ASYMPTOTIC ANALYSIS OF A DIFFUSE INTERFACE RELAXATION TO A NONLOCAL OPTIMAL PARTITION PROBLEM

作者：Du Qiang^*; Zhang Jingyan

来源：Discrete and Continuous Dynamical Systems, 2011, 29(4): 1443-1461.

DOI：10.3934/dcds.2011.29.1443

• Nonlocal optimal domain partition
• Diffuse interface
• Normalized gradient flow
• Asymptotic analysis

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 eigen values. When adopted to the gradient flow system, the ansatz gives an estimate of the asymptotic convergence rate in time to the equilibrium partitions.

• 出版日期2011-4