EVENT DETAILS AND ABSTRACT

Title: A Nonlinear Variational Wave Equation
Speaker: Professor Yuxi Zheng
Speaker Info: Indiana University-Bloomington
Brief Description:
Special Note: SPECIAL TIME
Abstract:

A nonlinear wave equation arises in a simplified liquid crystal model through the variational principle. The wave speed of the wave equation is a given function of the wave amplitude. It has been known for the equation that smooth initial data may develop singularities in finite time, a sequence of weak solutions may develop concentrations, while oscillations may persist. We formulate a viscous approximation of the equation and establish the global existence of smooth solutions for the viscously perturbed equation. For monotone wave speed functions in the equation, we find an invariant region in the phase space in which we discover: (a) smooth data evolve smoothly forever; (b) both the viscous regularization and the smooth solutions obtained through data mollification and step (a) for not-as-smooth initial data yield weak solutions to the Cauchy problem of the nonlinear variational wave equation. The main tool is the Young measure theory and related techniques.