Dynamics of Partial Differential Equations

Volume 17 (2020)

Number 3

A steady model on Navier–Stokes equations with a free surface

Pages: 185 – 227

DOI: https://dx.doi.org/10.4310/DPDE.2020.v17.n3.a1


Boling Guo (Institute of Applied Physics and Computational Mathematics, Beijing, China)

Yunrui Zheng (School of Mathematics, Shandong University, Shandong, Jinan, China)


We consider the evolution of viscous fluids in a 2D horizontally periodic slab bounded above by a free top surface and below by a fixed flat bottom. The dynamics of the fluid are governed by the incompressible stationary Navier–Stokes equations under the influence of gravity and the effect of surface tension. We develop the existence and uniqueness of solutions in low regularity Sobolev spaces on $[0, T]$ for any $T \gt 0$. Our methods are mainly based on linear estimates of a geometric formulation of an $\varepsilon$–approximate system.


free boundary problems, Navier–Stokes equations, surface tension

2010 Mathematics Subject Classification

35Q30, 35R35, 76D45

Y. Zheng is the corresponding author: yunrui_zheng@sdu.edu.cn

Y. Zheng is partially supported by NSF of China under Grant 11901350, and by the Fundamental Research Funds of Shandong University under Grant 11140079614046.

Received 31 March 2019

Published 7 July 2020