abstract: I report on joint work with T. Lamm. We consider local solutions to nonlinear parabolic equations with initial data which are small bounded perturbations, perturbations in BMO, or in Lipschitz spaces of more regular data, depending on the type of the equations. This builds on the study of the Navier Stokes equations in BMO-1 with D. Tataru. It relies on the Carleson measure characterization of BMO.