Abstract
We prove that for antisymmetric vector field Omega with small L^2-norm there exists a gauge A in L^infty cap {dot{W}}^{1/2,2}({mathbb {R}}^1,GL(N)) such that div12(AΩ-d12A)=0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\ ext {div}}_{\\frac{1}{2}} (A\\Omega - d_{\\frac{1}{2}} A) = 0. \\end{aligned}$$\\end{document}This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.
Highlights
The GL(N)-gauge transform allows for regularity theory and the study of weak convergence [27]; it is an important tool for energy quantization, see [16]
The conservation laws that we obtain in the current paper are more similar in the spirit to those found in the paper [27] for harmonic maps and concern nonlocal systems
(1.3) where the antisymmetric potential acts in general as a nonlocal operator. We hope this technique to be as useful for the question of concentration compactness and energy quantization for systems as it was in the local case in [16]; a question we will study in a future work
Summary
The conservation laws that we obtain in the current paper are more similar in the spirit to those found in the paper [27] for harmonic maps and concern nonlocal systems (1.3) where the antisymmetric potential acts in general as a nonlocal operator. We hope this technique to be as useful for the question of concentration compactness and energy quantization for systems as it was in the local case in [16]; a question we will study in a future work.
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