The consecutive lifting-projection flow as an approximation of Boltzmann and Landau flow

TL;DR

The paper introduces the consecutive lifting-projection flow as a new approximation method for Boltzmann and Landau equations, offering analytical solutions, stability, and a framework for developing efficient numerical schemes.

Contribution

It presents a novel LP flow framework that linearizes the collision operator, preserves physical invariants, and unifies existing discretizations for kinetic equations.

Findings

LP flow preserves mass, momentum, and energy.

Converges to Maxwellian equilibrium.

Enables explicit analytical solutions and stable numerical schemes.

Abstract

We introduce the consecutive lifting-projection (LP) flow as a novel approximation framework for the spatially homogeneous Boltzmann and Landau equations. The key idea is to lift the nonlinear collision operator to a higher dimensional linear Kac master equation on spheres, evolve this lifted equation in time, and project the solution back to the lower dimensional velocity space. The resulting LP flow is a tangent flow to the original kinetic dynamics and admits a clear semigroup structure. We show that the consecutive LP flow preserves mass, momentum, and energy, satisfies an entropy dissipation property, and converges to the correct Maxwellian equilibrium. In addition, the lifting removes the intrinsic nonlinearity of the collision operator and enables explicit analytical representations of the solution. For Maxwell molecules, we provide an error estimate quantifying the accuracy over finite time intervals. The framework provides a concise and general methodology for constructing reliable numerical solvers in kinetic theory. It unifies existing explicit discretizations, which helps understanding numerical stability and clarifying the trade-off between conservation and positivity. More importantly, it enables the development of new schemes. In particular, we propose the Green's function method, which is not only unconditionally stable, but also perfectly compatible with fast spectral discretizations.