Article Contents

2022, Volume 42, Issue 11: 5453-5486. Doi: 10.3934/dcds.2022109

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

1.
Maxwell Institute for Mathematical Sciences and School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, Scotland
2.
School of Mathematics, University of Birmingham, Birmingham B15 2TT, England
3.
School of Mathematics, University of Edinburgh, Mayfield Rd, Edinburgh, EH9 3FD, Scotland

^*Corresponding author: Manh Hong Duong
^*Corresponding author: Manh Hong Duong

Received: July 2021

Revised: June 2022

Early access: August 2022

Published: November 2022

D. Adams was supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. M. Hong Duong was supported by EPSRC Grants EP/W008041/1 and EP/V038516/1. G. dos Reis acknowledges support from the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UIDB/00297/2020 and UIDP/00297/2020 (Centro de Matemática e Aplicações CMA/FCT/UNL)

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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.

Keywords:

Wasserstein gradient flows,
degenerate diffusions,
variational principle,
operator-splitting methods,
non-local partial differential equations,
optimal transport,
entropic regularisation.

Mathematics Subject Classification: Primary: 35K15, 35K55, Secondary: 65K05, 90C25.

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References

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