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.

中文翻译：

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具有自由表面的Navier–Stokes方程的稳定模型

我们考虑粘性流体在二维水平周期性平板中的演化，该二维平板中的上方是自由顶面，下方是固定的平底。在重力和表面张力的影响下，流体的动力学由不可压缩的固定Navier–Stokes方程控制。对于任何$ T \ gt 0 $，我们在$ [0，T] $的低规则Sobolev空间中开发解的存在性和唯一性。我们的方法主要基于$ \ varepsilon $-近似系统的几何公式​​的线性估计。