Fast reaction limit and forward-backward diffusion: A Radon–Nikodym approach

Jakub Skrzeczkowski

doi:10.5802/crmath.279

Équations aux dérivées partielles

Fast reaction limit and forward-backward diffusion: A Radon–Nikodym approach

Jakub Skrzeczkowski ¹

¹ Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Stefana Banacha 2, 02-097 Warsaw, Poland

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 189-203.

Résumé

We consider two singular limits: a fast reaction limit with a non-monotone nonlinearity and a regularization of the forward-backward diffusion equation. We derive pointwise identities satisfied by the Young measure generated by these problems. As a result, we obtain an explicit formula for the Young measure even without the non-degeneracy assumption used in the previous works. The main new idea is an application of the Radon–Nikodym theorem to decompose the Young measure.

Reçu le : 2021-05-25
Révisé le : 2021-08-12
Accepté le : 2021-10-05
Publié le : 2022-02-15

DOI : 10.5802/crmath.279

Classification : 35K57, 35B25, 35B36

Affiliations des auteurs :

Jakub Skrzeczkowski ¹

¹ Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Stefana Banacha 2, 02-097 Warsaw, Poland

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Jakub Skrzeczkowski},
     title = {Fast reaction limit and forward-backward diffusion: {A} {Radon{\textendash}Nikodym} approach},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {189--203},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.279},
     language = {en},
}

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Jakub Skrzeczkowski. Fast reaction limit and forward-backward diffusion: A Radon–Nikodym approach. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 189-203. doi : 10.5802/crmath.279. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.279/

Bibliographie
Cité par

[1] Graham Andrews; John MacLeod Ball Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity, J. Differ. Equations, Volume 44 (1982) no. 2, pp. 306-341 (Special issue dedicated to J. P. LaSalle) | DOI | MR | Zbl

[2] John Macleod Ball A version of the fundamental theorem for Young measures, PDEs and continuum models of phase transitions (Nice, 1988) (Michel Rascle, ed.) (Lecture Notes in Physics), Volume 344, Springer, 1989, pp. 207-215 | DOI | MR | Zbl

[3] Michiel Bertsch; Flavia Smarrazzo; Alberto Tesei Pseudoparabolic regularization of forward-backward parabolic equations: a logarithmic nonlinearity, Anal. PDE, Volume 6 (2013) no. 7, pp. 1719-1754 | DOI | MR | Zbl

[4] Michiel Bertsch; Flavia Smarrazzo; Alberto Tesei Pseudo-parabolic regularization of forward-backward parabolic equations: power-type nonlinearities, J. Reine Angew. Math., Volume 712 (2016), pp. 51-80 | DOI | MR | Zbl

[5] Dieter Bothe; Danielle Hilhorst A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., Volume 286 (2003) no. 1, pp. 125-135 | DOI | MR | Zbl

[6] Dieter Bothe; Michel Pierre; Guillaume Rolland Cross-diffusion limit for a reaction-diffusion system with fast reversible reaction, Commun. Partial Differ. Equations, Volume 37 (2012) no. 11, pp. 1940-1966 | DOI | MR | Zbl

[7] Emanuele Bottazzi Grid functions of nonstandard analysis in the theory of distributions and in partial differential equations, Adv. Math., Volume 345 (2019), pp. 429-482 | DOI | MR | Zbl

[8] Emanuele Bottazzi A grid function formulation of a class of ill-posed parabolic equations, J. Differ. Equations, Volume 271 (2021), pp. 39-75 | DOI | MR | Zbl

[9] Federica Bubba; Benoît Perthame; Camille Pouchol; Markus Schmidtchen Hele–Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues, Arch. Ration. Mech. Anal., Volume 236 (2020) no. 2, pp. 735-766 | DOI | MR | Zbl

[10] José A. Carrillo; Francis Filbet; Markus Schmidtchen Convergence of a finite volume scheme for a system of interacting species with cross-diffusion, Numer. Math., Volume 145 (2020) no. 3, pp. 473-511 | DOI | MR | Zbl

[11] Elaine C. M. Crooks; Danielle Hilhorst Self-similar fast-reaction limits for reaction-diffusion systems on unbounded domains, J. Differ. Equations, Volume 261 (2016) no. 3, pp. 2210-2250 | DOI | MR | Zbl

[12] Szymon Cygan; Anna Marciniak-Czochra; Grzegorz Karch; Kanako Suzuki Instability of all regular stationary solutions to reaction-diffusion-ODE systems (2021) (URL: https://arxiv.org/abs/2105.05023)

[13] Szymon Cygan; Anna Marciniak-Czochra; Grzegorz Karch; Kanako Suzuki Stable discontinuous stationary solutions to reaction-diffusion-ODE systems (2021) (In preparation, https://arxiv.org/abs/2111.01214)

[14] Esther S. Daus; Laurent Desvillettes; Ansgar Jüngel Cross-diffusion systems and fast-reaction limits, Bull. Sci. Math., Volume 159 (2020), 102824 | DOI | MR | Zbl

[15] Laurent Desvillettes; Thomas Lepoutre; Ayman Moussa Entropy, duality, and cross diffusion, SIAM J. Math. Anal., Volume 46 (2014) no. 1, pp. 820-853 | DOI | MR | Zbl

[16] Laurent Desvillettes; Thomas Lepoutre; Ayman Moussa; Ariane Trescases On the entropic structure of reaction-cross diffusion systems, Commun. Partial Differ. Equations, Volume 40 (2015) no. 9, pp. 1705-1747 | DOI | MR | Zbl

[17] Lawrence C. Evans A convergence theorem for a chemical diffusion-reaction system, Houston J. Math., Volume 6 (1980) no. 2, pp. 259-267 | MR | Zbl

[18] Lawrence C. Evans Weak convergence methods for nonlinear partial differential equations. Expository lectures from the CBMS regional conference held at Loyola University of Chicago, June 27-July 1, 1988, CBMS Regional Conference Series in Mathematics, 74, American Mathematical Society, 1990 (published for the Conference Board of the Mathematical Sciences, Washington, DC) | DOI | MR | Zbl

[19] Lawrence C. Evans; Manuel Portilheiro Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Models Methods Appl. Sci., Volume 14 (2004) no. 11, pp. 1599-1620 | DOI | MR | Zbl

[20] Danielle Hilhorst; Masayasu Mimura; Hirokazu Ninomiya Fast reaction limit of competition-diffusion systems, Handbook of differential equations: evolutionary equations. Vol. V (C. M. Dafermos, ed.) (Handbook of Differential Equations), Elsevier; North-Holland, 2009, pp. 105-168 | DOI | MR | Zbl

[21] Masato Iida; Harunori Monobe; Hideki Murakawa; Hirozaku Ninomiya Vanishing, moving and immovable interfaces in fast reaction limits, J. Differ. Equations, Volume 263 (2017) no. 5, pp. 2715-2735 | DOI | MR | Zbl

[22] Ansgar Jüngel The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, Volume 28 (2015) no. 6, pp. 1963-2001 | DOI | MR | Zbl

[23] Pauline Lafitte; Corrado Mascia Numerical exploration of a forward-backward diffusion equation, Math. Models Methods Appl. Sci., Volume 22 (2012) no. 6, 1250004 | DOI | MR | Zbl

[24] Yuan Lou; Wei-Ming Ni Diffusion, self-diffusion and cross-diffusion, J. Differ. Equations, Volume 131 (1996) no. 1, pp. 79-131 | DOI | MR | Zbl

[25] Anna Marciniak-Czochra; Grzegorz Karch; Kanako Suzuki Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl., Volume 99 (2013) no. 5, pp. 509-543 | DOI | MR | Zbl

[26] Corrado Mascia; Andrea Terracina; Alberto Tesei Two-phase entropy solutions of a forward-backward parabolic equation, Arch. Ration. Mech. Anal., Volume 194 (2009) no. 3, pp. 887-925 | DOI | MR | Zbl

[27] Ayman Moussa Some variants of the classical Aubin-Lions lemma, J. Evol. Equ., Volume 16 (2016) no. 1, pp. 65-93 | DOI | MR | Zbl

[28] Ayman Moussa; Benoît Perthame; Delphine Salort Backward parabolicity, cross-diffusion and Turing instability, J. Nonlinear Sci., Volume 29 (2019) no. 1, pp. 139-162 | DOI | MR | Zbl

[29] Hideki Murakawa; Hirokazu Ninomiya Fast reaction limit of a three-component reaction-diffusion system, J. Math. Anal. Appl., Volume 379 (2011) no. 1, pp. 150-170 | DOI | MR | Zbl

[30] François Murat A survey on compensated compactness, Contributions to modern calculus of variations (Bologna, 1985) (Pitman Research Notes in Mathematics Series), Volume 148, Longman Scientific & Technical, 1987, pp. 145-183 | MR

[31] Amy Novick-Cohen; Robert L. Pego Stable patterns in a viscous diffusion equation, Trans. Am. Math. Soc., Volume 324 (1991) no. 1, pp. 331-351 | DOI | MR | Zbl

[32] Pablo Pedregal Parametrized measures and variational principles, Progress in Nonlinear Differential Equations and their Applications, 30, Birkhäuser, 1997 | DOI | MR | Zbl

[33] Benoît Perthame; Jakub Skrzeczkowski Fast reaction limit with nonmonotone reaction function (2020) (to appear in Communications on Pure and Applied Mathematics, https://arxiv.org/abs/2008.11086)

[34] Pavel I. Plotnikov Equations with a variable direction of parabolicity and the hysteresis effect, Dokl. Akad. Nauk SSSR, Volume 330 (1993) no. 6, pp. 691-693 | MR | Zbl

[35] Pavel I. Plotnikov Passage to the limit with respect to viscosity in an equation with a variable direction of parabolicity, Differ. Uravn, Volume 30 (1994) no. 4, p. 665-674, 734 | MR | Zbl

[36] Filip Rindler Calculus of variations, Universitext, Springer, 2018 | DOI | MR | Zbl

[37] Franz Rothe Global solutions of reaction-diffusion systems, Lecture Notes in Mathematics, 1072, Springer, 1984 | DOI | MR | Zbl

[38] Filippo Santambrogio Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications, 87, Birkhäuser/Springer, 2015 (Calculus of variations, PDEs, and modeling,) | DOI | MR | Zbl

[39] Flavia Smarrazzo; Alberto Tesei Long-time behavior of solutions to a class of forward-backward parabolic equations, SIAM J. Math. Anal., Volume 42 (2010) no. 3, pp. 1046-1093 | DOI | MR | Zbl

[40] Flavia Smarrazzo; Alberto Tesei Degenerate regularization of forward-backward parabolic equations: the vanishing viscosity limit, Math. Ann., Volume 355 (2013) no. 2, pp. 551-584 | DOI | MR | Zbl

[41] Luc C. Tartar Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV (Pitman Research Notes in Mathematics Series), Volume 39, Pitman Publishing Inc, 1979, pp. 136-212 | MR | Zbl

[42] Andrea Terracina Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem, SIAM J. Math. Anal., Volume 43 (2011) no. 1, pp. 228-252 | DOI | MR | Zbl

[43] Andrea Terracina Non-uniqueness results for entropy two-phase solutions of forward-backward parabolic problems with unstable phase, J. Math. Anal. Appl., Volume 413 (2014) no. 2, pp. 963-975 | DOI | MR | Zbl

[44] Laurence C. Young Generalized surfaces in the calculus of variations, Ann. Math., Volume 43 (1942), pp. 84-103 | DOI | MR | Zbl

[45] Laurence C. Young Generalized surfaces in the calculus of variations. II, Ann. Math., Volume 43 (1942), pp. 530-544 | DOI | MR | Zbl

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