A fractional version of Rivi\`ere's GL(N)-gauge

TL;DR

This paper extends Rivi e's gauge theorem to a nonlocal fractional setting, establishing existence of gauges with specific divergence properties for small antisymmetric vector fields, and deriving conservation laws and stability results.

Contribution

It introduces a fractional gauge theorem for antisymmetric vector fields, generalizing Rivi e's classical result to nonlocal operators and providing new conservation laws.

Findings

Existence of fractional gauges for small antisymmetric fields.

Extension of Rivi e's theorem to nonlocal fractional operators.

Establishment of conservation laws and stability under weak convergence.

Abstract

We prove that for antisymmetric vectorfield $\Omega$ 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 ${\rm div}_{\frac12} (A\Omega - d_{\frac{1}{2}} A) = 0$. This extends a celebrated theorem by Rivi\`ere to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.