Title: The boundary value problem for the multidimensional Camassa-Holm equation: generalized flows and numerical challenges
Abstract: In this talk we will consider a multidimensional generalization of the Camassa-Holm equation, describing a shallow-water approximation for ideal fluid flow with a free boundary. Similarly to its one-dimensional counterpart, the multidimensional model has a variational interpretation: it can be formally obtained as the Euler-Lagrange equation for an action functional, the time integral of velocity H(div) norm. A natural question is then whether one can give a meaning to the solutions of the associated boundary value problem (in which the final configuration of fluid particles is provided instead of their initial velocity). We answer this question adapting the concept of generalized flow introduced by Brenier to solve the same problem for the incompressible Euler equations. We will present numerical approaches based on optimal transport to compute such generalized flows, and discuss related challenges and open problems.
