PDE analysis of thin liquid films: Steady states, long-time behavior, and blow-up

该奖项将支持PI对粘性液体薄膜数学模型的研究。这些问题的一个特点是不同的物理效应之间存在竞争，例如重力和表面张力。具体来说，描述薄膜演化方程的偏微分方程既具有线性稳定项（来自表面张力），导致光滑的薄膜轮廓，也具有使长尺度不稳定的项（例如来自重力）。因此，这些模型呈现出有限带的线性不稳定模态，导致有趣的稳定状态或长时间动力学和新的有限时间奇点。这些模型将用解析、数值和渐近技术来研究。流体薄膜中的图案形成是很常见的。例如，当液体洒在桌子上时，是什么决定了水滴的大小、分布和形状？油漆的成分是如何影响干燥后形成的不规则形状的？吸入剂的什么特性能使其最有效地在肺组织上扩散？在所有这些情况下，都存在“平滑”和“粗糙”物理效果之间的竞争。这些影响通常是不可避免的，通常会导致持久或短暂模式的形成。该奖项将支持在液体薄膜中形成和发展这种模式的数学工作。理解这种现象不仅对数学和科学有意义，而且对工程、工业和医学也有意义。

This award will support the PI's studies of mathematical models for thin films of viscous liquids. A characteristic feature of these problems is that there is a competition between different physical effects, such as gravity and surface tension. Specifically, the partial differential equations that describe the thin film evolution equation have both linearly stabilizing terms (from surfacetension) that lead to smooth film profiles and terms (from gravity, for example) that destabilize long length scales. The models thus exhibit a finite band of linearly unstable modes, leading to interesting steady states or long-time dynamics and to new finite-time singularities. These models willbe studied with analytical, numerical, and asymptotic techniques.Pattern formation in fluid films is very familiar. For example, when a liquidis spilled on a table, what determines the size, distribution, and shape of the droplets that form? How does the composition of a paint affectthe irregularities that form when it dries? What properties of an inhalant will allow it to spread most effectively on lung tissue? In all these situations there is a competition between "smoothing" and "roughening" physical effects. These effects are often unavoidable andtypically lead to the formation of persistent or transient patterns. The award will support mathematical work on the formation and developmentof such patterns in thin liquid films. Understanding such phenomena is of mathematical and scientific interest, as well as of engineering, industrial, and medical use.