Abstract
In this talk, we discuss how, based on ideas by Yann Brenier, it is possible to construct Lagrangian numerical methods for PDEs from fluid mechanics, such as Wasserstein gradient flows associated with an internal energy, or Euler flows/Hamiltonian flows for the same energy. This class includes incompressible Euler equations, compressible (barotropic) fluids and fluid-structure interactions.
To construct these schemes, the internal energy is replaced by its Moreau-Yosida regularization in the LL2 sense, which can be efficiently computed as a semi-discrete optimal transport problem. Using an argument based on modulated energy exploiting the convexity of the problem in the Eulerian variables, quantitative estimates of convergence to regular solutions of the partial differential equation system under consideration can be established.
