Georgia Institute of Technology College of Sciences

In this talk, we will discuss a sequence of recent progresses on the global well-posedness of energy conservative Holder continuous weak solutions for a class of nonlinear variational wave equations and the Camassa-Holm equation, etc. A typical feature of solutions in these equations is the formation of cusp singularity and peaked soliton waves (peakons), even when initial data are smooth. The lack of Lipschitz continuity of solutions gives the major difficulty in studying the well-posedness and behaviors of solutions. Several collaboration works with Alberto Bressan will be discussed, including the uniqueness by characteristic method, Lipschitz continuous dependence on a Finsler type optimal transport metric and a generic regularity result using Thom's transversality theorem.