Special Colloquium-Blow up for the critical Zakharov-Kuznetsov equation

Speaker: Justin Holmer (Brown University and NSF)
Date and Time: Tuesday, April 24th, 1pm-2pm
Place: Gelman B02
Title: Blow up for the critical Zakharov-Kuznetsov equation
Abstract: The generalized Zakharov-Kuznetsov (ZK) equation is a higher dimensional version of the generalized Korteweg-de Vries (gKdV) equation. While the KdV equation and its generalizations have long been studied, a question about existence of blow-up solutions in gKdV has posed challenges and is far from being answered. One of the main obstacles is that unlike other dispersive models (such as the nonlinear Schrödinger or wave equations), the gKdV equation does not have a suitable variance quantity, which is the key to showing existence of finite time singularities. Only at the dawn of this century the works of Martel and Merle established the existence of finite-time blow-up solutions in the critical gKdV equation.  

In this talk we discuss the existence of finite time blow-up solutions in the corresponding two-dimensional critical ZK equation. We prove sharp pointwise decay estimates for the linear part of ZK, give applications to   linearized ZK solutions and some applications to well-posedness and soliton stability problems. 

This is a joint work with Luiz Farah (UFMG, Brazil), Svetlana Roudenko (GWU) and Kai Yang (GWU).