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.
Original language | English |
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Pages (from-to) | 5453-5486 |
Number of pages | 34 |
Journal | Discrete and Continuous Dynamical Systems - Series A |
Volume | 42 |
Issue number | 11 |
Early online date | 31 Aug 2022 |
DOIs | |
Publication status | Published - Nov 2022 |
Keywords
- Wasserstein gradient flows
- degenerate diffusions
- variational principle
- operator-splitting methods
- non-local partial differential equations
- optimal transport
- entropic regularisation