January  2015, 35(1): 483-512. doi: 10.3934/dcds.2015.35.483

The singular limit problem in a phase separation model with different diffusion rates $^*$

1. 

Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan 430071, China

Received  August 2013 Revised  June 2014 Published  August 2014

In this paper we study the singularly perturbed parabolic system of competing species. This problem exhibit a ``phase separation" phenomena when the interaction between different species is very strong. We are concerned with the case where different species may have different diffusion rates. We identify its singular limit with the heat flow (i.e. gradient flow) of harmonic maps into a metric space with non-positive curvature, by establishing the system of differential inequalities satisfied by this heat flow and uniqueness of the solution to the corresponding initial-boundary value problem.
Citation: Kelei Wang. The singular limit problem in a phase separation model with different diffusion rates $^*$. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 483-512. doi: 10.3934/dcds.2015.35.483
References:
[1]

H. Brezis and A. Ponce, Kato's inequality when $\Delta u$ is a measure,, C. R. Math. Acad. Sci. Paris, 338 (2004), 599.  doi: 10.1016/j.crma.2003.12.032.  Google Scholar

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L. Caffarelli, A. Karakhanyan and F. Lin, The geometry of solutions to a segregation problem for non-divergence systems,, J. Fixed Point Theory Appl., 5 (2009), 319.  doi: 10.1007/s11784-009-0110-0.  Google Scholar

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L. Caffarelli and F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries,, Journal of the American Mathematical Society, 21 (2008), 847.  doi: 10.1090/S0894-0347-08-00593-6.  Google Scholar

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L. Caffarelli and F. Lin, Nonlocal heat flows preserving the $L^2$ energy,, Discrete and Continuous Dynamical Systems (DCDS-A), 23 (2009), 49.  doi: 10.3934/dcds.2009.23.49.  Google Scholar

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L. Caffarelli and S. Salsa, A geometric Approach to Free Boundary Problems,, Graduate Studies in Mathematics, (2005).   Google Scholar

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M. Conti, S. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction diffusion systems,, Indiana Univ. Math. J., 54 (2005), 779.  doi: 10.1512/iumj.2005.54.2506.  Google Scholar

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M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems,, Adv. Math., 195 (2005), 524.  doi: 10.1016/j.aim.2004.08.006.  Google Scholar

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E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system,, European J. Appl. Math., 10 (1999), 97.  doi: 10.1017/S0956792598003660.  Google Scholar

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E. N. Dancer, K. Wang and Z. Zhang, Dynamics of strongly competing systems with many species,, Trans. Amer. Math. Soc., 364 (2012), 961.  doi: 10.1090/S0002-9947-2011-05488-7.  Google Scholar

[11]

E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species,, Journal of Differential Equations, 251 (2011), 2737.  doi: 10.1016/j.jde.2011.06.015.  Google Scholar

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E. N. Dancer, K. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture,, Journal of Functional Analysis, 262 (2012), 1087.  doi: 10.1016/j.jfa.2011.10.013.  Google Scholar

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E. N. Dancer, K. Wang and Z. Zhang, Addendum to "The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture", [J. Funct. Anal., 262 (2012), 1087.  doi: 10.1016/j.jfa.2012.10.009.  Google Scholar

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E. N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction,, Journal of Differential Equations, 182 (2002), 470.  doi: 10.1006/jdeq.2001.4102.  Google Scholar

[15]

E. De Giorgi, New problems on minimizing movements,, in Boundary Value Problems for PDE and Applications (eds. C. Baiochhi and J.L. Lions), (1993), 81.   Google Scholar

[16]

L. Evans, A chemical diffusion-reaction free boundary problem,, Nonlinear Anal., 6 (1982), 455.  doi: 10.1016/0362-546X(82)90059-1.  Google Scholar

[17]

N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability,, Calculus of Variations and Partial Differential Equations, 39 (2010), 101.  doi: 10.1007/s00526-009-0303-9.  Google Scholar

[18]

M. Gromov and R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one,, Publications Mathématiques de L'IHÉS, 76 (1992), 165.   Google Scholar

[19]

W. Littman, G. Stampacchia and H. Weinberger, Regular points for elliptic equations with discontinuous coefficients,, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 17 (1963), 43.   Google Scholar

[20]

F. Lin and X. Yang, Geometric Measure Theory: An Introduction,, Advanced Mathematics (Beijing/Boston), (2002).   Google Scholar

[21]

U. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps,, Commun. Anal. Geom., 6 (1998), 199.   Google Scholar

[22]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition,, Comm. Pure Appl. Math., 63 (2010), 267.   Google Scholar

[23]

W. Rudin, Real and Complex Analysis,, Tata McGraw-Hill Education, (2006).   Google Scholar

[24]

K. Wang and Z. Zhang, Some new results in competing systems with many species,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 739.  doi: 10.1016/j.anihpc.2009.11.004.  Google Scholar

show all references

References:
[1]

H. Brezis and A. Ponce, Kato's inequality when $\Delta u$ is a measure,, C. R. Math. Acad. Sci. Paris, 338 (2004), 599.  doi: 10.1016/j.crma.2003.12.032.  Google Scholar

[2]

L. Caffarelli, A. Karakhanyan and F. Lin, The geometry of solutions to a segregation problem for non-divergence systems,, J. Fixed Point Theory Appl., 5 (2009), 319.  doi: 10.1007/s11784-009-0110-0.  Google Scholar

[3]

L. Caffarelli and F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries,, Journal of the American Mathematical Society, 21 (2008), 847.  doi: 10.1090/S0894-0347-08-00593-6.  Google Scholar

[4]

L. Caffarelli and F. Lin, Nonlocal heat flows preserving the $L^2$ energy,, Discrete and Continuous Dynamical Systems (DCDS-A), 23 (2009), 49.  doi: 10.3934/dcds.2009.23.49.  Google Scholar

[5]

L. Caffarelli and S. Salsa, A geometric Approach to Free Boundary Problems,, Graduate Studies in Mathematics, (2005).   Google Scholar

[6]

J. Cannon and C. Hill, On the movement of a chemical reaction interface,, Indiana Univ. Math. J., 20 (1970), 429.  doi: 10.1512/iumj.1971.20.20037.  Google Scholar

[7]

M. Conti, S. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction diffusion systems,, Indiana Univ. Math. J., 54 (2005), 779.  doi: 10.1512/iumj.2005.54.2506.  Google Scholar

[8]

M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems,, Adv. Math., 195 (2005), 524.  doi: 10.1016/j.aim.2004.08.006.  Google Scholar

[9]

E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system,, European J. Appl. Math., 10 (1999), 97.  doi: 10.1017/S0956792598003660.  Google Scholar

[10]

E. N. Dancer, K. Wang and Z. Zhang, Dynamics of strongly competing systems with many species,, Trans. Amer. Math. Soc., 364 (2012), 961.  doi: 10.1090/S0002-9947-2011-05488-7.  Google Scholar

[11]

E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species,, Journal of Differential Equations, 251 (2011), 2737.  doi: 10.1016/j.jde.2011.06.015.  Google Scholar

[12]

E. N. Dancer, K. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture,, Journal of Functional Analysis, 262 (2012), 1087.  doi: 10.1016/j.jfa.2011.10.013.  Google Scholar

[13]

E. N. Dancer, K. Wang and Z. Zhang, Addendum to "The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture", [J. Funct. Anal., 262 (2012), 1087.  doi: 10.1016/j.jfa.2012.10.009.  Google Scholar

[14]

E. N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction,, Journal of Differential Equations, 182 (2002), 470.  doi: 10.1006/jdeq.2001.4102.  Google Scholar

[15]

E. De Giorgi, New problems on minimizing movements,, in Boundary Value Problems for PDE and Applications (eds. C. Baiochhi and J.L. Lions), (1993), 81.   Google Scholar

[16]

L. Evans, A chemical diffusion-reaction free boundary problem,, Nonlinear Anal., 6 (1982), 455.  doi: 10.1016/0362-546X(82)90059-1.  Google Scholar

[17]

N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability,, Calculus of Variations and Partial Differential Equations, 39 (2010), 101.  doi: 10.1007/s00526-009-0303-9.  Google Scholar

[18]

M. Gromov and R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one,, Publications Mathématiques de L'IHÉS, 76 (1992), 165.   Google Scholar

[19]

W. Littman, G. Stampacchia and H. Weinberger, Regular points for elliptic equations with discontinuous coefficients,, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 17 (1963), 43.   Google Scholar

[20]

F. Lin and X. Yang, Geometric Measure Theory: An Introduction,, Advanced Mathematics (Beijing/Boston), (2002).   Google Scholar

[21]

U. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps,, Commun. Anal. Geom., 6 (1998), 199.   Google Scholar

[22]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition,, Comm. Pure Appl. Math., 63 (2010), 267.   Google Scholar

[23]

W. Rudin, Real and Complex Analysis,, Tata McGraw-Hill Education, (2006).   Google Scholar

[24]

K. Wang and Z. Zhang, Some new results in competing systems with many species,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 739.  doi: 10.1016/j.anihpc.2009.11.004.  Google Scholar

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