Weak geodesic flow and global solutions of the Hunter-Saxton equation. (English) Zbl 1140.35388
Summary: We show how global weak solutions of the Hunter-Saxton equation can be naturally constructed using the geometric interpretation of the equation as the Euler equation for the geodesic flow on an \(L^2\)-sphere. The approach involves forming a weak extension of the geodesic flow and relating it to a corresponding weak formulation of the Hunter-Saxton equation.
MSC:
35G25 | Initial value problems for nonlinear higher-order PDEs |
58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |
53C22 | Geodesics in global differential geometry |