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On quasilinear Maxwell equations in two dimensions

Robert Schippa and Roland Schnaubelt
Vol. 4 (2022), No. 2, 313–365
DOI: 10.2140/paa.2022.4.313
Abstract

New sharp Strichartz estimates for the Maxwell system in two dimensions with rough permittivity and nontrivial charges are proved. We use the FBI transform to carry out the analysis in phase space. For this purpose, the Maxwell equations are conjugated to a system of half-wave equations with rough coefficients. For this system, Strichartz estimates are proved using methods similar to those in previous works by Tataru on scalar wave equations with rough coefficients. We use the estimates to improve the local well-posedness theory for quasilinear Maxwell equations in two dimensions.

Keywords
Maxwell equations, Strichartz estimates, quasilinear wave equations, rough coefficients, half wave equation, Kerr nonlinearity, FBI transform, phase space analysis
Mathematical Subject Classification
Primary: 35B65, 35L45
Secondary: 35Q61
Milestones
Received: 17 May 2021
Accepted: 26 January 2022
Published: 16 October 2022
Authors
Robert Schippa
Department of Mathematics
Karlsruhe Institute of Technology
Karlsruhe
Germany
Roland Schnaubelt
Department of Mathematics
Karlsruhe Institute of Technology
Karlsruhe
Germany