Diffusion, Dissipation Enhancement, and Mixing

扩散和混合是在从微流体到气象学到宇宙学等应用中出现的关键物理现象。扩散倾向于分散集群，并试图分散它们，而混合则编织集群，并倾向于将一个集群的层穿插到另一个集群。该项目旨在量化它们的相互作用，并确定它如何影响物理和化学流体系统的动力学。提高对它们如何相互作用的理解在科学领域有直接和间接的影响，例如对趋化性、包括合金在内的二元混合物的相分离和地球磁场的研究。该研究将在学生和博士后同事的积极参与下进行，他们将在广泛的领域接受培训。这个项目将研究几个关于增强耗散、扩散和混合的问题。在不可压缩的流体中，搅拌引起了细丝的混合，并促进了小尺度的形成。另一方面，扩散有效地阻尼了小尺度，这两种现象之间的平衡会导致耗散增强——溶液比正常情况下耗散能量更快的趋势。该项目的四个目标是：构建简单而明确的流族，以任意大的量增强耗散；利用耗散增强来解决非线性偏微分方程中有物理意义的问题；根据混合速率获得耗散增强的精确估计；并研究了剩余扩散率，以及耗散增强对磁场长期行为的影响。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Diffusion and mixing are key physical phenomena that arise in applications ranging from micro-fluids to meteorology, to cosmology. Diffusion tends to spread out clusters and tries to disperse them, while mixing weaves the clusters and tends to intersperse layers of one cluster with another. This project seeks to quantify their interaction and determine how it impacts the dynamics of physical and chemical fluid systems. Improving the understanding of how they interact has both direct and indirect consequences in scientific areas, such as the study of chemotaxis, of phase separation of binary mixtures including alloys, and of the earth's magnetic field. The research will be carried out with active involvement from students and postdoctoral associates, who will be trained across a broad range of fields.This project will examine several issues concerning enhanced dissipation, diffusion, and mixing. In incompressible fluids, stirring induces mixing by filamentation and facilitates the formation of small scales. Diffusion, on the other hand, efficiently damps small scales and the balance between these two phenomena can lead to enhanced dissipation - the tendency of solutions to dissipate energy faster than normal. Four goals of the project are to construct simple and explicit family of flows that enhance dissipation by an arbitrarily large amount; to use dissipation enhancement to address physically meaningful questions concerning nonlinear partial differential equations; to obtain sharp estimates for the dissipation enhancement in terms of the mixing rates; and to investigate residual diffusivity, and the effect of dissipation enhancement on the long-time behavior of magnetic fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.