Local Existence of Solutions to 3D Prandtl Equations with a Special Structure

In this chapter, we shall consider the 3D Prandtl equation in a periodic domain and prove the local existence and uniqueness of solutions by the energy method in a polynomial weighted Sobolev space. Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the Crocco transform has always been used with the general outer flow \(U\neq \text{constant}\) , this Crocco transform is not needed here for 3D Prandtl equations.