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.
翻译:在本文中,我们为非本地的、退化的保守保守和分裂的进化方程这一一般类别制定了自然操作分解的变异方案。分化方案由两个阶段组成:保守(运输)阶段和分散(扩散)阶段。第一阶段完全使用特性方法和DiPerna-Lion理论来解决,第二阶段则使用JKO型变异方案解决,这种方案将某种Kantorovich最佳运输成本功能的能量功能降到最低。此外,我们还引入了一种对方案进行正反常规化的办法。我们证明这两种方案都与进化方程的薄弱解决方案相融合。我们通过提供若干例子,包括动能Fokker-Planck方程和(正规化的)Vlasov-Poisson-Fokker-Planck方程,来说明我们工作的一般性。