A wavelet particle approximation for McKean–Vlasov and 2D-Navier–Stokes statistical solutions
Résumé
Letting the initial condition of a PDE be random is interesting when considering complex phenomena. For 2D-Navier–Stokes equations, it is for instance an attempt to take into account the turbulence arising with high velocities and low viscosities. The solutions of these PDEs are random and their laws are called statistical solutions. We start by studying McKean–Vlasov equations with initial conditions parameterized by a real random variable θ, and link their weak measure solutions to the laws of nonlinear SDEs, for which the drift coefficients are expressed as conditional expectations in the diffusions' laws given θ. We propose an original stochastic particle method to compute the first-order moments of the statistical solutions, obtained by approximating the conditional expectations by wavelet regression estimators. We establish a convergence rate that improves the ones obtained for existing methods with Nadaraya–Watson kernel estimators. We then carry over these results to 2D-Navier–Stokes equations and compute some physical quantities of interest, like the mean velocity vector field. Numerical simulations illustrate the method and allow us to test its robustness.