A Note on the Well-posedness of Prandtl Equations in Dimension Two

In this paper, we investigate the local-in-time well-posedness for the two-dimensional Prandtl equations in weighted Sobolev spaces under the Oleinik's monotonicity condition.Due to the loss of tangential derivative caused by vertical velocity appearing in convective term, we add with artificial horizontal viscosity term to construct an approximate system that can obtain local-in-time well-posedness results easily. For this approximate system, we construct a new weighted norm for the vorticity to derive a positive life time(independent of artificial viscosity coefficient) and obtain uniform bound for vorticity in this weighted norm.Then, based on compactness argument, we prove the solution of approximate system converging to the solution of original Prandtl equations, and hence, obtain the local-in-time well-posedness result for the Prandtl equations with any large initial data, which improves the recent work [10].