WASSERSTEIN DISTANCES FOR VORTICES APPROXIMATION OF EULER-TYPE EQUATIONS

Source abstract

We establish the convergence of a vortex system towards equations similar to the 2D Euler equation in vorticity formulation. The only but important difference is that we use singular kernel of the type x ⊥ /|x| α+1 , with α < 1, instead of the Biot–Savard kernel x ⊥ /|x| 2 . This paper follows a previous work of Jabin and the author about the particles approximation of Vlasov equation in Ref. 13. Here we study a different mean-field equation, simplify the proofs and weaken non-physical initial conditions. The simplification is due to the introduction of the infinite Wasserstein distance. The results are obtained for L 1 ∩ L ∞ vorticities without any sign assumption, in the periodic setting, on the whole space and on the half space (with Neumann boundary conditions). A vortex-blob result is also given, that is valid for short times in the true vortex case.