Abstract
The convergence to equilibrium for renormalised solutions to nonlinear reaction–diffusion systems is studied. The considered reaction–diffusion systems arise from chemical reaction networks with mass action kinetics and satisfy the complex balanced condition. By applying the so-called entropy method, we show that if the system does not have boundary equilibria, i.e. equilibrium states lying on the boundary of \({\mathbb {R}}_+^N\), then any renormalised solution converges exponentially to the complex balanced equilibrium with a rate, which can be computed explicitly up to a finite-dimensional inequality. This inequality is proven via a contradiction argument and thus not explicitly. An explicit method of proof, however, is provided for a specific application modelling a reversible enzyme reaction by exploiting the specific structure of the conservation laws. Our approach is also useful to study the trend to equilibrium for systems possessing boundary equilibria. More precisely, to show the convergence to equilibrium for systems with boundary equilibria, we establish a sufficient condition in terms of a modified finite-dimensional inequality along trajectories of the system. By assuming this condition, which roughly means that the system produces too much entropy to stay close to a boundary equilibrium for infinite time, the entropy method shows exponential convergence to equilibrium for renormalised solutions to complex balanced systems with boundary equilibria.
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Ardenson, D.F.: A proof of the global attractor conjecture in the single linkage class case. SIAM J. Appl. Math. 71, 1487–1508 (2011)
Ardenson, D.F.: A short note on the Lyapunov function for complex-balanced chemical reaction networks, online notes. https://www.math.wisc.edu/~anderson/CRNT_Lyapunov.pdf
Arnold, A., Markowich, P., Toscani, G., Unterreiter, A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations. Commun. Partial Differ. Equ. 26, 43–100 (2001)
Boltzmann, L.: Gastheorie, Leipzig, J. A. Barth (1896)
Boltzmann, L.: Neuer Beweis zweier Sätze über das Wärmegleichgewicht unter mehratomigen Gasmolekülen. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien 95, 153–164 (1887)
Cañizo, J.A., Desvillettes, L., Fellner, K.: Improved duality estimates and applications to reaction–diffusion equations. Commun. Partial Differ. Equ. 39, 1185–1204 (2014)
Chipot, M.: Elements of Nonlinear Analysis. Birkhäuser Advanced Texts. Birkhäuser Verlag, Basel (2000)
Craciun, G., Nazarov, F., Pantea, C.: Persistence and permanence of mass-action and power-law dynamical systems. SIAM J. Appl. Math. 73, 305–329 (2013)
Craciun, G.: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture (2015). arXiv:1501.02860
Craciun, G., Dickenstein, A., Shiu, A., Sturmfels, B.: Toric dynamical systems. J. Symb. Comput. 44(11), 1551–1565 (2009)
Desvillettes, L., Fellner, K.: Exponential decay toward equilibrium via entropy methods for reaction–diffusion equations. J. Math. Anal. Appl. 319, 157–176 (2006)
Desvillettes, L., Fellner, K.: Entropy methods for reaction–diffusion equations: slowly growing a priori bounds. Rev. Mat. Iberoam. 24, 407–431 (2008)
Desvillettes, L., Fellner, K.: Exponential convergence to equilibrium for a nonlinear reaction–diffusion systems arising in reversible chemistry. Syst. Model. Optim. IFIP AICT 443, 96–104 (2014)
Desvillettes, L., Fellner, K., Pierre, M., Vovelle, J.: About global existence of quadratic systems of reaction–diffusion. J. Adv. Nonlinear Stud. 7, 491–511 (2007)
Desvillettes, L., Fellner, K., Tang, B.Q.: Trend to equilibrium for reaction–diffusion systems arising from complex balanced chemical reaction networks. SIAM J. Math. Anal. 49(4), 2666–2709 (2017)
Eliaš, J.: Mathematical Model of the Role and Temporal Dynamics of Protein p53 After Drug-Induced DNA Damage. PhD Thesis, Pierre and Marie Curie University (2015)
Eliaš, J.: Trend to equilibrium for a reaction-diffusion system modelling reversible enzyme reaction. Bull. Math. Biol. 80(1), 104–129 (2018). arXiv:1610.07172
Fellner, K., Laamri, E.-H.: Exponential decay towards equilibrium and global classical solutions for nonlinear reaction-diffusion systems. J. Evol. Equ. 16(3), 681–704 (2016)
Feinberg, M.: Lectures on Chemical Reaction Networks. University of Wisconsin-Madison. https://crnt.osu.edu/LecturesOnReactionNetworks (1979)
Feinberg, M.: Chemical reaction network structure and the stability of complex isothermal reactors. I. The deficiency zero and deficiency one theorems. Chem. Eng. Sci. 42, 2229–2268 (1987)
Feinberg, M., Horn, F.J.M.: Dynamics of open chemical systems and the algebraic structure of the underlying reaction network. Chem. Eng. Sci. 29, 775–787 (1974)
Fellner, K., Laamri, E.-H.: Exponential decay towards equilibrium and global classical solutions for nonlinear reaction–diffusion systems. J. Evol. Equ. 16, 681–704 (2016)
Fellner, K., Latos, E., Suzuki, T.: Global classical solutions for mass-conserving, (super)-quadratic reaction–diffusion systems in three and higher space dimensions. Discrete Contin. Dyn. Syst. Ser. B. 21(10), 3441–3462 (2016)
Fellner, K., Latos, E., Tang, B.Q.: Well-posedness and exponential equilibration of a volume-surface reaction–diffusion system with nonlinear boundary coupling. Ann. Inst. H. Poincaré Anal. Non Linéaire 35(3), 643–673 (2018)
Fellner, K., Prager, W., Tang, B.Q.: The entropy method for reaction–diffusion systems without detailed balance: first order chemical reaction networks. Kinet. Relat. Models 10(4), 1055–1087 (2017)
Fellner, K., Tang, B.Q.: Explicit exponential convergence to equilibrium for mass action reaction–diffusion systems with detailed balance condition. Nonlinear Anal. 159, 145–180 (2017)
Fischer, J.: Global existence of renormalized solutions to entropy-dissipating reaction–diffusion systems. Arch. Ration. Mech. Anal. 218, 553–587 (2015)
Fischer, J.: Weak-strong uniqueness of solutions to entropy-dissipating reaction–diffusion equations. Nonlinear Anal. 159, 181–207 (2017)
Fontbona, J., Jourdain, B.: A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations. Ann. Probab. 44, 131–170 (2016)
Gröger, K.: Asymptotic behavior of solutions to a class of diffusion–reaction equations. Math. Nachr. 112(1), 19–33 (1983)
Gröger, K.: On the existence of steady states of certain reaction–diffusion systems. Arch. Ration. Mech. Anal. 92(4), 297–306 (1986)
Gröger, K.: Free Energy Estimates and Asymptotic Behaviour of Reaction–Diffusion Processes. Preprint 20, Institut für Angewandte Analysis und Stochastik, Berlin (1992)
Glitzky, A., Gröger, K., Hünlich, R.: Free energy and dissipation rate for reaction–diffusion processes of electrically charged species. Appl. Anal. 60, 201–217 (1996)
Glitzky, A., Hünlich, R.: Energetic estimates and asymptotics for electro-reaction–diffusion systems. Z. Angew. Math. Mech. 77, 823–832 (1997)
Gopalkrishnan, M.: On the Lyapunov function for complex-balanced mass-action systems. arXiv:1312.3043 (2013)
Gopalkrishnan, M., Miller, E., Shiu, A.: A geometric approach to the global attractor conjecture. SIAM J. Appl. Dyn. Syst. 13, 758–797 (2014)
Gentil, I., Zegarlinski, B.: Asymptotic behaviour of a general reversible chemical reaction–diffusion equation. Kinet. Relat. Models 3, 427–444 (2010)
Gross, L.: Logarithmic sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)
Haskovec, J., Hittmeir, S., Markowich, P., Mielke, A.: Decay to equilibrium for energy-reaction–diffusion systems. SIAM J. Math. Anal. 50(1), 1037–1075 (2018)
Horn, F.J.M.: Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal. 49, 172–186 (1972)
Horn, F.J.M.: The dynamics of open reaction systems. In: SIAM-AMS Proceedings, vol. VIII, pp. 125–137. SIAM, Philadelphia (1974)
Horn, F.J.M., Jackson, R.: General mass action kinetics. Arch. Ration. Mech. Anal. 47, 81–116 (1972)
Kirane, M.: On stabilization of solutions of the system of parabolic differential equations describing the kinetics of an autocatalytic reversible chemical reaction. Bull. Inst. Math. Acad. Sin. 18(4), 369–377 (1990)
Michel, P., Mischler, S., Perthame, B.: General relative entropy inequality: an illustration on growth models. J. Math. Pures Appl. 84, 1235–1260 (2005)
Mielke, A., Haskovec, J., Markowich, P.A.: On uniform decay of the entropy for reaction–diffusion systems. J. Dyn. Differ. Equ. 27, 897–928 (2015)
Pantea, C.: On the persistence and global stability of mass-action systems. SIAM J. Math. Anal. 44, 1636–1673 (2012)
Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Springer, Berlin (1992)
Perthame, B.: Transport Equations in Biology. Birkhäuser, Basel (2007)
Pierre, M.: Weak solutions and supersolutions in \(L^1\) for reaction–diffusion systems. J. Evol. Equ. 3, 153–168 (2003)
Pierre, M.: Global existence in reaction–diffusion systems with control of mass: a survey. Milan J. Math. 78(2), 417–455 (2010)
Pierre, M., Schmitt, D.: Blowup in reaction–diffusion systems with dissipation of mass. SIAM Rev. 42(1), 93–106 (2000)
Pierre, M., Suzuki, T., Zou, R.: Asymptotic behavior of solutions to chemical reaction–diffusion systems. J. Math. Anal. Appl. 450(1), 152–168 (2017)
Quittner, P., Souplet, P.: Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States. Springer, Berlin (2007)
Rothe, F.: Global Solutions of Reaction–Diffusion Systems. Lecture Notes in Mathematics. Springer, Berlin (1984)
Siegel, D., Johnston, M.D.: Linearization of complex balanced reaction systems (2008). https://www.researchgate.net/publication/253758553_Linearization_of_Complex_Balanced_Chemical_Reaction_Systems
Siegel, D., MacLean, D.: Global stability of complex balanced mechanisms. J. Math. Chem. 27, 89–110 (2000)
Toscani, G., Villani, C.: On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Stat. Phys. 98(5–6), 1279–1309 (2000)
Volpert, A.I.: Differential equations on graphs. Mat. Sb. 88(130), 578–588 (1972) (in Russian). Math. USSR-Sb. 17, 571–582 (1972) (in English)
Wegscheider, R.: Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme. Monatshefte für Chemie 32, 849–906 (1901)
Willett, D.: A linear generalization of Gronwall’s inequality. Proc. Am. Math. Soc. 16, 774–778 (1965)
Wegscheider, R.: Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reaktionskinetik homogener Systeme. Zeitschrift für physikalische Chemie 39(1), 257–303 (1902)
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Open access funding provided by University of Graz. This work is partially supported by International Research Training Group IGDK 1754 and NAWI Graz. Moreover, the authors would like to acknowledge the referees for the careful reading of the paper and the suggested improvements.
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Fellner, K., Tang, B.Q. Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction–diffusion systems. Z. Angew. Math. Phys. 69, 54 (2018). https://doi.org/10.1007/s00033-018-0948-3
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DOI: https://doi.org/10.1007/s00033-018-0948-3