Séance Séminaire

Séminaire ACSIOM

An optimal transportation metric for singular solutions of a nonlinear water wave equation

In this talk we construct a global, continuous flow of solutions to the Camassa-Holm equation on the space $H^1$. The solutions are conservative, in the sense that the total energy remains a.e. constant in time. Distances defined in term of convex norm perform well in connection with linear problems, but occasionally fail when nonlinear features become dominant. The new approach is thus based on the construction of a distance, related on a optimal transportation problem, which provides the ideal tool to measure continuous dependence on the initial data for solutions to the Camassa-Holm equation. Using this new distance functional, we can construct arbitrary solutions as the uniform limit of multi-peakon solutions, and prove a general uniqueness result.