An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 8.

In this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric setting. We use a constructive method, alternating minimizing movements for the Wasserstein distance and for the Fisher-Rao distance, and prove existence of weak solutions for general scalar reaction-diffusion-advection equations. We extend the approach to systems of multiple interacting species, and also consider an application to a very degenerate diffusion problem involving a Gamma-limit. Moreover, some numerical simulations are included.

DOI : 10.1051/cocv/2018001
Classification : 35K15, 35K57, 35K65, 47J30
Mots-clés : Unbalanced optimal transport, Wasserstein-Fisher-Rao, Hellinger-Kantorovich, JKO scheme, reaction-diffusion-advection equations
Gallouët, Thomas 1 ; Laborde, Maxime 1 ; Monsaingeon, Léonard 1

1 
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Gallouët, Thomas; Laborde, Maxime; Monsaingeon, Léonard. An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 8. doi : 10.1051/cocv/2018001. http://www.numdam.org/articles/10.1051/cocv/2018001/

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