Wave Propagation in Nonlinear Acoustics, Viscoelasticity, and Heat Transfer

本研究探讨非线性声学、黏弹性及双曲扩散中非线性波动方程解的定性行为。 本计画主要研究四个领域的大数据解的存在性、唯一性及长时间行为：（1）拟线性波动方程（2）非线性粘弹性问题：（3）作为非稳态热传导模型的阻尼拟线性波动方程;（4）发展了一种新的固体力学非局部理论，称为周向力学，它是经典连续介质力学的扩展，可以有效地模拟材料的断裂。 在其他方法中，PI提出了一种基于物理考虑的原始方法来研究拟线性双曲方程解的存在性。这种方法使人们能够显示解决方案的存在性与任意大的初始数据从小的初始数据的解决方案的存在。所需的成分（有限传播速度，唯一性，能量恒等式）可用于许多双曲系统，因此该方法具有潜在的广泛适用性。非线性波传播现象出现在弹性体的振动中，在声压，等离子体和半导体理论中，以及量子力学中。由于波的相互作用不遵循叠加原理，非线性波动方程的研究带来了很大的困难。 相反，波可以产生新的波，在有限的时间内爆炸，或者在无穷远处消失。该项目涉及开发新的数学工具，以确定在这些相互作用中起主导作用的因素，并预测解决方案的长期行为。所研究的问题模型在科学和医学中的重要现象：碎石模型出现在用于打破肾结石的冲击波传播;双曲线热传导是相关的相变转变和超导性;和周波是非常有前途的预测材料科学中的骨折。 这些领域的进展将有利于学术研究和工业应用。 PI将她的研究成果融入了跨学科课程“城市数学”的开发中，吸引了本科生学习偏微分方程。 在该计划下，学生与当地企业和研究中心合作，开发和分析使用真实的数据的数学模型。该项目还涉及研究生和博士后助理的研究。

This research investigates the qualitative behavior of solutions to nonlinear wave equations in nonlinear acoustics, viscoelasticity, and hyperbolic diffusion. The project studies existence, uniqueness, and long-time behavior of large-data solutions in four problem areas: (1) quasilinear wave equations (the Kuznetsov equation, together with its simplified version, the Westervelt equation) with degeneracy of the differential operator; (2) problems in nonlinear viscoelasticity; (3) the damped quasilinear wave equation as a model for unsteady heat conduction; and (4) development of a new non-local theory in solid mechanics, called peridynamics, an extension of classical continuum mechanics to allow effective modeling of fracture in materials. Among other methods, the PI proposes an original approach based on physical considerations to study existence of solutions for quasilinear hyperbolic equations. This approach enables one to show existence of solutions with arbitrarily large initial data starting from existence of solutions with small initial data. The ingredients needed (finite speed of propagation, uniqueness, energy identity) are available for many hyperbolic systems, hence the approach has potentially wide applicability.Nonlinear wave propagation phenomena appear in vibrations of elastic bodies, in theories of acoustic pressure, plasmas, and semiconductors, and in quantum mechanics. The investigation of nonlinear wave equations poses great difficulty since the interaction of waves does not follow the principle of superposition. Instead, the waves can generate new waves, blow up in finite time, or vanish at infinity. The project involves developing new mathematical tools to determine the factors that play a dominant role in these interactions and to predict the long time behavior of solutions. The problems under study model important phenomena in science and medicine: the lithotripsy model appears in shock wave propagation used in breaking up kidney stones; hyperbolic heat conduction is relevant in phase change transitions and superconductivity; and peridynamics is very promising in predicting fractures in material science. Progress in these areas will benefit both academic research and industrial applications. The PI has integrated her research efforts into the development of an interdisciplinary course, Math in the City, which attracts undergraduate students to study partial differential equations. Under this program, students work in collaboration with local businesses and research centers to develop and analyze mathematical models that use real data. The project also involves graduate students and a postdoctoral associate in the research.