Remarks on the ill-posedness of the Prandtl equation

In the lines of the recent paper [5], we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary shear flows, we show that for some linearizations of the Prandtl equation and some C∞ initial data, local in time C∞ solutions do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides ill-posedness in time, we also establish some ill-posedness in space, that casts some light on the results obtained by Oleinik for monotonic data.