Wasserstein Distances for Vortices Approximation of Euler-Type Equations

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 L1 ∩ 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.