Stability of the Prandtl boundary layer equation under various boundary conditions

By the Crocco transformation, the boundary layer system of the viscous incompressible flow is transferred to a strong degenerate parabolic equation with a nonlinear boundary value condition, referred as the Prandtl boundary layer equation. The key technique in this paper involves applying the reciprocal transformation to convert the Prandtl boundary layer equation into a degenerate parabolic equation in divergent form. The main challenge arises on account of that the reciprocal transformation renders the initial value condition unbounded. To address this, a new unknown function $u_1$ is introduced, and the partial differential equation for $u_1$ is derived. For this new equation, the existence of these BV entropy solutions are proved by the parabolically regularized method, the maximal value principle is used to obtain the $L^{\infty }$-estimate. Under certain restrictions on the data of the Prandtl system, the stability of entropy solutions is demonstrated using different boundary value conditions. Consequently, under the Oleǐnik assumption and the monotonicity condition, the two-dimensional Prandtl boundary layer system is shown to be well-posed through the inverse Crocco transformation.