Analysis of Viscoelastic and Compressible Flows

屈服应力流体是在流动之前需要受到临界应力的液体。这些例子基本上包括所有从管子中挤出或用刀涂抹的东西，包括化妆品、洗发水和番茄酱等常见的家庭用品，以及各种各样的工业化合物。对这些流体的研究很重要，例如，在食品、化妆品、制药工业以及生物应用中。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，番茄酱在第二次食用时会比第一次更容易流动。这种复杂的屈服应力行为被称为“触变”。有几种方法来模拟触变流体。该项目是在研究者的工作基础上进行的，该研究表明，当松弛时间变得很长时，可以通过考虑某些粘弹性流体模型的极限来获得触变行为的基本特征。这种模型自然地开辟了应用渐近分析的数学方法的可能性，这依赖于一个小参数的存在。研究者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。项目的第二部分涉及可控性问题，也就是说，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。这是一个自然工程问题，从数学的角度来看，它提出了偏微分方程理论中的基本和具有挑战性的问题。研究者对可压缩流动的方程和粘弹性流体模型中出现的类似数学结构的其他方程进行了研究。与可控性问题密切相关的一个问题是向后唯一性问题。我们是通过改变一个系统现在的状态来改变它未来的状态，还是未来部分独立于现在？对于由偏微分方程给出的系统，在这个意义上“创造未来”并不总是可能的。研究者开发了一种技术，可以用来证明某些偏微分方程组的反向唯一性。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：当松弛时间较大时，触变屈服应力行为作为粘弹性流动的极限而出现。这个大的松弛时间自然为渐近分析提供了一个小的参数。研究者最近分析了在这个极限中出现的快速、缓慢和屈服动力学的不同动力学机制。在空间非均匀流动中，这些机制共存于由尖锐边界分隔的空间区域。在这里，他分析了这些边界作为时间函数的发展。第二个主题涉及可控性问题。可压缩流的方程包含了输运方程和抛物方程的耦合。研究者研究这个系统的可控性和其他数学上类似的问题。将先前线性化问题的结果推广到非线性情况。该分析基于抛物线部分的Carleman估计和输运方程的特征方法，以及允许以迭代方式构建控制的适当分裂方法。这种分割方法的设计是主要的挑战。偏微分方程的后向唯一性问题与可控性问题密切相关。研究者开发了一种基于phragmen - lindelef定理的建立向后唯一性的方法。该方法可应用于一维线性化可压缩流动和一维阻尼波动方程。研究者的目标是将这项技术扩展到更高空间维度的问题上。在分析的基本成分是某些解决估计的推导，这是基于匹配渐近的严格应用。

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.