Analysis of Viscoelastic and Compressible Flows

屈服应力流体是需要承受临界应力才能流动的液体。例如，基本上所有从管子里挤出来或用刀涂抹的东西，包括化妆品、洗发水和番茄酱等常见的家用物质，以及各种各样的工业化合物。例如，对这些流体的研究在食品、化妆品和制药工业以及生物应用中都是重要的。对于许多这样的流体，屈服应力不是固定的，而是取决于流动历史。一个熟悉的例子是，人们普遍认为，番茄酱会比第一次更容易地流出第二道菜。这种复杂的屈服应力行为被称为“触变性”。有几种方法可以对触变流体进行建模。这个项目是研究者的后续工作，研究表明，当松弛时间变得很长时，通过考虑某些粘弹性流体模型的限制，可以获得触变行为的本质特征。这种模型自然打开了应用数学方法进行渐近分析的可能性，这种方法依赖于一个小参数的存在。研究人员探索了这些方法的系统应用，以研究触变屈服应力流体的行为。该项目的第二部分涉及可控性问题，即是否可以通过控制机制将系统从给定的一类输入状态驱动到期望的状态。这是一个自然的工程问题，从数学的角度来看，它在偏微分方程组理论中提出了基本的和具有挑战性的问题。研究人员针对模拟可压缩流动的方程和在模拟粘弹性流体时出现的类似数学结构的其他方程研究了这一问题。与可控性问题密切相关的一个问题是向后唯一性。我们是通过改变一个系统的当前状态来改变它的未来状态，还是说未来在一定程度上独立于现在？对于偏微分方程式所给出的系统，在这个意义上并不总是有可能“发明未来”。研究人员开发了一种技术，可以用来证明某些偏微分方程组的向后唯一性。该项目包括一名研究生和一名博士后。该项目涉及以下领域：当松弛时间较长时，触变屈服应力行为作为粘弹性流动的极限出现。这种大的松弛时间自然为渐近分析提供了一个小参数。这位研究人员最近分析了在这一极限下出现的快、慢和屈服动力学的不同动力学机制。在空间上不均匀的流动中，这些区域共存于由尖锐边界分隔的空间区域。在这里，他分析了这些边界的发展作为时间的函数。第二个话题涉及可控性问题。可压缩流动方程包括输运方程和抛物线方程的耦合。研究人员研究了这个系统的可控性和其他数学上类似的问题。将已有的线性化问题的结果推广到非线性情形。分析的基础是抛物线部分的Carleman估计和输运方程的特征线法，以及允许以迭代方式构造控制的适当的分裂方法。这种拆分方法的设计是主要的挑战。偏微分方程解的向后唯一性问题与可控性问题密切相关。这位研究人员开发了一种建立向后唯一性的方法，该方法基于弗拉格曼-林德洛夫定理。该方法适用于一维线性化的可压缩流动和一维的阻尼波方程。这位研究人员的目标是将这项技术扩展到更高空间维度的问题。分析中的基本成分是某些预解估计的推导，这些估计是基于匹配渐近性的严格应用。

Yield stress fluids are liquids that need to be subjected to a critical stress before they will flow. Examples include essentially everything that is squeezed from a tube or spread with a knife, including common household substances such as cosmetics, shampoo, and ketchup, as well as a wide array of industrial compounds. The study of these fluids is important, for instance, in food, cosmetics, and pharmaceutical industries as well as for biological applications. For many such fluids the yield stress is not fixed, but depends on the flow history. A familiar example is the common experience that ketchup will flow more readily for a second helping than it did the first time. This type of complex yield stress behavior is referred to as "thixotropic." There are several approaches to modeling thixotropic fluids. This project follows up on work of the investigator that showed that the essential features of thixotropic behavior can be obtained by considering the limit of certain models of viscoelastic fluids when a relaxation time becomes very long. This kind of model naturally opens up the possibility of applying mathematical methods of asymptotic analysis, which rely on the presence of a small parameter. The investigator explores the systematic application of these methods to study the behavior of thixotropic yield stress fluids. The second part of the project concerns questions of controllability, that is, whether a system can be driven from a given class of input states to a desired state by a control mechanism. This is a natural engineering question that, from a mathematical point of view, poses fundamental and challenging problems in the theory of partial differential equations. The investigator studies this issue for the equations modeling compressible flow and for other equations of a similar mathematical structure that arise in modeling viscoelastic fluids. A question closely related to controllability issues is that of backward uniqueness. Do we change the future state of a system by altering its present state or is the future partly independent of the present? For systems given by partial differential equations, it is not always possible to "invent the future" in this sense. The investigator has developed a technique that can be used to prove backward uniqueness for certain systems of partial differential equations. A graduate student and a postdoctoral student are included in the project. The project addresses the following areas: Thixotropic yield stress behavior arises as a limit of viscoelastic flow when a relaxation time is large. This large relaxation time naturally provides a small parameter for asymptotic analysis. The investigator has recently analyzed the distinct dynamics regimes of fast, slow and yielded dynamics that arise in this limit. In spatially inhomogeneous flows, these regimes coexist in spatial regions separated by sharp boundaries. Here he analyzes the development of these boundaries as functions of time. The second topic concerns questions of controllability. The equations of compressible flow involve the coupling of a transport equation and a parabolic equation. The investigator studies the controllability of this system and other mathematically similar problems. Prior results for linearized problems are extended to the nonlinear situation. The analysis is based on Carleman estimates for the parabolic part and the method of characteristics for the transport equation, along with a suitable splitting method that allows the construction of controls in an iterative fashion. The design of this splitting method is the principal challenge. The problem of backward uniqueness for partial differential equations is closely related to questions of controllability. The investigator has developed a method of establishing backward uniqueness that is based on the Phragmen-Lindeloef theorem. This method can be applied to one-dimensional linearized compressible flow and to the one-dimensional damped wave equation. The investigator aims to extend the technique to problems in higher space dimensions. The essential ingredient in the analysis is the derivation of certain resolvent estimates, which are based on a rigorous application of matched asymptotics.