Strichartz estimates for Maxwell equations in media: the partially anisotropic case

Abstract

We prove Strichartz estimates for solutions to Maxwell equations in three dimensions with rough permittivities, which have less than three different eigenvalues. To this end, Maxwell equations are conjugated to half-wave equations in phase space. We use the Strichartz estimates in a known combination with energy estimates to show the new well-posedness results for quasilinear Maxwell equations.

Keywords:

• Maxwell equations
• Strichartz estimates
• quasilinear wave equations
• rough coefficients
• Kerr nonlinearity

MSC Subject Classification (2010):

• Primary: 35L45, 35B65
• Secondary: 35Q61

Acknowledgments

I would like to thank Rainer Mandel (KIT) and Roland Schnaubelt (KIT) for discussions on related topics and thanks to the anonymous referee for helpful remarks, which improved the presentation.

Disclosure statement

There is no potential competing interest.

Notes

1 We first truncate the frequencies and then take the inverse.

2 Strictly speaking, the frequency projection has to be enlarged slightly at every instance of inserting. We do not keep track of this to lighten the notation.

3 We omit the frequency truncation in the coefficients to lighten the notation.

4 Note that these are just Fourier multiplier.

Additional information

Funding

Financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173 and Korea Institute of Advanced Study, grant No. MG093901 is gratefully acknowledged.