The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces

Hartmut Pecher

doi:10.3934/dcds.2019199

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2019, Volume 39, Issue 8: 4875-4893. Doi: 10.3934/dcds.2019199

This issue Previous Article Regularity and weak comparison principles for double phase quasilinear elliptic equations Next Article Prescribed energy connecting orbits for gradient systems

The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces

Hartmut Pecher

Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany

Received: November 2018

Revised: January 2019

Published: May 2019

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Abstract

The Chern-Simons-Higgs and the Chern-Simons-Dirac systems in Lorenz gauge are locally well-posed in suitable Fourier-Lebesgue spaces $ \hat{H}^{s, r} $. Our aim is to minimize $ s = s(r) $ in the range $ 1<r \le 2 $. If $ r \to 1 $ we show that we almost reach the critical regularity dictated by scaling. In the classical case $ r = 2 $ the results are due to Huh and Oh. Crucial is the fact that the decisive quadratic nonlinearities fulfill a null condition.

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Mathematics Subject Classification: 35Q40, 35L70.

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