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Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum
Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria
1. | Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison |
2. | Department of Mathematics, University of Wisconsin-Madison |
3. | Department of Mathematics, West Virginia University |
In this paper we study the rate of convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. We first analyze a three-species system with boundary equilibria in some stoichiometric classes, and whose right hand side is bounded above by a quadratic nonlinearity in the positive orthant. We prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.
References:
[1] |
D. F. Anderson,
Global asymptotic stability for a class of nonlinear chemical equations, SIAM J. Appl. Math., 68 (2008), 1464-1476.
doi: 10.1137/070698282. |
[2] |
D. F. Anderson and A. Shiu,
The dynamics of weakly reversible population processes near facets, SIAM J. Appl. Math., 70 (2010), 1840-1858.
doi: 10.1137/090764098. |
[3] |
D. F. Anderson,
A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508.
doi: 10.1137/11082631X. |
[4] |
A. Arnold, P. Markowich, G. Toscani and A. Unterreiter,
On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.
doi: 10.1081/PDE-100002246. |
[5] |
W. X. Chen, C. M. Li and E. S. Wright,
On a nonlinear parabolic system-modeling chemical reactions in rivers, Communications On Pure And Applied Analysis, 4 (2005), 889-899.
doi: 10.3934/cpaa.2005.4.889. |
[6] |
M. Choulli, L. Kayser and E. M. Ouhabaz,
Observations on Gaussian upper bounds for Neumann heat kernels, Bulletin of the Australian Mathematical Society, 92 (2015), 429-439.
doi: 10.1017/S0004972715000611. |
[7] |
G. Craciun, A. Dickenstein, A. Shiu and B. Sturmfels,
Toric dynamical systems, Journal of Symbolic Computation, 44 (2009), 1551-1565.
doi: 10.1016/j.jsc.2008.08.006. |
[8] |
G. Craciun, F. Nazarov and C. Pantea,
Persistence and permanence of mass-action and power-law dynamical systems, SIAM J. Appl. Math., 73 (2013), 305-329.
doi: 10.1137/100812355. |
[9] |
G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, (2016), arXiv: 1501.02860. Google Scholar |
[10] |
L. Desvillettes and K. Fellner,
Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176.
doi: 10.1016/j.jmaa.2005.07.003. |
[11] |
L. Desvillettes and K. Fellner,
Entropy methods for reaction-diffusion equations: Slowly growing a-priori bounds, Rev. Mat. Iberoamericana, 24 (2008), 407-431.
doi: 10.4171/RMI/541. |
[12] |
L. Desvillettes and K. Fellner, Exponential convergence to equilibrium for a nonlinear reaction-diffusion systems arising in reversible chemistry, System Modelling and Optimization, IFIP AICT, 443 (2014), 96-104. Google Scholar |
[13] |
L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle,
About global existence for quadratic systems of reaction-diffusion, J. Adv. Nonlinear Stud., 7 (2007), 491-511.
doi: 10.1515/ans-2007-0309. |
[14] |
L. Desvillettes, K. Fellner and B. Q. Tang,
Trend to equilibrium for reaction-diffusion system arising from complex balanced chemical reaction networks, SIAM J. Math. Anal., 49 (2017), 2666-2709.
doi: 10.1137/16M1073935. |
[15] |
M. Feinberg,
Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972/73), 187-194.
doi: 10.1007/BF00255665. |
[16] |
K. Fellner, W. Prager and B. Q. Tang, The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks, Kinet. Relat. Models, 10 (2017), 1055–1087, arXiv: 1504.08221.
doi: 10.3934/krm.2017042. |
[17] |
K. Fellner and B. Q. Tang, Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction-diffusion systems, Z. Angew. Math. Phys., 69 (2018), Paper No. 54, 30 pp.
doi: 10.1007/s00033-018-0948-3. |
[18] |
J. Fischer,
Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Arch. Ration. Mech. Anal., 218 (2015), 553-587.
doi: 10.1007/s00205-015-0866-x. |
[19] |
W. E. Fitzgibbon, J. Morgan and R. Sanders,
Global existence and boundedness for a class of inhomogeneous semilinear parabolic systems, Nonlin. Anal., 19 (1992), 885-899.
doi: 10.1016/0362-546X(92)90057-L. |
[20] |
M. Gopalkrishnan, E. Miller and A. Shiu,
A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn. Syst., 13 (2014), 758-797.
doi: 10.1137/130928170. |
[21] |
M. Gopalkrishnan, E. Miller and A. Shiu, A projection argument for differential inclusions, with applications to persistence of mass-action kinetics, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), Paper 025, 25 pp.
doi: 10.3842/SIGMA.2013.025. |
[22] |
F. Horn and R. Jackson,
General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.
doi: 10.1007/BF00251225. |
[23] |
F. Horn,
The dynamics of open reaction systems, Mathematical Aspects of Chemical and Biochemical Problems and Quantum Chemistry, SIAM-AMS Proceedings, Amer. Math. Soc., Providence, R.I., 8 (1974), 125-137.
|
[24] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Trans. Math. Monographs, AMS, 23 (1995). Google Scholar |
[25] |
A. Mielke, J. Haskovec and P. A. Markowich,
On uniform decay of the entropy for reaction-diffusion systems, J. Dynam. Differential Equations, 27 (2015), 897-928.
doi: 10.1007/s10884-014-9394-x. |
[26] |
M. Minchevaand and D. Siegel,
Stability of mass action reaction-diffusion systems, Nonlinear Anal., 56 (2004), 1105-1131.
doi: 10.1016/j.na.2003.10.025. |
[27] |
F. Mohamed, C. Pantea and A. Tudorascu,
Chemical reaction-diffusion networks: Convergence of the method of lines, J. Math. Chem., 56 (2018), 30-68.
doi: 10.1007/s10910-017-0779-z. |
[28] |
C. Pantea,
On the persistence and global stability of mass-action systems, SIAM J. Math. Anal., 44 (2012), 1636-1673.
doi: 10.1137/110840509. |
[29] |
M. Pierre, T. Suzuki and H. Umakoshi,
Asymptotic behavior in chemical reaction-diffusion systems with boundary equilibria, J. Appl. Anal. Comp., 8 (2018), 836-858.
doi: 10.11948/2018.836. |
[30] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[31] |
F. Rothe, Global Solutions of Reaction-Diffusion System, Lecture Notes in Mathematics, 1072. Springer-Verlag, Berlin, 1984.
doi: 10.1007/BFb0099278. |
[32] |
D. Siegel and D. MacLean,
Global stability of complex balanced mechanisms, J. Math. Chem., 27 (2000), 89-110.
doi: 10.1023/A:1019183206064. |
[33] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der mathematischen Wissenschaften, 258. Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[34] |
E. D. Sontag,
Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction, IEEE Trans. Automat. Control, 46 (2001), 1028-1047.
doi: 10.1109/9.935056. |
[35] |
M. E. Taylor, Partial Differential Equation Ⅲ. Nonlinear Equations, Springer Series Applied Mathematical Sciences, 117. Springer, New York, 2011.
doi: 10.1007/978-1-4419-7049-7. |
[36] |
P. Weidemaier,
Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L^p$-norm, Electron. Res. Announc. Amer. Math. Soc., 8 (2002), 47-51.
doi: 10.1090/S1079-6762-02-00104-X. |
show all references
References:
[1] |
D. F. Anderson,
Global asymptotic stability for a class of nonlinear chemical equations, SIAM J. Appl. Math., 68 (2008), 1464-1476.
doi: 10.1137/070698282. |
[2] |
D. F. Anderson and A. Shiu,
The dynamics of weakly reversible population processes near facets, SIAM J. Appl. Math., 70 (2010), 1840-1858.
doi: 10.1137/090764098. |
[3] |
D. F. Anderson,
A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508.
doi: 10.1137/11082631X. |
[4] |
A. Arnold, P. Markowich, G. Toscani and A. Unterreiter,
On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.
doi: 10.1081/PDE-100002246. |
[5] |
W. X. Chen, C. M. Li and E. S. Wright,
On a nonlinear parabolic system-modeling chemical reactions in rivers, Communications On Pure And Applied Analysis, 4 (2005), 889-899.
doi: 10.3934/cpaa.2005.4.889. |
[6] |
M. Choulli, L. Kayser and E. M. Ouhabaz,
Observations on Gaussian upper bounds for Neumann heat kernels, Bulletin of the Australian Mathematical Society, 92 (2015), 429-439.
doi: 10.1017/S0004972715000611. |
[7] |
G. Craciun, A. Dickenstein, A. Shiu and B. Sturmfels,
Toric dynamical systems, Journal of Symbolic Computation, 44 (2009), 1551-1565.
doi: 10.1016/j.jsc.2008.08.006. |
[8] |
G. Craciun, F. Nazarov and C. Pantea,
Persistence and permanence of mass-action and power-law dynamical systems, SIAM J. Appl. Math., 73 (2013), 305-329.
doi: 10.1137/100812355. |
[9] |
G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, (2016), arXiv: 1501.02860. Google Scholar |
[10] |
L. Desvillettes and K. Fellner,
Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176.
doi: 10.1016/j.jmaa.2005.07.003. |
[11] |
L. Desvillettes and K. Fellner,
Entropy methods for reaction-diffusion equations: Slowly growing a-priori bounds, Rev. Mat. Iberoamericana, 24 (2008), 407-431.
doi: 10.4171/RMI/541. |
[12] |
L. Desvillettes and K. Fellner, Exponential convergence to equilibrium for a nonlinear reaction-diffusion systems arising in reversible chemistry, System Modelling and Optimization, IFIP AICT, 443 (2014), 96-104. Google Scholar |
[13] |
L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle,
About global existence for quadratic systems of reaction-diffusion, J. Adv. Nonlinear Stud., 7 (2007), 491-511.
doi: 10.1515/ans-2007-0309. |
[14] |
L. Desvillettes, K. Fellner and B. Q. Tang,
Trend to equilibrium for reaction-diffusion system arising from complex balanced chemical reaction networks, SIAM J. Math. Anal., 49 (2017), 2666-2709.
doi: 10.1137/16M1073935. |
[15] |
M. Feinberg,
Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972/73), 187-194.
doi: 10.1007/BF00255665. |
[16] |
K. Fellner, W. Prager and B. Q. Tang, The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks, Kinet. Relat. Models, 10 (2017), 1055–1087, arXiv: 1504.08221.
doi: 10.3934/krm.2017042. |
[17] |
K. Fellner and B. Q. Tang, Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction-diffusion systems, Z. Angew. Math. Phys., 69 (2018), Paper No. 54, 30 pp.
doi: 10.1007/s00033-018-0948-3. |
[18] |
J. Fischer,
Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Arch. Ration. Mech. Anal., 218 (2015), 553-587.
doi: 10.1007/s00205-015-0866-x. |
[19] |
W. E. Fitzgibbon, J. Morgan and R. Sanders,
Global existence and boundedness for a class of inhomogeneous semilinear parabolic systems, Nonlin. Anal., 19 (1992), 885-899.
doi: 10.1016/0362-546X(92)90057-L. |
[20] |
M. Gopalkrishnan, E. Miller and A. Shiu,
A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn. Syst., 13 (2014), 758-797.
doi: 10.1137/130928170. |
[21] |
M. Gopalkrishnan, E. Miller and A. Shiu, A projection argument for differential inclusions, with applications to persistence of mass-action kinetics, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), Paper 025, 25 pp.
doi: 10.3842/SIGMA.2013.025. |
[22] |
F. Horn and R. Jackson,
General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.
doi: 10.1007/BF00251225. |
[23] |
F. Horn,
The dynamics of open reaction systems, Mathematical Aspects of Chemical and Biochemical Problems and Quantum Chemistry, SIAM-AMS Proceedings, Amer. Math. Soc., Providence, R.I., 8 (1974), 125-137.
|
[24] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Trans. Math. Monographs, AMS, 23 (1995). Google Scholar |
[25] |
A. Mielke, J. Haskovec and P. A. Markowich,
On uniform decay of the entropy for reaction-diffusion systems, J. Dynam. Differential Equations, 27 (2015), 897-928.
doi: 10.1007/s10884-014-9394-x. |
[26] |
M. Minchevaand and D. Siegel,
Stability of mass action reaction-diffusion systems, Nonlinear Anal., 56 (2004), 1105-1131.
doi: 10.1016/j.na.2003.10.025. |
[27] |
F. Mohamed, C. Pantea and A. Tudorascu,
Chemical reaction-diffusion networks: Convergence of the method of lines, J. Math. Chem., 56 (2018), 30-68.
doi: 10.1007/s10910-017-0779-z. |
[28] |
C. Pantea,
On the persistence and global stability of mass-action systems, SIAM J. Math. Anal., 44 (2012), 1636-1673.
doi: 10.1137/110840509. |
[29] |
M. Pierre, T. Suzuki and H. Umakoshi,
Asymptotic behavior in chemical reaction-diffusion systems with boundary equilibria, J. Appl. Anal. Comp., 8 (2018), 836-858.
doi: 10.11948/2018.836. |
[30] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[31] |
F. Rothe, Global Solutions of Reaction-Diffusion System, Lecture Notes in Mathematics, 1072. Springer-Verlag, Berlin, 1984.
doi: 10.1007/BFb0099278. |
[32] |
D. Siegel and D. MacLean,
Global stability of complex balanced mechanisms, J. Math. Chem., 27 (2000), 89-110.
doi: 10.1023/A:1019183206064. |
[33] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der mathematischen Wissenschaften, 258. Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[34] |
E. D. Sontag,
Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction, IEEE Trans. Automat. Control, 46 (2001), 1028-1047.
doi: 10.1109/9.935056. |
[35] |
M. E. Taylor, Partial Differential Equation Ⅲ. Nonlinear Equations, Springer Series Applied Mathematical Sciences, 117. Springer, New York, 2011.
doi: 10.1007/978-1-4419-7049-7. |
[36] |
P. Weidemaier,
Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L^p$-norm, Electron. Res. Announc. Amer. Math. Soc., 8 (2002), 47-51.
doi: 10.1090/S1079-6762-02-00104-X. |

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