On a Fractional Ginzburg–Landau Equation and 1/2-Harmonic Maps into Spheres

By V. Millot, Yannick Sire. 2013

123 citations recorded by the source corpus.

This paper is devoted to the asymptotic analysis of a fractional version of the Ginzburg–Landau equation in bounded domains, where the Laplacian is replaced by an integro-differential operator related to the square root Laplacian as defined in Fourier space. In the singular limit $${\varepsilon \to 0}$$ε→0 , we show that solutions with uniformly bounded energy converge weakly to sphere valued 1/2-harmonic maps, that is, the fractional analogues of the usual harmonic maps. In addition, the convergence holds in smooth functions spaces away from a countably $${\fancyscript{H}^{n-1}}$$Hn-1 -rectifiable closed set of finite (n−1)-Hausdorff measure. The proof relies on the representation of the square root Laplacian as a Dirichlet-to-Neumann operator in one more dimension, and on the analysis of a boundary version of the Ginzburg–Landau equation. Besides the analysis of the fractional Ginzburg–Landau equation, we also give a general partial regularity result for stationary 1/2-harmonic maps in an arbitrary dimension.

Subfields: Applied Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/3d5d5d12ac38ffafbd66147568688564a285bb4a

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In Applied Mathematics

In Control and Systems Engineering

Citations in the corpus

Fractional div-curl quantities and applications to nonlocal geometric equations (2017, methodology): “Another approach was given by Millot–Sire [26] who interpreted the halfharmonic map equation (3.”
Blow-up analysis of a nonlocal Liouville-type equation (2015, methodology): “We would like to recall a characterization of 12 -harmonic maps of S 1 into submanifolds of Rn , which has been already observed in [Da Lio 2015] and then in [Millot and Sire 2015].”
A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients (2016, background): “A striking example of this is a model for anomalous transport processes and diffusion, leading to fractional PDEs [1], but other examples are readily available for generalizing many classical problems in mathematical physics, including the fractional Burgers’ equation [2, 3], the fractional Fokker–P”