Global semigroup of conservative solutions of the nonlinear variational wave equation
Helge Holden and Xavier Raynaud
Abstract: We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation utt−c(u)(c(u)ux)x=0. We allow for initial data u|t=0 and ut|t=0 that contain measures. We assume that 0<κ−1≤c(u)≤κ. Solutions of this equation may experience concentration of the energy density (ut2+c(u)2ux2)dx into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may only focus on a set of times of zero measure or at points where c'(u) vanishes. A new numerical method to construct conservative solutions is provided and illustrated on examples.