Nonlinear Waves in Continuum Mechanics

对材料变形和流动的数学研究的动机是应用于地下水、石油和天然气等流体的地下流动，雪崩动力学，包括滚圆桶和滑槽中颗粒材料的处理，微观流体滴的运动，以及冲击波在信号传播中的作用，例如在光纤中。将这些应用联系起来的数学涉及到用非线性方程对连续过程进行建模，以及研究这些方程的类波解。最近的实验和新模型为对这些复杂系统的新见解提供了机会。该研究项目将包括对研究生进行数学和物理方面的培训，重点是在科学会议上清晰地展示结果。这个研究连续介质力学非线性模型的项目集中在四个领域，其中最近的发展为进步提供了机会。在磁化和多孔介质流动模型的背景下，将对色散和耗散激波进行研究。新的两相流模型具有奇异的非线性特性，激发了分析和数值模拟的创新。软弹性基底上液滴接触线的研究涉及到非对称液滴模型的建立，该模型可以用变换和对偶积分方程的方法求解。微观液滴运动和控制的推测机制是诱导基底弹性模量的空间变化，从而为液滴提供驱动力。颗粒流动模型，特别是多分散粒度，将使用非线性偏微分方程理论进行分析。这些方程将结合颗粒材料中应力分布的最新描述，并结合已建立的动态机制，量化不同大小的颗粒如何在不同的层中积累。

This research into the mathematics of deformation and flow of materials is motivated by applications to subsurface flows of fluids such as groundwater, oil and gas, avalanche dynamics, including the processing of granular materials in tumblers and chutes, the motion of microscopic fluid droplets, and the role of shock waves in the propagation of signals, for example in optical fibers. The mathematics linking these applications concerns the modeling of continuous processes with nonlinear equations, and the study of wave-like solutions of the equations. Recent experiments and new models have opened up opportunities for fresh insights into these complex systems. The research project will include graduate student training in mathematics and physics, with an emphasis on clear presentation of results at scientific conferences. This research project on nonlinear models of continuum mechanics focuses on four areas where recent developments provide opportunities for progress. The investigation of dispersive and dissipative shock waves will take place in the context of models of magnetization and porous media flow. New models of two phase flow possess singular nonlinearities that stimulate innovations in analysis and numerical simulation. The study of contact lines of droplets on soft elastic substrates involves the formulation of models for asymmetric drops that can be solved by the method of transforms and dual integral equations. A conjectured mechanism for the motion and control of microscopic droplets is to induce spatial variations in the elastic modulus of the substrate, thereby providing a driving force on the droplet. Granular flow models, specifically with poly-disperse grain sizes, will be analyzed using the theory of nonlinear partial differential equations. The equations will incorporate recent descriptions of stress distribution in granular materials, coupled to an established dynamic mechanism that quantifies how grains of different sizes accumulate in distinct layers.