Nonlinear Partial Differential Equations of Mechanics

该项目涉及研究物理系统的数学模型，如雪崩的动力学，水波的运动，地下流体的流动，以及凝胶上小液滴的运动。选择这些模型是因为它们新颖的数学兴趣，以及系统的物理相关性。部分研究，特别是颗粒状雪崩和液滴，将数学与物理实验联系起来。该项目所包含的应用包括碳封存，这是一种将温室气体二氧化碳以液体形式储存在地下深层盐水含水层中的方法;通过称为durotaxis的过程进行生物细胞的运动;制定土壤，农业散装谷物，制药材料等颗粒材料的模型。数学挑战是找到有助于稳定表示颗粒流的模型，例如在排空粮仓和料斗时避免容纳结构失效。 本计画研究非线性偏微分方程，其性质可应用于各种力学上。一方面是探讨色散方程的基本性质，推广的KdV方程，通过Whitham方程。这里的兴趣是在方程的Whitham系统未能双曲，或其中的非线性退化，超曲面。数值模拟展示了一系列的非线性波的相互作用，将探讨分析。多孔介质中的流动和碳封存中的重力流的新模型表现出最近在学生项目中探索的不寻常的效果。接下来的步骤旨在建立这些波效应背后的严格数学。颗粒材料的研究将建立在最近的突破，我们的理解，如何制定连续模型的时间依赖方程是适定的。这个问题在这个领域已经存在了至少30年。继续进行的研究将在新的框架中规定本构律，并在数值模拟、分子动力学表示和物理实验中对照原型特殊流动对其进行测试。最后，一个关于硬旋转中液滴运动的学生项目在相应的静态问题上取得了进展，并准备转向动态，其中液滴通过底层柔性基底的变形而形成的脊冲浪。该项目与水、凝胶和空气界面上的接触线和表面张力的物理实验同时进行。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns research into mathematical models of physical systems, such as the dynamics of avalanches, motion of water waves, the underground flow of fluids, and the motion of small liquid drops on a gel. The models are chosen for their novel mathematical interest, and for the physical relevance of the system. Part of the research, notably granular avalanches and droplets, relates the mathematics to physical experiments. Applications contained in this project include carbon sequestration, a means of storing the greenhouse gas carbon dioxide in liquid form in deep underground salt water aquifers; the motion of cells in biology by a process called durotaxis; the formulation of models of granular materials such as soils, agricultural bulk grains, pharmaceutical materials. The mathematical challenge is to find models that facilitate the stable representation of granular flow, for example in draining a grain bin and hopper avoiding failure of the containing structure. The research in this project on nonlinear partial differential equations concerns properties of the equations modeling a variety of applications in mechanics. One aspect is to explore fundamental properties of dispersive equations, generalizations of the KdV equation, via the Whitham equations. The interest here is in equations for which the Whitham system fails to be hyperbolic, or for which the nonlinearity is degenerate, on hypersurfaces. Numerical simulations exhibit a range of nonlinear wave interactions that will be explored analytically. Novel models of flow in porous media and gravity currents in carbon sequestration exhibit unusual effects that were recently explored in student projects. The next steps aim to establish the rigorous mathematics behind these wave effects. Research on granular materials will build on a recent breakthrough in our understanding of how to formulate continuum models for which the time dependent equations are well-posed. This has been a problem in the field for at least thirty years. The continuing research will specify constitutive laws in the new framework and test them against prototype special flows, in numerical simulations, molecular dynamics representations and physical experiments. Finally, a student project on the motion of droplets in durotaxis has made progress on the corresponding static problem and is ready to move onto dynamics, in which the droplet surfs a ridge created through deformation of the underlying flexible substrate. This project is proceeding in tandem with physical experiments on contact lines and surface tension, at the interfaces between water, a gel, and air.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.