Asymptotic analysis ofa diffuse interface relaxation to a nonlocal optimal partitionproblem

Qiang Du; Jingyan Zhang

doi:10.3934/dcds.2011.29.1443

Article Contents

2011, Volume 29, Issue 4: 1443-1461. Doi: 10.3934/dcds.2011.29.1443

This issue Previous Article Symbolic extensions and partially hyperbolic diffeomorphisms Next Article On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow

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

Received: October 31, 2009

Revised: June 30, 2010

Published: September 2011

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Abstract

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.

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Mathematics Subject Classification: 35J47, 35K50, 30E05, 49R50.

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