Asymptotic analysis of a diffuse interface relaxation to a nonlocal optimal partition problem

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October 2011, 29(4): 1443-1461. doi: 10.3934/dcds.2011.29.1443

Asymptotic analysis of a diffuse interface relaxation to a nonlocal optimal partition problem

Qiang Du ^1, and Jingyan Zhang ^2,

1.	Department of Mathematics, Pennsylvania State University, University Park, PA 16802
2.	Department of Mathematics, Pennsylvania Sate University, University Park, PA 16802, United States

Received November 2009 Revised July 2010 Published December 2010

Abstract
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We present some asymptotic analysis of a diffuse interface relaxation to a nonlocal optimal domain partition problem and the associated nonlocal interfacial motion when the interfacial width is approaching to zero. Motivated by careful numerical calculations, we first discuss several assumptions on the steady state solutions of the coupled system of differential equations which are supported by numerical results. These assumptions allow us to construct a suitable ansatz to the solutions which not only captures the leading order behavior but also provides sufficient estimates on the next order behavior so that more accurate estimates can be shown for interesting physical quantities such as energies and eigenvalues. When adopted to the gradient flow system, the ansatz gives an estimate of the asymptotic convergence rate in time to the equilibrium partitions.

Keywords: Normalized gradient flow, Asymptotic analysis., Diffuse interface, Nonlocal optimal domain partition.

Mathematics Subject Classification: 35J47, 35K50, 30E05, 49R5.

Citation: Qiang Du, Jingyan Zhang. Asymptotic analysis of a diffuse interface relaxation to a nonlocal optimal partition problem. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1443-1461. doi: 10.3934/dcds.2011.29.1443

References:

[1]	W. Bao, Ground states and dynamics of multi-component Bose-Einstein condensates, SIAM Multiscale Model. Simulat., 2 (2004), 210-236. doi: 10.1137/030600209.
[2]	W. Bao and Q. Du, Comuputing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697. doi: 10.1137/S1064827503422956.
[3]	H. Berestycki, T. C. Lin, J. Wei and C. Zhao, On phase-separation model: Asymptotics and qualitative properties, preprint, 2010.
[4]	V. Bonnaillie-Noel, B. Helffer and G. Vial, Numerical simulations for nodal domains and spectral minimal partitions, ESAIM: COCV, 16 (2010), 221-246. doi: 10.1051/cocv:2008074.
[5]	L. A. Cafferelli and F. H. Lin, An optimal partition problem for eigenvalues, Journal of Scientific Computing, 31 (2007), 5-18. doi: 10.1007/s10915-006-9114-8.
[6]	L. A. Cafferelli and F. H. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc., 21 (2008), 847-862. doi: 10.1090/S0894-0347-08-00593-6.
[7]	L. A. Cafferrelli and F. H. Lin, Nonlocal heat flows preserving the $L^2$ energy, Discrete and Continuous Dynamical Systems A, 23 (2009), 49-64.
[8]	S.-M. Chang, C.-S. Lin, T.-C. Lin and W.-W. Lin, Segregated nodal domains of two dimensional multispecies Bose-Einstein condensates, Phys. D, 196 (2004), 341-361. doi: 10.1016/j.physd.2004.06.002.
[9]	S.-M. Chang, W.-W. Lin and S.-F. Shieh, Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate, J. of Computational Physics, 202 (2005), 367-390 doi: 10.1016/j.jcp.2004.07.012.
[10]	M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X.
[11]	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.
[12]	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.
[13]	Q. Du and F.-H. Lin, Numerical approximations of a norm preserving nonlinear gradient flow and applications to an optimal partition problem, Nonlinearity, 22 (2009), 67-83. doi: 10.1088/0951-7715/22/1/005.
[14]	T.-C. Lin and J. Wei, Ground state of n coupled nonlinear schrodinger equations in $R^n$, $n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.
[15]	J. Wei and T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition, Nonlinearity, 21 (2008), 305-317. doi: 10.1088/0951-7715/21/2/006.

show all references

References:

[1]	W. Bao, Ground states and dynamics of multi-component Bose-Einstein condensates, SIAM Multiscale Model. Simulat., 2 (2004), 210-236. doi: 10.1137/030600209.
[2]	W. Bao and Q. Du, Comuputing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697. doi: 10.1137/S1064827503422956.
[3]	H. Berestycki, T. C. Lin, J. Wei and C. Zhao, On phase-separation model: Asymptotics and qualitative properties, preprint, 2010.
[4]	V. Bonnaillie-Noel, B. Helffer and G. Vial, Numerical simulations for nodal domains and spectral minimal partitions, ESAIM: COCV, 16 (2010), 221-246. doi: 10.1051/cocv:2008074.
[5]	L. A. Cafferelli and F. H. Lin, An optimal partition problem for eigenvalues, Journal of Scientific Computing, 31 (2007), 5-18. doi: 10.1007/s10915-006-9114-8.
[6]	L. A. Cafferelli and F. H. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc., 21 (2008), 847-862. doi: 10.1090/S0894-0347-08-00593-6.
[7]	L. A. Cafferrelli and F. H. Lin, Nonlocal heat flows preserving the $L^2$ energy, Discrete and Continuous Dynamical Systems A, 23 (2009), 49-64.
[8]	S.-M. Chang, C.-S. Lin, T.-C. Lin and W.-W. Lin, Segregated nodal domains of two dimensional multispecies Bose-Einstein condensates, Phys. D, 196 (2004), 341-361. doi: 10.1016/j.physd.2004.06.002.
[9]	S.-M. Chang, W.-W. Lin and S.-F. Shieh, Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate, J. of Computational Physics, 202 (2005), 367-390 doi: 10.1016/j.jcp.2004.07.012.
[10]	M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X.
[11]	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.
[12]	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.
[13]	Q. Du and F.-H. Lin, Numerical approximations of a norm preserving nonlinear gradient flow and applications to an optimal partition problem, Nonlinearity, 22 (2009), 67-83. doi: 10.1088/0951-7715/22/1/005.
[14]	T.-C. Lin and J. Wei, Ground state of n coupled nonlinear schrodinger equations in $R^n$, $n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.
[15]	J. Wei and T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition, Nonlinearity, 21 (2008), 305-317. doi: 10.1088/0951-7715/21/2/006.

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