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On the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law. (English) Zbl 1230.49008
Summary: We establish the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law on any smooth domain in space dimensions N3. These equations arise in modeling microphase separation in diblock copolymers. The only assumptions that guarantee our convergence result are (i) well-preparedness of the initial data and (ii) smoothness of the limiting interface. Our method makes use of the “Gamma-convergence” of a gradient flows scheme initiated by Sandier and Serfaty and the constancy of multiplicity of the limiting interface due to its smoothness. For the case of radially symmetric initial data without well-preparedness, we give a new and short proof of the result of M. Henry for all space dimensions. Finally, we establish transport estimates for solutions of the Ohta-Kawasaki equation characterizing the transport mechanism.
MSC:
49J45Optimal control problems involving semicontinuity and convergence; relaxation
35Q93PDEs in connection with control and optimization
35B25Singular perturbations (PDE)
35K30Higher order parabolic equations, initial value problems
35B40Asymptotic behavior of solutions of PDE
49S05Variational principles of physics