Transport Equations, Mixing, and Fluid dynamics

对流-扩散方程在物理学、生物学和工程学的许多领域都具有重要的意义。它们描述了演化系统，其中标量同时被给定的速度场扩散和平流。在许多应用中，例如，在流体动力学的背景下，这些速度场是高度不规则的。然而，由于扩散算子的正则化作用，数学模型往往是适定的。其中之一与剪切流引起的流体混合特性有关。在这个例子中，有一个有趣的相互作用之间的（不规则的）运输的剪切流和正则化扩散，这导致，经过一段时间后，出现了一个占主导地位的长度尺度持续在随后的演变，并确定混合和耗散率。严格理解相关的长度尺度和混合率是可取的。预计某些稳定性估计，比较对流扩散方程和那些纯对流方程的解决方案，将发挥重要作用，在这些混合过程的数学分析。更一般的对流扩散方程的稳定性估计将被导出。这些将深入了解解决方案如何依赖于系数和初始数据的变化。在推导这些估计时，将特别注意证明方法的某种“灵活性”。更具体地说，稳定性估计不仅适用于连续型对流扩散方程，而且适用于离散型对流扩散方程。因此，新的结果应随后应用于适当的有限体积法给出的近似解，以估计数值方案产生的误差。

Advection-diffusion equations are of fundamental importance in many areas of physics, biology and engineering. They describe evolutionary systems, in which a scalar quantity is simultaneously diffused and advected by a given velocity field. In many applications, for instance, in the context of fluid dynamics, these velocity fields are highly irregular. Thanks to the regularizing effect of the diffusion operator, however, the mathematical model is often well-posed.In this project, several quantitative aspects shall be investigated. One of those is related to the mixing properties in fluids that is caused by shear flows. In this example, there is an interesting interplay between the (irregular) transport by the shear flow and the regularizing diffusion, which leads, after a certain time, to the emergence of a dominant length scale which persists during the subsequent evolution and determines mixing and dissipation rates. A rigorous understanding of relevant length scales and mixing rates is desired.It is expected that certain stability estimates, that compare solutions to advection-diffusion equations and those to pure advection equations, will play an important role in the mathematical analysis of these mixing processes. More general stability estimates for advection-diffusion equations will be derived. These shall give a deep insight into how solutions depend on variations of coefficients and initial data. In the derivation of these estimates, special attention will be given to a certain "flexibility" in the method of proof. To be more specific, the stability estimates shall apply not only to the continuous advection-diffusion equations, but also to its discrete variants. As a consequence, the new results shall subsequently be applied to approximate solutions given by suitable finite volume methods in order to estimate the error generated by the numerical scheme.