A minimizing-movements approach to GENERIC systems

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.

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Existence of variational solutions for time dependent integrands via minimizing movements

Analysis ◽

10.1515/anly-2017-0047 ◽

2017 ◽

Vol 37 (4) ◽

Author(s):

Leah Schätzler

Keyword(s):

Time Dependent ◽

Solutions To Equations ◽

Variational Solutions ◽

Minimizing Movements

AbstractWe prove the existence of variational solutions to equations of the formwhere the function

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Minimizing Movements for Mean Curvature Flow of Partitions

SIAM Journal on Mathematical Analysis ◽

10.1137/17m1159294 ◽

2018 ◽

Vol 50 (4) ◽

pp. 4117-4148 ◽

Cited By ~ 2

Author(s):

Giovanni Bellettini ◽

Shokhrukh Yu. Kholmatov

Keyword(s):

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow ◽

Minimizing Movements

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Minimizing movements and evolution problems in Euclidean spaces

Annali di Matematica Pura ed Applicata (1923 -) ◽

10.1007/bf02505987 ◽

1999 ◽

Vol 176 (1) ◽

pp. 29-48

Author(s):

Massimo Gobbino

Keyword(s):

Euclidean Spaces ◽

Evolution Problems ◽

Minimizing Movements

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A tumor growth model of Hele-Shaw type as a gradient flow

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020019 ◽

2020 ◽

Vol 26 ◽

pp. 103

Author(s):

Simone Di Marino ◽

Lénaïc Chizat

Keyword(s):

Tumor Growth ◽

Optimal Transport ◽

Gradient Flow ◽

Convex Domains ◽

Wasserstein Metric ◽

Uniqueness Of Solutions ◽

Tumor Growth Model ◽

Entropic Regularization ◽

Minimizing Movements ◽

Convergence Of The Scheme

In this paper, we characterize a degenerate PDE as the gradient flow in the space of nonnegative measures endowed with an optimal transport-growth metric. The PDE of concern, of Hele-Shaw type, was introduced by Perthame et. al. as a mechanical model for tumor growth and the metric was introduced recently in several articles as the analogue of the Wasserstein metric for nonnegative measures. We show existence of solutions using minimizing movements and show uniqueness of solutions on convex domains by proving the Evolutional Variational Inequality. Our analysis does not require any regularity assumption on the initial condition. We also derive a numerical scheme based on the discretization of the gradient flow and the idea of entropic regularization. We assess the convergence of the scheme on explicit solutions. In doing this analysis, we prove several new properties of the optimal transport-growth metric, which generally have a known counterpart for the Wasserstein metric.

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