Local existence of analytical solutions to an incompressible Lagrangian stochastic model in a periodic domain

TL;DR

This paper proves the short-time existence of analytic solutions to a simplified incompressible Lagrangian stochastic model in a periodic domain, addressing coupling challenges with a pressure-like force.

Contribution

It establishes the first analytical existence results for this simplified kinetic Fokker-Planck equation in one dimension, using novel functional norms.

Findings

Proved short-time existence of analytic solutions in 1D

Addressed coupling with a pressure-type force

Utilized new techniques from singular model analysis

Abstract

We consider an incompressible kinetic Fokker Planck equation in the flat torus, which is a simplified version of the Lagrangian stochastic models for turbulent flows introduced by S.B. Pope in the context of computational fluid dynamics. The main difficulties in its treatment arise from a pressure type force that couples the Fokker Planck equation with a Poisson equation which strongly depends on the second order moments of the fluid velocity. In this paper we prove short time existence of analytic solutions in the one-dimensional case, for which we are able to use techniques and functional norms that have been recently introduced in the study of a related singular model.