Uniqueness and singularities of weak solutions to some nonlinear wave equations

Open Access
Author:
Zhang, Qingtian
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
May 04, 2016
Committee Members:
  • Alberto Bressan, Dissertation Advisor
  • Alberto Bressan, Committee Chair
  • Chun Liu, Committee Member
  • Anna L Mazzucato, Committee Member
  • Francesco Costanzo, Outside Member
  • Toan Nguyen, Committee Member
  • Yuxi Zheng, Committee Member
Keywords:
  • Camassa-Holm
  • Singularity
  • Uniqueness
  • weak solution
  • p-system
  • variational wave equation
  • well-posedness
Abstract:
Wave phenomenon is very common in physics. Understanding wave dynamics is very important and helpful to reveal physical principles. In this dissertation I consider three kinds of equations: Camassa-Holm equation, variational wave equation, and p-system. Camassa-Holm equation is an approximate model for the shallow water phenonmenon. Variational wave equation is derived from a physical model of liquid crystal, and p-system is for one-dimensional compressible gas dynamics. I mainly care about the analytical properties of weak solutions to these equations. In the first project, I proved the uniqueness of conservative weak solutions for the (two-component) Camassa- Holm equations and variational wave equation by the method of characteristics. They all have H1 energy, so the characteristic equations have H ̈older continuous right-hand-side. In general, the ODEs with Ho ̈lder continuous right-hand-side do not have unique solutions. We resolve this difficulty by using an additional balance law, related to energy, which is a change of coordinate based on the fact of energy concentration. And we proved uniqueness by rewriting the quasilinear PDE into a equivalent semilinear system with Lipschitz continuous right-hand- side. The main difference for the variational wave equation is that it is a second-order equation. It has two families of waves and they may interact with each other. We show that the total amount of interaction is bounded on bounded time intervals. Then by carefully choosing new characteristic variables, we can reduce the PDE to an equivalent first-order semilinear system with Lipschitz right-hand-side. In this way, we obtain uniqueness of conservative weak solutions. I also proved the global well-posedness of cubic Camassa-Holm equation. In the second project, we studied the generic singular behavior of these two equations. Here the ‘generic behavior’ refers to the properties which are true for all solutions starting from an open dense set of initial data.In particular, we showed that a generic solution is piecewise smooth. The proof mainly relies on Thom’s transversality Lemma from differential geometry. We also calculated the detailed local asymptotic expansion of solutions near the singular points and found that the Camassa-Holm equation and the two-component Camassa-Holm equation produce very different types of singularities. In the third project, we studied solutions to one-dimensional conservation laws with large BV data. Specially, we considered the p-system, modeling the isentropic gas dynamics. We consider two cases: when the density is uniformly away from zero and when the density may approach zero. In both cases, we construct examples of front-tracking approximate solutions where the total variation becomes arbitrarily large.