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A fractional version of Rivière's GL(n)-gauge
Francesca Da Lio1, Katarzyna Mazowiecka2, Armin Schikorra3
1Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland.
Annali Di Matematica Pura Ed Applicata
|July 25, 2022
Summary
We prove that for antisymmetric vector fields with small norms, a gauge exists to simplify the field. This extends Rivière
Area of Science:
- Mathematical Physics
- Differential Geometry
- Nonlocal Analysis
Background:
- Rivière's theorem addresses gauge existence for antisymmetric vector fields.
- Nonlocal equations present unique mathematical challenges.
- Understanding vector field behavior is crucial in various physics applications.
Purpose of the Study:
- To extend Rivière's theorem to the nonlocal case.
- To establish conservation laws for nonlocal equations with antisymmetric potentials.
- To investigate the stability of these systems under weak convergence.
Main Methods:
- Analysis of antisymmetric vector fields in a nonlocal setting.
- Application of gauge transformations.
- Exploration of conservation laws and stability criteria.
Main Results:
- Existence of a gauge for small-norm antisymmetric vector fields in the nonlocal context.
- Identification of conservation laws for specific nonlocal equations.
- Demonstration of stability under weak convergence.
Conclusions:
- The study successfully extends classical results to nonlocal settings.
- Conservation laws and stability are established for a new class of equations.
- Provides a foundation for further research in nonlocal mathematical physics.
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