Analysis and computation of partial differential equations in Mechanics and related fields

本项目主要研究连续介质力学及其相关领域，如统计力学和概率论中出现的偏微分方程的分析。理论和计算两个方面都得到了解决。其目的是帮助提高我们对多粒子系统中物理现象的理解，并影响现实生活中的应用。考虑了三个主要的调查领域。(a)不可压缩流体力学：研究了线性化流动和螺旋对称流动的边界层。(b)弹性：首席研究员继续在多面体域中建立弹性模型，并发展适当的数值方法，特别是广义有限元法；她正在研究如何从边界测量中获得包裹体的大小估计值。(c)演化方程的解法：研究人员正在进一步发展格林的抛物线方程，特别是福克-普朗克方程的函数方法；她还继续研究变速散射问题的波包解决方法，并将其应用于地震成像。共同的主题，如边界和界面对连续系统的影响的调查，以及特定技术的使用，如缩放和本地化，使该项目成为一个有凝聚力的研究计划。由于所研究问题的复杂性，特别是偏微分方程的非线性、不适定性和不稳定性以及奇异几何，需要使用精细的分析工具，特别是微局部和谐波分析。该项目的目的是通过对潜在的数学模型进行严格的分析，并通过设计高效而准确的计算工具来模拟它们，从而提高我们对流体和弹性固体力学中发生的复杂现象的认识。其中一些现象，如流体中的湍流，是经常发生的，但它们仍然缺乏透彻的认识。该项目的一个相关特征是理论和计算方法之间的相互作用，每种方法都提供了自己的研究途径，可以揭示同一现象的不同方面。项目每个部分的进展都有可能影响现实生活中的应用。容器壁上的粘性流动产生的涡度增强了流体的混合和输送，例如在气候和环境建模以及工业过程中的应用（项目a部分）。例如，在血液流动的建模（a部分）中出现了螺旋对称流动。界面问题自然出现在各种应用中，例如确定复合材料的弹性特性和生物过程的建模（项目的第(b)部分）。弹性波成像作为一种非侵入性医疗诊断工具，用于探测地球内部，即用于地震成像，用于地震预测(本项目（b)和(c)部分）。Fokker-Planck类型的抛物方程在应用中出现的概率，例如，等离子体物理学和经济学（项目的(c)部分）。该项目为研究生和本科生提供了培训机会。

Mazzucato1312727 This project focuses on the analysis of partial differential equations arising in continuum mechanics and related fields, such as statistical mechanics and probability. Both theoretical and computational aspects are addressed. Its aim is to help advance our understanding of physical phenomena in many-particle systems and impact real-life applications. Three main areas of investigation are considered.(a) Incompressible fluid mechanics: boundary layers for linearized flows and helically-symmetric flows are studied.(b) Elasticity: the principal Investigator continues to model elasticity in polyhedral domains and to develop suitable numerical methods, in particular the Generalized Finite Element Method (GFEM); she is investigating how to obtain size estimates of inclusions from boundary measurements.(c) Solution methods for evolution equations: the investigator is further developing Green's function methods for parabolic equations, in particular Fokker-Planck equations; she also continues her work on a wave-packet solution method for variable-speed scattering problems, with applications to seismic imaging.Common themes, such as the investigation of the effect of boundaries and interfaces on continuum systems, and the use of specific techniques, such as scaling and localization, make the project a cohesive research program. Employing refined analytical tools, microlocal and harmonic analysis in particular, is warranted by the complexity of the problems studied, which feature nonlinearities in the partial differential equations, ill-posedness and instabilities, and singular geometries. The aim of this project is to advance our knowledge of complex phenomena occurring in the mechanics of fluids and elastic solids, by utilizing a rigorous analysis of the underlying mathematical models and by devising efficient, yet accurate, computational tools to simulate them. Some of these phenomena, such as turbulence in fluids, are a common occurrence, yet they still lack a thorough understanding. A relevant trait of the project is the interplay between theoretical and computational methods, with each providing its own avenue for investigation that can shed light on different aspects of the same phenomenon. Progress on each part of the project has the potential to impact real-life applications. Vorticity created by viscous flows at container walls enhances mixing and transport in fluids with applications for example to climate and environmental modeling, and industrial processes (part a) of the project). Helically-symmetric flows arise, for instance, in modeling of blood flow (part a)). Problems with interfaces appear naturally in a variety of applications, such as determining the elastic properties of composite materials and modeling of biological processes (part(b) of the project). Imaging by elastic waves is used as a non-invasive medical diagnostic tool and in probing the earth's interior, that is, in seismic imaging, for earthquake prediction (parts (b) and (c) of the project). Parabolic equations of the Fokker-Planck type arise in probability with applications, for example, to plasma physics and economics (part (c) of the project). The project provides training opportunities for both graduate and undergraduate students.