arXiv subjects

Xiaoding Yang

Publications and source records attributed to Xiaoding Yang.

5 recordsLinked to original sources

arXiv2026

Linear Stability and Jacobi Kernels of Three-Dimensional Sessile Drops

We consider the linear stability of three-dimensional sessile drops with a free contact line. The equilibrium surface is an axisymmetric solution of the Young--Laplace equation with gravity, fixed volume, and prescribed contact angle. We derive the constrained second variation of the gravity--capillary energy and formulate the associated Jacobi problem. Although variational stability gives nonnegativity, the second variation is necessarily degenerate because horizontal translations preserve the energy. Our main result identifies this degeneracy completely. Under the pressure--volume nondegeneracy condition $(dV/d\lambda\neq 0)$, we prove that the kernel of the constrained Jacobi operator is exactly the two-dimensional space generated by horizontal translations. The proof combines the geometric structure of the Jacobi operator with a Fourier-mode analysis: the axisymmetric mode is ruled out by the pressure--volume condition, the first mode gives translations, and all higher modes are excluded by comparison. This provides the precise linear nondegeneracy underlying stability of droplet dynamics modulo translations.

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arXiv2026

Global Well-Posedness of sessile drop problem: 2D Navier-Stokes Flow

We prove the global well-posedness of small perturbations of two-dimensional sessile droplet equilibria for the incompressible Navier--Stokes equations with surface tension, Navier-slip boundary conditions, and a dynamic contact-point law. The main difficulty is the construction of solutions in the presence of the horizontal translational degeneracy of the equilibrium manifold. To remove this degeneracy, we work in a moving polar coordinate system determined by an orthogonality condition. We then establish local well-posedness through a Galerkin construction of pressureless weak solutions, recovery of the pressure, higher-order estimates, and a contraction argument. Combining this local theory with the global energy--dissipation estimates obtained in our previous work yields a unique global solution and the corresponding exponential decay estimate.

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arXiv2026

Global dynamic stability of contact lines in fluids: 2-D droplet problem

In this paper, we investigate the dynamics of an incompressible viscous Navier-Stokes fluid evolving above a one-dimensional flat surface. The fluid is subject to a uniform gravitational field and capillary forces acting along the free boundary. The interface between the fluid and the surrounding air is a free surface whose motion is driven by gravity, surface tension, and the fluid velocity field. The triple-phase intersections where the fluid, the air above the vessel, and the solid vessel wall meet are referred to as contact points, and the angles formed there are called contact angles. The model under consideration incorporates boundary conditions that allow for full motion of the contact points and dynamic contact angles. Under these conditions, \cite{Yang} established the existence of equilibrium configurations for the model. These equilibria consist of a quiescent fluid occupying a domain whose upper boundary can be represented as the graph of a function in polar coordinates, minimizing a gravity-capillary energy functional subject to a fixed mass constraint. The equilibrium contact angles may take any value in $(0,\pi)$ depending on the choice of capillary parameters. In the present work, we develop a framework of a priori estimates for this model. We prove that, for initial data sufficiently close to equilibrium, the system admits global solutions that converge exponentially fast to a (horizontally) shifted equilibrium state.

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arXiv2025

The steady state of gravity-capillary problem with inclined walls

The gravity-capillary problem with inclined walls is a problem that describes an open fluid flowing over an angled wall. It has broad applications in science and engineering. In this paper, we study the steady states of the two-dimensional inclined-wall problem. The steady-state configurations are characterized as solutions of the Euler-Lagrange equation associated with a prescribed energy functional, subject to a fixed contact-angle boundary condition. By parameterizing the free surface using an appropriately chosen maximal point, we construct solutions to this Euler-Lagrange equation via a shooting method, with the fluid volume serving as the shooting parameter. The construction is valid for arbitrary contact angles and arbitrary inclined angles of the walls.

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arXiv2024

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

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_t\zeta(\pm\ell,t))=[\![\gamma]\!]\mp\sigma\frac{\p_1\zeta}{(1+|\p_1\zeta|^2)^{1/2}}(\pm\ell,t) \end{equation*} in 2D domain, where $\zeta(x_1,t)$ is a free surface with two contact points $\zeta(\pm\ell,t)$, $[\![\gamma]\!]$ and $\sigma$ 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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