Mathematical Sciences: Hydrodynamic Interface Motion

9409484 Bertozzi固体表面上粘性薄膜的动力学由薄膜厚度的四阶非线性扩散方程来模拟(Greenspan，J.of Fluid Mech)。1978年，第84卷，第125页)。该项目涉及该问题的三个方面，奇点形成、弱解和模式形成。当薄膜厚度变为零时，就会出现奇点。在物理上，奇点对应于流体中的拓扑转变。本研究将通过数值计算和分析这种奇点形成的四阶退化扩散方程相结合的方法来进行研究。这项工作可能包括与M.Brenner、L.Kadanoff、T.Dupont和A.Bernoff的合作。这位研究人员正与库兰特研究所的玛丽·普格一起研究弱解问题。这项合作最近建立了薄膜方程弱解的更尖锐的存在性和长期行为结果。它的存在性属于正则性类，它只包括一族唯一的“源型”解，类似于多孔介质方程的“Barenblatt”解。初步的数值计算表明，一般情况下，弱解会迅速收敛到“源型”解。计划完成数值计算，并将这些弱解与滑移模型建议的散布液滴问题联系起来。最后，在与M.Brenner的一个联合项目中，将研究图形形成中的问题，如重力和温度驱动的薄膜中的指进不稳定性。***

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. ***