Local well-posedness and global stability of one-dimensional shallow water equations with surface tension and constant contact angle

Abstract

We consider the one-dimensional shallow water problem with capillary surfaces and moving contact {lines}. An energy-based model is derived from the two-dimensional water wave equations, where we explicitly discuss the case of a stationary force balance at a moving contact line and highlight necessary changes to consider dynamic contact angles. The moving contact line becomes our free boundary at the level of shallow water equations, and the depth of the shallow water degenerates near the free boundary, which causes singularities for the derivatives and degeneracy for the viscosity. This is similar to the physical vacuum of compressible flows in the literature. The equilibrium, the global stability of the equilibrium, and the local well-posedness theory are established in this paper.

Figures (2)

• Figure 1: Sketch of fluid film height $h=h(x,t)$, horizontally uniform velocity field $u=u(x,t)$, contact lines $a=a(t)$, $b=b(t)$, contact angle $0\le\vartheta\ll 1$.
• Figure 2: Equilibrium solutions $h_{\mathrm{s}}$ for increasing $R/\lambda\in\{\tfrac{1}{4},\tfrac{1}{2},1,2,4,8,16\}$ (blue lines) and parabolic shapes with same $R/\lambda$ (red dashed lines).

Theorems & Definitions (14)

• Remark 1
• Theorem 1: Informal statement of main theorems
• Lemma 1: Hardy's inequality, $L^p$ version
• Lemma 2: Weighted Poincare's inequality
• Theorem 2: Asymptotic Stability
• Remark 2
• Lemma 3
• proof
• Lemma 4
• proof