Formation and local stability of a two-dimensional Prandtl boundary layer system in fluid dynamics

Abstract

By the linearized method, Oleǐnik-Samokhin had constructed solutions to distinguish between two cases of the behavior of the fluid at the initial stage of its motion past the surface in Oleinǐk and Samokhin (1999). In this paper, we consider the development of the boundary layer about a body that gradually starts to move in a resting fluid. In the analytical frame, the formation of the layer shows that, when the layer starts to move, the local solutions of the Prandtl boundary layer system is positive. By the Crocco transformation, the Prandtl boundary layer system reduces a degenerate parabolic equation with a nonlinear boundary value condition. Then by the reciprocal transformation, we can obtain a divergence type parabolic equation. We quote a new kind of BV entropy solution matching up with this divergence type parabolic equation when t≥t0, where t0 is a small enough positive constant. By imposing some close connections between the velocity of the outflow and the geometric characteristic of the spatial domain, using Kružkov’s bi-variables method, the local stability of BV entropy solutions is proved. Thus, by the Crocco inverse transformation, in the analytical frame, we obtain the existence and the uniqueness of the global solution of the Prandtl boundary layer system for a regular spatial domain.