Local Well-Posedness for Compressible Capillary-Gravity Water Waves with Acute Contact Angles

Abstract

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 $\pi/2$, 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.