The classical no-slip boundary condition of the Navier-Stokes equations fails to describe the spreading motion of a droplet on a substrate due to the missing small-scale physics near the contact line. In this thesis, we introduce a novel regularization of the thin-film equation to model droplet spreading. The solution of the regularized thin-film equation -- the Geometric Thin-Film Equation is studied and characterized. Two robust numerical solvers are discussed, notably, a fast and mesh-free numerical scheme for simulating thin-film flows in two and three spatial dimensions. Moreover, we prove the regularity and convergence of the numerical solutions. The existence and uniqueness of the solution of the Geometric Thin-Film Equation with respect to a wide range of measure-valued initial conditions are also discussed.

翻译：Navier-Stokes 方程的经典无滑移边界条件由于在接触线附近缺失小尺度物理描述，无法准确描述液滴在基底上的铺展运动。本论文引入了一种薄膜方程的新型正则化方法，用于模拟液滴铺展过程。我们对正则化薄膜方程——几何薄膜方程的解进行了研究与表征。文中讨论了两种鲁棒的数值求解器，特别是一种用于模拟二维与三维空间薄膜流动的快速无网格数值格式。此外，我们证明了数值解的正则性与收敛性。同时，也探讨了几何薄膜方程在广泛测度值初始条件下解的存在性与唯一性。