Abstract kinetic equations with positive collision operators

TL;DR

This paper studies abstract kinetic equations with positive collision operators, establishing conditions for unique solutions of boundary problems and applying results to Fokker-Planck and other parabolic equations of the forward-backward type.

Contribution

It proves that if the operator JL is similar to a self-adjoint operator, then associated boundary problems have unique solutions, extending to various parabolic equations.

Findings

Unique solutions exist under similarity conditions for JL.

Application to Fokker-Planck equations demonstrates practical relevance.

Framework covers a class of forward-backward parabolic equations.

Abstract

We consider "forward-backward" parabolic equations in the abstract form $Jd \psi / d x + L \psi = 0$, $ 0< x < \tau \leq \infty$, where $J$ and $L$ are operators in a Hilbert space $H$ such that $J=J^*=J^{-1}$, $L=L^* \geq 0$, and $\ker L = 0$. The following theorem is proved: if the operator $B=JL$ is similar to a self-adjoint operator, then associated half-range boundary problems have unique solutions. We apply this theorem to corresponding nonhomogeneous equations, to the time-independent Fokker-Plank equation $ \mu \frac {\partial \psi}{\partial x} (x,\mu) = b(\mu) \frac {\partial^2 \psi}{\partial \mu^2} (x, \mu)$, $ 0<x<\tau$, $ \mu \in \R$, as well as to other parabolic equations of the "forward-backward" type. The abstract kinetic equation $ T d \psi/dx = - A \psi (x) + f(x)$, where $T=T^*$ is injective and $A$ satisfies a certain positivity assumption, is considered also.