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On the 2D Ericksen-Leslie equations with anisotropic energy and external forces. (English) Zbl 1503.35152

Summary: In this paper we consider the 2D Ericksen-Leslie equations which describe the hydrodynamics of nematic liquid crystal with external body forces and anisotropic energy modeling the energy of applied external control such as magnetic or electric field. Under general assumptions on the initial data, the external data and the anisotropic energy, we prove the existence and uniqueness of global weak solutions with finitely many singular times. If the initial data and the external forces are sufficiently small, then we establish that the global weak solution does not have any singular times and is regular as long as the data are regular.

MSC:

35Q35 PDEs in connection with fluid mechanics
76A15 Liquid crystals
76W05 Magnetohydrodynamics and electrohydrodynamics
35D30 Weak solutions to PDEs
35D35 Strong solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35K45 Initial value problems for second-order parabolic systems
35K55 Nonlinear parabolic equations
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