Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems

TL;DR

This paper introduces an operator-splitting variational scheme for non-local, degenerate conservative-dissipative equations, combining exact transport solutions with variational diffusion steps, and proves convergence to weak solutions.

Contribution

It develops a novel operator-splitting scheme with entropic regularization for complex evolutionary equations, ensuring convergence and broad applicability.

Findings

Scheme converges to weak solutions

Applicable to kinetic Fokker-Planck and Vlasov-Poisson-Fokker-Planck equations

Includes entropic regularization for improved stability

Abstract

In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation.