Mean field limit of point vortices with environmental noises to deterministic 2D Navier-Stokes equations

Abstract

We consider point vortex systems on the two dimensional torus perturbed by environmental noise. It is shown that, under a suitable scaling of the noises, weak limit points of the empirical measures are solutions to the vorticity formulation of deterministic 2D Navier-Stokes equations.

Scaling Limit of Moderately Interacting Particle Systems with Singular Interaction and Environmental Noise

“…After choosing some N -related parameters in our proof, we can compute the convergence rate. We follow some ideas in the proof of [16,Proposition 3.3] and assume ǫ is fixed in this section. Lemma 3.1.…”

“…For the second term J 2 N , following the method used in the proof of [16,Proposition 3.3], for some M > 0, we have…”

“…[7,8,40]); in these papers, however, the noise part was not rescaled with respect to the number of particles and thus the limit equations of empirical measures are stochastic partial differential equations (SPDEs), unlike in the classical case (1.2) where the limit equations are deterministic. Following the idea of the very recent paper [16], we shall seek for a certain rescaling of the noise in (1.1) and prove, for some singular interaction kernels (including the Biot-Savart and Poisson kernels), strong convergence of mollified empirical measures to solutions of deterministic nonlinear Fokker-Planck equations.…”

“…There are two main differences from (1.3): the noise in the above system is of environmental type, and the cut-off F A in the interaction part is replaced by convolution with ρ ǫ (see Remark 1.4 below for some comments on the latter). As in [16], for particle systems (1.4) with environmental noise, one difficulty is to show that the noise parts in the equations of empirical measures vanish as N → ∞, a fact that follows easily in the case of independent Brownian noises as in (1.3). To overcome this difficulty (see Lemma 3.1), we shall make nontrivial use of estimates on Boltzmann entropy obtained in Lemma 2.4, where in the proof we need to compute the divergence of the drift vector field corresponding to the interaction part, and it seems difficult to get useful estimates for the cut-off in (1.3).…”

Coagulation dynamics under environmental noise: scaling limit to SPDE

“…[28,29,35,27,14]. Also in those regimes, particle systems subject to environmental noise, with even "singular" interactions, have been studied, see [3,17,19] among others, where the interaction can occur in the drift of the SDEs and not just in auxiliary variables (like mass). Localizing the range of interaction for diffusions is a non-trivial task, in particular, in this work (as well as in [13]) we have to utilize some techiniques developed in [21] in the spirit of the classical Itô-Tanaka trick, to handle the convergence of the nonlinear terms, see Section 3.…”