The purpose of this paper is to introduce a Minimizing Movement approach to scalar reaction–diffusion equations of the form
| $$ |
with parameters Λ, Σ > 0 and no-flux boundary condition
| $$ |
which is built on their gradient-flow-like structure in the space $$ of finite nonnegative Radon measures on $$, endowed with the recently introduced Hellinger-Kantorovich distance HKΛ,Σ. It is proved that, under natural general assumptions on $$ and $$, the Minimizing Movement scheme
| $$ |
for
| $$ |
yields weak solutions to the above equation as the discrete time step size τ ↓ 0. Moreover, a superdifferentiability property of the Hellinger-Kantorovich distance HKΛ,Σ, which will play an important role in this context, is established in the general setting of a separable Hilbert space; that result will constitute a starting point for the study of the differentiability of HKΛ,Σ along absolutely continuous curves which will be carried out in a subsequent paper.
Keywords: Optimal transport, gradient flows, Minimizing Movements, reactiondiffusion equations, Hellinger-Kantorovich distance
@article{COCV_2021__27_1_A20_0,
author = {Flei{\ss}ner, Florentine Catharina},
title = {A minimizing {Movement} approach to a class of scalar reaction{\textendash}diffusion equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2020090},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2020090/}
}
TY - JOUR AU - Fleißner, Florentine Catharina TI - A minimizing Movement approach to a class of scalar reaction–diffusion equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2020090/ DO - 10.1051/cocv/2020090 LA - en ID - COCV_2021__27_1_A20_0 ER -
%0 Journal Article %A Fleißner, Florentine Catharina %T A minimizing Movement approach to a class of scalar reaction–diffusion equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2020090/ %R 10.1051/cocv/2020090 %G en %F COCV_2021__27_1_A20_0
Fleißner, Florentine Catharina. A minimizing Movement approach to a class of scalar reaction–diffusion equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 18. doi: 10.1051/cocv/2020090
[1] , and , Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31 (1993) 387–438.
[2] , and , Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in mathematics. ETH Zürich, Birkhäuser (2005).
[3] , Local Minimization, Variational Evolution and Γ-Convergence. Vol. 2094 of Lecture Notes in Mathematics. Springer (2012).
[4] and A tumor growth model of Hele-Shaw type as a gradient flow. ESAIM: COCV 26 (2020) 103.
[5] , , and , An interpolating distance between optimal transport and Fisher–Rao metrics. Found. Comput. Math. 18 (2018) 1–44.
[6] , , and , Unbalanced optimal transport: dynamic and Kantorovich formulations. J. Funct.Anal. 274 (2018) 3090–3123.
[7] , New problems on minimizing movements, in Boundary Value Problems for PDE and Applications, edited by and . Masson (1993) 81–98.
[8] , and , Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 68 (1980) 180–187.
[9] , and , Evolution equations with lack of convexity. Nonlinear Anal. 9 (1985) 1401–1443.
[10] , Gamma-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach. ESAIM: COCV 25 (2019) 28.
[11] , A note on the differentiability of the Hellinger-Kantorovich distances. Preprint (2020). | arXiv
[12] and , Reverse approximation of gradient flows as minimizing movements: a conjecture by De Giorgi. Annali della Scuola Normale di Pisa - Classe di Scienze 20 (2020) 677–720.
[13] and , Modern Methods in the Calculus of Variations: Lp Spaces. Springer Science & Business Media (2007).
[14] and , A JKO splitting scheme for Kantorovich–Fisher–Rao gradient flows. SIAM J. Math. Anal. 49 (2017) 1100–1130.
[15] and , The geometry of optimal transportation. Acta Math. 177 (1996) 113–161.
[16] , and , Free energy and the Fokker-Planck equation. Physica D 107 (1997) 265–271.
[17] , and , The variational formulation of the Fokker-Planck Equation. SIAM J. Math. Anal. 29 (1998) 1–17.
[18] , and , A fitness-driven cross-diffusion system from population dynamics as a gradient flow. J. Differ. Equ. 261 (2016) 2784–2808.
[19] , and , et al., A new optimal transport distance on the space of finite Radon measures. Adv. Differ. Equ. 21 (2016) 1117–1164.
[20] and , Nonlinear Fokker-Planck equations with reaction as gradient flows of the free energy. J. Funct. Anal. 278 (2020) 108310.
[21] , and , On geodesic λ-convexity with respect to the Hellinger-Kantorovich distance, in preparation.
[22] , and , Optimal transport in competition with reaction: the Hellinger–Kantorovich distance and geodesic curves. SIAM J. Math. Anal. 48 (2016) 2869–2911.
[23] , and , Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures. Invent. Math. 211 (2018) 969–1117.
[24] , and , Curves of maximal slope and parabolic variational inequalities on nonconvex constraints. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16 (1989) 281–330.
[25] , Differential, energetic, and metric formulations for rate-independent processes. Springer (2011).
[26] , A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24 (2011) 1329.
[27] and , Reverse approximation of energetic solutions to rate-independent processes. NoDEA Nonlinear Differ. Equ. Appl. 16 (2009) 17–40.
[28] , and , Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J. Eur. Math. Soc. 18 (2016) 2107–2165.
[29] , The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26 (2001) 101–174.
[30] , Topics in optimal transportation. Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2003).
[31] , Optimal transport. Old and new. Vol. 338 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin (2009).
Cité par Sources :
