arXiv 2407.178952024-07math.AP

接触线全局适定性：二维纳维-斯托克斯流

Global Well-Posedness of Contact Lines: 2D Navier-Stokes Flow

Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Department of Mathematical Sciences, Carnegie Mellon University（卡内基梅隆大学数学科学系）；Department of Mathematics, Lehigh University（利哈伊大学数学系）；School of Mathematics, Shandong University（山东大学数学学院）

Yan Guo, Ian Tice, Lei Wu, Xiaoding Yang, Yunrui Zheng

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AI总结：

本文基于Guo-Tice的先验估计，研究了满足接触线动态定律的粘性流体模型的全局适定性，通过构造无压弱解和人工正则化处理毛细力项，建立了二维自由表面的适定性理论。

摘要：

基于Guo-Tice在[Guo-Tice, J. Eur. Math. Soc. (2024)]中提出的全局先验估计，本文建立了满足接触线动态定律的粘性流体模型的适定性。该动态定律为$\mathscr{W}(\p_tζ(\pm\ell,t))=[\![γ]\!]\mpσ\frac{\p_1ζ}{(1+|\p_1ζ|^2)^{1/2}}(\pm\ell,t)$，其中$ζ(x_1,t)$是自由表面，具有两个接触点$ζ(\pm\ell,t)$，$[\![γ]\!]$和$σ$是描述固-液-气自由能的常数，$\mathscr{W}$是接触点速度响应函数。受能量耗散结构的启发，本文通过构造带有时间依赖基函数的Galerkin近似和毛细力项的人工正则化，建立了耦合速度和自由界面的线性化问题的无压弱解，从而证明了该模型在二维域中的全局适定性。

英文摘要：

Based on the global a priori estimates in [Guo-Tice, J. Eur. Math. Soc. (2024)], we establish the well-posedness of a viscous fluid model satisfying the dynamic law for the contact line \begin{equation*} \mathscr{W}(\p_tζ(\pm\ell,t))=[\![γ]\!]\mpσ\frac{\p_1ζ}{(1+|\p_1ζ|^2)^{1/2}}(\pm\ell,t) \end{equation*} in 2D domain, where $ζ(x_1,t)$ is a free surface with two contact points $ζ(\pm\ell,t)$, $[\![γ]\!]$ and $σ$ are constants characterizing the solid-fluid-gas free energy, and the increasing $\mathscr{W}$ is the contact point velocity response function. Motivated by the energy-dissipation structure, our construction relies on the construction of a pressureless weak solution for the coupled velocity and free interface for the linearized problems, via a Galerkin approximation with a time-dependent basis and an artificial regularization for the capillary operator.

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