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Zbl 1230.49008
Le, Nam Q.
On the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law. (English)
[J] SIAM J. Math. Anal. 42, No. 4, 1602-1638 (2010). ISSN 0036-1410; ISSN 1095-7154/e

Summary: We establish the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law on any smooth domain in space dimensions $N \leq 3$. 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 2000:: *49J45 Optimal control problems inv. semicontinuity and convergence
35Q93
35B25 Singular perturbations (PDE)
35K30 Higher order parabolic equations, initial value problems
35B40 Asymptotic behavior of solutions of PDE
49S05 Variational principles of physics

Keywords: Ohta-Kawasaki equation; nonlocal Mullins-Sekerka law; microphase separation in diblock copolymers; Gamma-convergence of gradient flows; transport estimate; De Giorgi conjecture

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