Local wellposedness of Maxwell systems with retarded material laws in low regularity

Abstract We develop a complete local wellposedness theory for a Maxwell system on $$\mathbb {R}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> and a large class of nonlinear material laws which are nonlocal in time. Such constitutive relations are typical for nonlinear optics. The problem was treated before in the Sobolev space $$H^s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> </mml:math> for $$s&gt;3/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> by means of energy methods. Using a recently shown Strichartz estimate, we can lower this level of regularity to $$s&gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . In this context ’charge-type’ terms would spoil the analysis. We avoid them by the Helmholtz projection for the divergence operator with coefficients, which requires mapping properties of the projection also in $$H^{\alpha , q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> </mml:math> with $$q\ne 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> .