Uniform Approximation of 2 Dimensional Navier--Stokes Equation by Stochastic Interacting Particle Systems

Abstract

We consider an interacting particle system modeled as a system of N stochastic differential equations driven by Brownian motions. We prove that the (mollified) empirical process converges, uniformly in time and space variables, to the solution of the two-dimensional Navier-Stokes equation written in vorticity form. The proofs follow a semigroup approach.

Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel

“…The proofs of these two results are similar to the proofs of Propositions 2.1 and 2.2 in [20] (the kernel plays no role here). We present them in Appendix A.2 and note that this is where the restriction (A α ) on α appears.…”

“…The approximation of nonlinear Fokker-Planck equations of the type (1.1) has been widely covered in the literature, but such convergence of a mollified empirical measure in strong topologies, for singular kernels, are quite new. In this sense, we extend and improve previous results of [20]. The extension to several singular models, in particular to singular Riesz kernels and to attractive kernels (e.g.…”

“…with a density that is smooth enough (see [17,Lemma 2.9] for a related result). The reader may also find interesting comments on this assumption in Remark 1.2 of [20].…”

“…Here r is fixed in (A K iii ) and r is such that 1 r + 1 r = 1. Sections 4.1, 4.2 an 4.3 are devoted to the proof of Theorem 2.5, generalising the semigroup approach presented in [20].…”

“…This technique permits to approximate non-linear PDEs by smoothed empirical measures in strong functional topologies. It has already found many applications: see Flandoli and Leocata [17] for a PDE-ODE system related to aggregation phenomena; Simon and Olivera [52] for non-local conservation laws; Flandoli et al [20] for the 2d Navier-Stokes equation; and Olivera et al [43] for the 2d Keller-Segel systems.…”