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Well-posedness for the fractional Landau-Lifshitz equation without Gilbert damping. (English) Zbl 1260.35229
Summary: The main purpose is to consider the well-posedness of the fractional Landau- Lifshitz equation without Gilbert damping. The local existence of classical solutions is obtained by combining Kato’s method and vanishing viscosity method, by carefully choosing the working space. Since this equation is strongly degenerate and nonlocal and no regularizing effect is available, it is a challenging problem to extend this smooth solution to global. Instead, we give some regularity criteria to show that the solution is global if some additional regularity is assumed, which seems minimal in the sense of dimensional analysis. Finally, we introduce the commutator and show the global existence of weak solutions by vanishing viscosity method.

MSC:
35Q82 PDEs in connection with statistical mechanics
58E20 Harmonic maps, etc.
82D45 Statistical mechanics of ferroelectrics
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