Solution of linearized Fokker - Planck equation for incompressible fluid

TL;DR

This paper derives and solves the spectrum of the linearized Fokker-Planck operator for incompressible fluids, showing damping solutions and expressing eigenfunctions as combinations of standard Fokker-Planck eigenfunctions.

Contribution

It constructs an algebraic spectrum equation for the linearized Fokker-Planck operator in incompressible fluids and demonstrates how to compute and interpret its roots and eigenfunctions.

Findings

All roots have positive real parts indicating damping.

Eigenfunctions are linear combinations of standard Fokker-Planck eigenfunctions.

Pressure can be eliminated from the dynamics equations.

Abstract

In this work we construct algebraic equation for elements of spectrum of linearized Fokker - Planck differential operator for incompressible fluid. We calculate roots of this equation using simple numeric method. For all these roots real part is positive, that is corresponding solutions are damping. Eigenfunctions of linearized Fokker - Planck differential operator for incompressible fluid are expressed as linear combinations of eigenfunctions of usual Fokker - Planck differential operator. Poisson's equation for pressure is derived from incompressibility condition. It is stated, that the pressure could be totally eliminated from dynamics equations. The Cauchy problem setup and solution method is presented. The role of zero pressure solutions as eigenfunctions for confluent eigenvalues is emphasized.