Global semigroup of conservative solutions of the nonlinear variational wave equation

2009/10/28 by Helge Holden, Holden, Helge, Xavier Raynaud +1 · 1 citation
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP #msc:35L70 #msc:49K20

paper · pdf · doi:10.48550/arxiv.0910.5247

arxiv created 2009/10/28 · arxiv updated 2009/12/01

Abstract

We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u_tt-c(u)(c(u)u_x)_x=0. We allow for initial data u|_t=0 and u_t|_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 (u_t²+c(u)²u_x²)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.

Cited by

• Poiseuille flow of hyperbolic Ericksen-Leslie system in dimension two

Related

• Wave breaking for the nonlinear variational wave equation
• Global conservative weak solutions for a class of nonlinear dispersive wave equations beyond wave breaking
• Generic Regularity of Conservative Solutions to a Nonlinear Wave Equation
• Globally conservative weak solutions for a class of two-component nonlinear dispersive wave equations beyond wave breaking
• Global semigroup of conservative weak solutions of the two-component Novikov equation