Nonlinear Waves in Continuum Mechanics

对材料变形和流动数学的研究受到地下水、石油和天然气等流体的地下流动、雪崩动力学的应用的推动，包括转鼓和溜槽中颗粒材料的处理、微观液滴的运动，以及冲击波在信号传播中的作用，例如在光纤中。将这些应用联系在一起的数学涉及到用非线性方程对连续过程进行建模，以及研究方程的波状解。最近的实验和新的模型为深入了解这些复杂的系统提供了新的机会。该研究项目将包括数学和物理方面的研究生培训，重点是在科学会议上清楚地展示结果。这项关于连续介质力学非线性模型的研究项目集中在四个领域，在这些领域，最近的发展提供了取得进展的机会。色散和耗散冲击波的研究将在磁化和多孔介质流动模型的背景下进行。新的两相流模型具有奇异的非线性，这激发了分析和数值模拟方面的创新。液滴在弹性软基上接触线的研究涉及到非对称液滴模型的建立，该模型可以用变换法和对偶积分方程组来求解。一种推测的微观液滴运动和控制的机制是引起衬底弹性模数的空间变化，从而为液滴提供驱动力。颗粒流模型，特别是具有多分散颗粒尺寸的颗粒流模型，将使用非线性偏微分方程组理论进行分析。这些方程将包括最近对颗粒材料中应力分布的描述，以及一个已建立的动力学机制，该机制量化了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.