On quasilinear Maxwell equations in two dimensions

Abstract

New sharp Strichartz estimates for the Maxwell system in two dimensions with rough permittivity and non-trivial 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 which Strichartz estimates are similarly derived as in previous work 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.

Strichartz estimates for Maxwell equations in media: The fully anisotropic case

“…For sake of simplicity, we suppose that µ = 1 3×3 . Contrary to the isotropic Kerr case analyzed in [17,15], we cannot automatically deduce energy estimates by symmetrization. It turns out that this requires additional symmetries of the permittivity.…”

“…We note that the derivative loss ρ + 2−s 4 in ( 10) is the same as in [21] for scalar wave equations. We actually improve the loss in (9), but so far we cannot reach the sharp regularity loss established in [22,23] for the wave equation or [17,15] in the 2D or isotropic Maxwell case. In contrast to these works up to now we cannot use deeper properties of the Hamilton flow of the problem because of the strong system character in the fully isotropic case.…”

“…It is the first result of this kind for cases of fully anisotropic Maxwell systems. This improvement is smaller as in [23] or [15,17]. (For instance, for scalar quasilinear wave equations one gains 1/3 derivatives compared to energy arguments by the results in [23].)…”

“…In two space dimensions [17] we established sharp Strichartz estimates in the fully anisotropic case…”

“…x with implicit constant depending on (ε, µ) C 1 , T and δ, where we set L p T L q x = L p (0, T ; L q x ). The different form of the charge term stems from an argument involving Duhamel's formula, see (54) and also [15,17]. Theorem 1.1 then follows by interpolating the above display with the standard energy estimate…”