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

We develop a complete local wellposedness theory for a Maxwell system on ${\mathbb{R}}^3$ 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 $H^s$ for $s>3/2$ by means of energy methods. Using a recently shown Strichartz estimate, we can lower this level of regularity to $s>1$. In this context ’charge-type’ terms would spoil the analysis. We avoid them by the Helmholtz projection for the divergence operator with coeffients, which requires mapping properties of the projection also in $H^{\alpha ,q}$ with $q\ne 2$.