Communications in Mathematical Sciences

Volume 18 (2020)

Number 3

Stability and back flow of boundary layers for wind-driven oceanic current

Pages: 593 – 612



Shengbo Gong (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China, and HUAWEI Technologies Co. Ltd., Shanghai, China)

Xiang Wang (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)

Yaguang Wang (School of Mathematical Sciences, Center of Applied Mathematics, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai, China)


The proposal of this paper is to study the well-posedness and properties of solutions to the boundary layer problem for wind-driven oceanic current, which differs from the classical Prandtl boundary layer equations with a nonlocal integral term arising from the Coriolis force. First, under Oleinik’s monotonic condition [O.A. Oleinik, Dokl. Akad. Nauk SSSR, 150(4):28–31, 1963] on the tangential velocity field, we obtain the local well-posedness of the boundary layer problem by using the Crocco transformation. Secondly, we show that the back flow point appears at the physical boundary in a finite time under certain constraint on the growth rate of the tangential velocity when both the initial tangential velocity and the upstream velocity are monotonically increasing with respect to the normal variable of the boundary, even if the momentum of the outer flow is favorable for the classical Prandtl equations, in the sense with this favorable condition there will be no back flow in the two-dimensional Prandtl boundary layer. This shows that the factor of the Coriolis force stimulates the appearance of the back flow of the boundary layer.


boundary layer, Navier–Stokes–Coriolis equations, well-posedness, back flow

2010 Mathematics Subject Classification

35B40, 35Q30, 76D05, 76D10

Received 2 July 2019

Accepted 19 October 2019

Published 30 June 2020