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On moving Ginzburg-Landau vortices. (English) Zbl 1063.35081

Summary: We establish a quantization property for the heat equation of Ginzburg-Landau functional in \(\mathbb{R}^4\) which models moving vortices of surface types. It asserts that if the energy is sufficiently small on a parabolic ball in \(\mathbb{R}^4\times\mathbb{R}_+\), then there is no vortice in the parabolic ball of \(\frac 12\) radius. This extends a recent result of F.-H. Lin and T. Rivière [Commun. Pure Appl. Math. 54, No. 7, 826–850 (2001; Zbl 1029.35127)] in \(\mathbb{R}^3\).

MSC:

35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35B45 A priori estimates in context of PDEs
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