OPERATOR-SPLITTING SCHEMES FOR DEGENERATE, NON-LOCAL, CONSERVATIVE-DISSIPATIVE SYSTEMS

In this paper, we develop a natural operator-splitting variational scheme for a general class of non-lo cal, degenerate conservative-dissipative evo-lutionary equations. The splitting-scheme consists of two phases: a conserva -tive (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions the-ory while the second phase is solved approximately using a JKO-type varia-tional 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 gen-erality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation.