具有锐角接触角的可压缩毛细重力水波的局部适定性
Local Well-Posedness for Compressible Capillary-Gravity Water Waves with Acute Contact Angles
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中文总结 AI 辅助
本文研究具有锐角接触角的可压缩毛细重力水波的局部适定性，通过建立几何结构、奇异性分析与先验能量估计，得到了首个含接触点的可压缩欧拉方程自由边值问题的局部适定性结果。
中文摘要 AI 辅助
本文旨在研究二维有界角域内可压缩欧拉方程的局部适定性，该角域具有锐角接触角。此构型描述了自由表面在两点处与固定底部相交，流体受重力场作用，且流体与空气的界面受毛细力影响。当接触角小于π/2时，我们建立了解的局部存在理论，其中接触点处存在耗散效应。主要分析挑战源于接触点奇异性，这使得先前处理可压缩自由边界问题的方法不再适用。为克服这一问题，我们首先建立可压缩欧拉方程的几何结构，该方法最初由Shatah和Zeng（2008）针对不可压缩流体提出。此外，我们对该角域内的波动方程进行奇异性分析，以确保角附近计算的有效性。最后，基于几何结构和奇异性分析，我们得到先验能量估计。利用这些估计，我们还证明了该系统在几何形式下的局部适定性。据我们所知，这是首个涉及接触点的可压缩欧拉方程自由边值问题的相关结果。
英文摘要
Our purpose is to investigate the local well-posedness of the compressible Euler equations in a two-dimensional bounded corner domain with acute contact angles. This configuration describes a free surface intersecting the fixed bottom at two points, where the fluid is subject to a gravitational field and the interface between the fluid and air is influenced by capillary forces. When the contact angles are less than $π/2$, we establish a local existence theory for the solution, with dissipation effects occurring at the contact points. The main analytical challenge arises from contact point singularities, which renders previous methods for dealing with compressible free boundary problems inadequate. To overcome this, we first establish the geometric structure for the compressible Euler equations, an approach originally introduced by Shatah and Zeng \cite{Shatah2008} for incompressible fluids. Additionally, we provide a singularity analysis for the wave equations in the corner domain, which ensures the validity of calculations near the corner. Finally, based on the geometric structure and singularity analysis, we obtain a priori energy estimates. Using these estimates, we also prove the local well-posedness of the system in a geometric formulation. To our knowledge, this is the first result addressing compressible Euler equations with a free boundary that involves contact points.
