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 and Continuous Dynamical Systems, 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), 599-604. doi: 10.1016/j.crma.2003.12.032.

[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), 319-351. doi: 10.1007/s11784-009-0110-0.

[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), 847-862. doi: 10.1090/S0894-0347-08-00593-6.

[4]

L. Caffarelli and F. Lin, Nonlocal heat flows preserving the $L^2$ energy, Discrete and Continuous Dynamical Systems (DCDS-A), 23 (2009), 49-64. doi: 10.3934/dcds.2009.23.49.

[5]

L. Caffarelli and S. Salsa, A geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, 68, American Mathematical Society, Providence, RI, 2005.

[6]

J. Cannon and C. Hill, On the movement of a chemical reaction interface, Indiana Univ. Math. J., 20 (1970), 429-454. doi: 10.1512/iumj.1971.20.20037.

[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), 779-815. doi: 10.1512/iumj.2005.54.2506.

[8]

M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560. doi: 10.1016/j.aim.2004.08.006.

[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), 97-115. doi: 10.1017/S0956792598003660.

[10]

E. N. Dancer, K. Wang and Z. Zhang, Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961-1005. doi: 10.1090/S0002-9947-2011-05488-7.

[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), 2737-2769. doi: 10.1016/j.jde.2011.06.015.

[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), 1087-1131. doi: 10.1016/j.jfa.2011.10.013.

[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), 1087-1131], Journal of Functional Analysis, 264 (2013), 1125-1129. doi: 10.1016/j.jfa.2012.10.009.

[14]

E. N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction, Journal of Differential Equations, 182 (2002), 470-489. doi: 10.1006/jdeq.2001.4102.

[15]

E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for PDE and Applications (eds. C. Baiochhi and J.L. Lions), RMA Res. Notes Appl. Math., 29, Masson, Paris, 1993, 81-98.

[16]

L. Evans, A chemical diffusion-reaction free boundary problem, Nonlinear Anal., 6 (1982), 455-466. doi: 10.1016/0362-546X(82)90059-1.

[17]

N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability, Calculus of Variations and Partial Differential Equations, 39 (2010), 101-120. doi: 10.1007/s00526-009-0303-9.

[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), 165-246.

[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), 43-77.

[20]

F. Lin and X. Yang, Geometric Measure Theory: An Introduction, Advanced Mathematics (Beijing/Boston), Science Press, Beijing; International Press, Boston, 2002.

[21]

U. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Commun. Anal. Geom., 6 (1998), 199-253.

[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), 267-302.

[23]

W. Rudin, Real and Complex Analysis, Tata McGraw-Hill Education, 2006.

[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), 739-761. doi: 10.1016/j.anihpc.2009.11.004.

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), 599-604. doi: 10.1016/j.crma.2003.12.032.

[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), 319-351. doi: 10.1007/s11784-009-0110-0.

[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), 847-862. doi: 10.1090/S0894-0347-08-00593-6.

[4]

L. Caffarelli and F. Lin, Nonlocal heat flows preserving the $L^2$ energy, Discrete and Continuous Dynamical Systems (DCDS-A), 23 (2009), 49-64. doi: 10.3934/dcds.2009.23.49.

[5]

L. Caffarelli and S. Salsa, A geometric Approach to Free Boundary Problems, Graduate Studies in Mathematics, 68, American Mathematical Society, Providence, RI, 2005.

[6]

J. Cannon and C. Hill, On the movement of a chemical reaction interface, Indiana Univ. Math. J., 20 (1970), 429-454. doi: 10.1512/iumj.1971.20.20037.

[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), 779-815. doi: 10.1512/iumj.2005.54.2506.

[8]

M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560. doi: 10.1016/j.aim.2004.08.006.

[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), 97-115. doi: 10.1017/S0956792598003660.

[10]

E. N. Dancer, K. Wang and Z. Zhang, Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961-1005. doi: 10.1090/S0002-9947-2011-05488-7.

[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), 2737-2769. doi: 10.1016/j.jde.2011.06.015.

[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), 1087-1131. doi: 10.1016/j.jfa.2011.10.013.

[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), 1087-1131], Journal of Functional Analysis, 264 (2013), 1125-1129. doi: 10.1016/j.jfa.2012.10.009.

[14]

E. N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction, Journal of Differential Equations, 182 (2002), 470-489. doi: 10.1006/jdeq.2001.4102.

[15]

E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for PDE and Applications (eds. C. Baiochhi and J.L. Lions), RMA Res. Notes Appl. Math., 29, Masson, Paris, 1993, 81-98.

[16]

L. Evans, A chemical diffusion-reaction free boundary problem, Nonlinear Anal., 6 (1982), 455-466. doi: 10.1016/0362-546X(82)90059-1.

[17]

N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability, Calculus of Variations and Partial Differential Equations, 39 (2010), 101-120. doi: 10.1007/s00526-009-0303-9.

[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), 165-246.

[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), 43-77.

[20]

F. Lin and X. Yang, Geometric Measure Theory: An Introduction, Advanced Mathematics (Beijing/Boston), Science Press, Beijing; International Press, Boston, 2002.

[21]

U. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Commun. Anal. Geom., 6 (1998), 199-253.

[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), 267-302.

[23]

W. Rudin, Real and Complex Analysis, Tata McGraw-Hill Education, 2006.

[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), 739-761. doi: 10.1016/j.anihpc.2009.11.004.

[1]

Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673

[2]

Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6961-6978. doi: 10.3934/dcds.2019239

[3]

Huiqiang Jiang. Regularity of a vector valued two phase free boundary problems. Conference Publications, 2013, 2013 (special) : 365-374. doi: 10.3934/proc.2013.2013.365

[4]

Navnit Jha. Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization. Conference Publications, 2013, 2013 (special) : 355-363. doi: 10.3934/proc.2013.2013.355

[5]

Quan Wang, Hong Luo, Tian Ma. Boundary layer separation of 2-D incompressible Dirichlet flows. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 675-682. doi: 10.3934/dcdsb.2015.20.675

[6]

Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657

[7]

Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure and Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521

[8]

Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747

[9]

Pavel Krejčí, Elisabetta Rocca, Jürgen Sprekels. Phase separation in a gravity field. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 391-407. doi: 10.3934/dcdss.2011.4.391

[10]

Avner Friedman. Free boundary problems arising in biology. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013

[11]

Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44

[12]

Nikolaos S. Papageorgiou, Vicenţiu D. Rǎdulescu, Youpei Zhang. Anisotropic singular double phase Dirichlet problems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4465-4502. doi: 10.3934/dcdss.2021111

[13]

Tian Ma, Shouhong Wang. Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 459-472. doi: 10.3934/dcds.2004.10.459

[14]

Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807

[15]

Nam-Yong Lee, Bradley J. Lucier. Preconditioned conjugate gradient method for boundary artifact-free image deblurring. Inverse Problems and Imaging, 2016, 10 (1) : 195-225. doi: 10.3934/ipi.2016.10.195

[16]

Eun Heui Kim. Boundary gradient estimates for subsonic solutions of compressible transonic potential flows. Conference Publications, 2007, 2007 (Special) : 573-579. doi: 10.3934/proc.2007.2007.573

[17]

Donatella Danielli, Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Communications on Pure and Applied Analysis, 2005, 4 (2) : 357-366. doi: 10.3934/cpaa.2005.4.357

[18]

Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006

[19]

Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025

[20]

Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (103)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]