Mathematical Sciences: Hydrodynamic Interface Motion

[409484]固体表面上的粘性薄膜的动力学用薄膜厚度的四阶非线性扩散方程来模拟（Greenspan， J. of Fluid Mech. 1978, vol. 84, p. 125）。该项目涉及该问题的三个方面，奇点形成、弱解和模式形成。当薄膜的厚度趋于零时，奇点就出现了。物理上奇点对应于流体中的拓扑跃迁。本文将采用数值与分析相结合的方法对四阶退化扩散方程奇点形成的起始点进行研究。这项工作可能包括与M. Brenner， L. Kadanoff， T. Dupont和A. Bernoff的合作。这名研究人员正与Courant研究所的玛丽·皮尤（Mary Pugh）一起研究弱解问题。这项合作最近为薄膜方程的弱解建立了更清晰的存在性和长时间行为结果。存在性是在一个正则类中，它只包含一组独特的“源型”解，类似于多孔介质方程的“Barenblatt”解。初步数值计算表明，一般情况下，弱解迅速收敛于“源型”解。计划完成数值计算，并将这些弱解与所建议的滑移模型联系起来。最后，在与M. Brenner的联合项目中，将研究薄膜中重力和温度驱动的指指不稳定性等模式形成问题。＊＊＊

9409484 Bertozzi The dynamics of a thin viscous film on a solid surface are modeled by a fourth order nonlinear diffusion equation for the film thickness (Greenspan, J. of Fluid Mech. 1978, vol. 84, p. 125). The project involves three aspects of this problem, singularity formation, weak solutions, and pattern formation. A singularity occurs when the thickness of the film goes to zero. Physically the singularity corresponds to a topological transition in the fluid. The study will be made via a combination of numerics and analysis the onset of such singularity formation forth order degenerate diffusion equations. This work may include collaborations with M. Brenner, L. Kadanoff, T. Dupont, and A. Bernoff. The investigator is working with Mary Pugh at the Courant Institute on the weak solution problem. This collaboration recently established sharper existence and long time behavior results for weak solutions to the thin film equation. The existence is in a regularity class that just includes a family of unique `source type' solutions analogous to the `Barenblatt' solutions of the porous media equation. Preliminary numerical computations show that in general weak solutions converge rapidly onto the `source type' solution. It is planned to complete the numerics as well as to relate these weak solutions to slip models' suggested for the spreading drop problem. Lastly, in a joint project with M. Brenner, problems in pattern formation such a gravity and temperature driven fingering instabilities in thin films will be studied. ***