outlining the primary proof strategy.

Title :  Propagation of rough initial data for Navier-Stokes equation

Abstract: In this talk, we will present a quantitative study of a weak solution for an initial value problem of the compressible Navier-Stokes equation in the class of BV functions. The key tool in the proof is the ``effective Green's function'', which is an interpolation between heat kernel for BV coefficient and Green's function for linearized Navier-Stokes equation. This is a joint work with Xiongtao Zhang (Wuhan University).

Title :  Stability and instability for line solitary waves of Zakharov-Kuznetsov equation 

Abstract: The stability of the one soliton of Korteweg-de Vries equation on the energy space was proved by Benjamin, Pego-Weinstein and Martel-Merle. We regard the one soliton of Korteweg-de Vries equation as a line solitary wave of Zakharov-Kuznetsov equation which is one of the higher dimensional models of Korteweg-de Vries equation. In this talk, we discuss the orbital stability and the asymptotic stability for line solitary waves of Zakharov-Kuznetsov equation with the periodic boundary condition for the transverse direction.

Title :  Linear stability of elastic $2$-line solitons for the KP-II equation

Abstract : The KP-II equation was derived by Kadomtsev and Petviashvili to explain stability of line solitary waves of shallow water. Using Darboux transformations, we study linear stability of $2$-line solitons whose line solitons interact elastically with each other.  Time evolution of resonant continuous eigenfunctions is described by a damped wave equation in the transverse variable which is supposed to be a linear approximation of the local phase shifts of modulating line solitons.