Asymptotic analysis ofa diffuse interface relaxation to a nonlocal optimal partitionproblem
Qiang Du, Jingyan Zhang · Discrete and Continuous Dynamical Systems · 2010
We present some asymptotic analysis of a diffuseinterface relaxation to a nonlocal optimal domain partition problemand the associated nonlocal interfacial motion when the interfacialwidth is approaching to zero.Motivated by careful numerical calculations,we first discuss several assumptions on thesteady state solutions of the coupled system ofdifferential equations which are supportedby numerical results. These assumptionsallow us to construct a suitable ansatz to the solutionswhich not only captures the leadingorder behavior but also provides sufficient estimates on the nextorder behavior so that more accurate estimates can be shown forinteresting physical quantities such as energies and eigenvalues.When adopted to the gradient flow system, the ansatzgives an estimate of the asymptotic convergence rate in timeto the equilibrium partitions.