Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure

Tong Yang,Cheng-Jie Liu,Ya-Guang Wang

doi:10.3934/dcdss.2016082

Abstract

The global existence of weak solutions to the three space dimensional Prandtl equations is studied under some constraint on its structure. This is a continuation of our recent study on the local existence of classical solutions with the same structure condition. It reveals the sufficiency of the monotonicity condition on one component of the tangential velocity field and the favorable condition on pressure in the same direction that leads to global existence of weak solutions. This generalizes the result obtained by Xin-Zhang on the two-dimensional Prandtl equations to the three-dimensional setting.

Highlights

Except the recent work [5] about the classical solution with special structure and those in the analytic framework [12, 3], most of the mathematical studies on this fundamental system in boundary layer theory are limited to the problem in two space dimensions, cf. [1, 4, 7, 9, 10, 13, 14] and the references therein
In [5], we obtain the local well-posedness of classical solutions to the problem (1) under some constraint on the structure of the solution, in order to avoid the appearance of secondary flow ([8]) in boundary layers
Assuming that for the Euler flow given in (2), U (t, x, y) > 0, in the class of boundary layers that the direction of tangential velocity field is invariant in the normal variable z, and the x−component of velocity u(t, x, y, z) is strictly increasing in z, ∂zu > 0, in [5] we have constructed a classical solution to the problem (1), and it is linearly stable with respect to any three-dimensional perturbation

Summary

Introduction

Consider the initial boundary value problem for the Prandtl boundary layer equations in three space variables,. Assuming that for the Euler flow given in (2), U (t, x, y) > 0, in the class of boundary layers that the direction of tangential velocity field is invariant in the normal variable z, and the x−component of velocity u(t, x, y, z) is strictly increasing in z, ∂zu > 0, in [5] we have constructed a classical solution to the problem (1), and it is linearly stable with respect to any three-dimensional perturbation. The authors recently observed in [6] that for the shear flow (us(t, z), vs(t, z), 0) of the three-dimensional Prandtl equations, the special solution structure (3) is the only stable case. Under the above assumption (3) of special solution structure, the original problem (1) of three-dimensional Prandtl equations is reduced to the following one for two unknown functions (u, w):.

For this reduced problem under the assumption that

Ut U

Letting n