Transport Equations, Mixing, and Fluid dynamics

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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.