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

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

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.
Citation: Qiang Du, Jingyan Zhang. Asymptotic analysis of a diffuse interface relaxation to a nonlocal optimal partition problem. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1137/030600209. Google Scholar

[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. doi: 10.1137/S1064827503422956. Google Scholar

[3]

H. Berestycki, T. C. Lin, J. Wei and C. Zhao, On phase-separation model: Asymptotics and qualitative properties,, preprint, (2010). Google Scholar

[4]

V. Bonnaillie-Noel, B. Helffer and G. Vial, Numerical simulations for nodal domains and spectral minimal partitions,, ESAIM: COCV, 16 (2010), 221. doi: 10.1051/cocv:2008074. Google Scholar

[5]

L. A. Cafferelli and F. H. Lin, An optimal partition problem for eigenvalues,, Journal of Scientific Computing, 31 (2007), 5. doi: 10.1007/s10915-006-9114-8. Google Scholar

[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. doi: 10.1090/S0894-0347-08-00593-6. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.physd.2004.06.002. Google Scholar

[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. doi: 10.1016/j.jcp.2004.07.012. Google Scholar

[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. doi: 10.1016/S0294-1449(02)00104-X. Google Scholar

[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. doi: 10.1512/iumj.2005.54.2506. Google Scholar

[12]

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

[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. doi: 10.1088/0951-7715/22/1/005. Google Scholar

[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. doi: 10.1007/s00220-005-1313-x. Google Scholar

[15]

J. Wei and T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition,, Nonlinearity, 21 (2008), 305. doi: 10.1088/0951-7715/21/2/006. Google Scholar

show all references

References:
[1]

W. Bao, Ground states and dynamics of multi-component Bose-Einstein condensates,, SIAM Multiscale Model. Simulat., 2 (2004), 210. doi: 10.1137/030600209. Google Scholar

[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. doi: 10.1137/S1064827503422956. Google Scholar

[3]

H. Berestycki, T. C. Lin, J. Wei and C. Zhao, On phase-separation model: Asymptotics and qualitative properties,, preprint, (2010). Google Scholar

[4]

V. Bonnaillie-Noel, B. Helffer and G. Vial, Numerical simulations for nodal domains and spectral minimal partitions,, ESAIM: COCV, 16 (2010), 221. doi: 10.1051/cocv:2008074. Google Scholar

[5]

L. A. Cafferelli and F. H. Lin, An optimal partition problem for eigenvalues,, Journal of Scientific Computing, 31 (2007), 5. doi: 10.1007/s10915-006-9114-8. Google Scholar

[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. doi: 10.1090/S0894-0347-08-00593-6. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.physd.2004.06.002. Google Scholar

[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. doi: 10.1016/j.jcp.2004.07.012. Google Scholar

[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. doi: 10.1016/S0294-1449(02)00104-X. Google Scholar

[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. doi: 10.1512/iumj.2005.54.2506. Google Scholar

[12]

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

[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. doi: 10.1088/0951-7715/22/1/005. Google Scholar

[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. doi: 10.1007/s00220-005-1313-x. Google Scholar

[15]

J. Wei and T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition,, Nonlinearity, 21 (2008), 305. doi: 10.1088/0951-7715/21/2/006. Google Scholar

[1]

Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure & Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011

[2]

Eduard Feireisl. On weak solutions to a diffuse interface model of a binary mixture of compressible fluids. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 173-183. doi: 10.3934/dcdss.2016.9.173

[3]

Wenqing Hu, Chris Junchi Li. A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4951-4977. doi: 10.3934/dcds.2018216

[4]

K.H. Wong, C. Myburgh, L. Omari. A gradient flow approach for computing jump linear quadratic optimal feedback gains. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 803-808. doi: 10.3934/dcds.2000.6.803

[5]

Martin Gugat, Alexander Keimer, Günter Leugering, Zhiqiang Wang. Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks. Networks & Heterogeneous Media, 2015, 10 (4) : 749-785. doi: 10.3934/nhm.2015.10.749

[6]

Antonio DeSimone, Martin Kružík. Domain patterns and hysteresis in phase-transforming solids: Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation. Networks & Heterogeneous Media, 2013, 8 (2) : 481-499. doi: 10.3934/nhm.2013.8.481

[7]

Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006

[8]

Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel. Networks & Heterogeneous Media, 2010, 5 (4) : 783-812. doi: 10.3934/nhm.2010.5.783

[9]

Jun Wang, Qiuping Geng, Maochun Zhu. Existence of the normalized solutions to the nonlocal elliptic system with partial confinement. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2187-2201. doi: 10.3934/dcds.2019092

[10]

Michela Eleuteri, Elisabetta Rocca, Giulio Schimperna. On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2497-2522. doi: 10.3934/dcds.2015.35.2497

[11]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157

[12]

Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 741-766. doi: 10.3934/nhm.2012.7.741

[13]

Maria do Rosário de Pinho, Filipa Nunes Nogueira. On application of optimal control to SEIR normalized models: Pros and cons. Mathematical Biosciences & Engineering, 2017, 14 (1) : 111-126. doi: 10.3934/mbe.2017008

[14]

Luis A. Caffarelli, Fang Hua Lin. Analysis on the junctions of domain walls. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 915-929. doi: 10.3934/dcds.2010.28.915

[15]

Yuhong Dai, Ya-xiang Yuan. Analysis of monotone gradient methods. Journal of Industrial & Management Optimization, 2005, 1 (2) : 181-192. doi: 10.3934/jimo.2005.1.181

[16]

Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall. Networks & Heterogeneous Media, 2008, 3 (3) : 651-673. doi: 10.3934/nhm.2008.3.651

[17]

Anita T. Layton, J. Thomas Beale. A partially implicit hybrid method for computing interface motion in Stokes flow. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1139-1153. doi: 10.3934/dcdsb.2012.17.1139

[18]

Ken Ono. Parity of the partition function. Electronic Research Announcements, 1995, 1: 35-42.

[19]

Šárka Nečasová. Stokes and Oseen flow with Coriolis force in the exterior domain. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 339-351. doi: 10.3934/dcdss.2008.1.339

[20]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]