Topics in Applied PDE

该提案侧重于三个方向：流体流动的强烈结构，流体流动的混合，以及生物学，生态学和医学中的化学吸引力效应。流体流动具有高度的复杂性，容易形成强烈的结构。更好地理解流体流动中产生强烈特征的机制对于工程、天气预报和其他领域的许多应用都非常重要。该提案的第一个方向旨在研究流体力学的经典方程，重点关注强烈流体运动可以自发发展的情况。PI最近在这个方向上取得的新结果为理解这些复杂而重要的现象提供了希望。在第二个方向上，提出研究流体流动中最有效的混合方式。有效的混合在许多应用中都是至关重要的，从发动机的燃烧到生态。在第三个方向上，PI将研究化学传感和化学吸引在生物学中增强反应的作用。在一种情况下，它与身体的愈合有关，感染或受伤的组织释放出特殊的化合物，吸引免疫系统细胞对抗感染。化学吸引也可能是一种不良影响：已知一些肿瘤依靠这种机制生长。PI计划开发新的数学工具来分析这些更先进和更好的预测模型。该项目包括一个培训部分，各级初级研究人员将作为学者和教育工作者接受指导，并在项目负责人的指导下从事研究项目。该方案的第一个主要重点是研究经典流体力学方程的解，如不可压缩欧拉、Navier-Stokes和二维Boussinesq系统。最近侯和罗的数值模拟提出了三维欧拉方程解奇点形成的新场景。这种情况是轴对称的，奇点形成发生在边界。该场景也适用于二维无粘Boussinesq系统。受场景几何的启发，PI（与Vladimir Sverak联合）构建了涡度梯度双指数增长的二维欧拉方程的解示例。众所周知，这种增长是急剧的。PI打算利用在构建中获得的新见解来解决二维Boussinesq系统和三维欧拉方程的更复杂和长期开放的问题。所采用的方法将包括功能分析、傅立叶分析、偏微分方程技术和新颖的比较原理。第二个方向涉及在流体速度受限的情况下流体的混合效率。将寻求在各种约束和边界条件下混合速率的界限，以及对最佳混合流性质的洞察。这个领域是动力系统、偏微分方程和傅立叶分析的交叉领域。第三个方向涉及反应物质之间的化学吸引力对反应速率的影响。化学吸引力将由Keller-Segel和相关的非线性项来建模。PI建议获得关于此类系统中解的行为的定性结果，并推导出描述效应强度及其对各种关键参数的依赖的精确边界。对于具有一定自然势类的Fokker-Planck算子，也得到了收敛到平衡的尖锐界。

The proposal focuses on three directions: intense structures in fluid flows, mixing by fluid flows, and effects of chemical attraction in biology, ecology and medicine. Fluid flows exhibit high degree of complexity and can easily develop intense structures. Better fundamental understanding of the mechanisms of creation of intense features in fluid flows is very important for many applications in engineering, weather forecasting and other fields. The first direction of the proposal seeks to study classical equations of fluid mechanics focussing on situations where intense fluid motion can develop spontaneously. Recent novel results obtained by the PI in this direction provide hopes for an essential advance in understanding these complex and important phenomena. In the second direction, a study of most efficient ways of mixing in fluid flow is proposed. Efficient mixing is of critical importance in many applications, ranging from combustion in engines to ecology. In the third direction, the PI will study the role of chemical sensing and chemical attraction for enhancement of reactions in biology. One situation where it is relevant involves healing of the body, where infected or injured tissues releases special compounds which attract immune system cells to fight the infection. Chemical attraction can also be an undesirable effect: some tumors are known to rely on this mechanism for their growth. The PI plans to develop new mathematical tools to analyze these more advanced and better predictive models. The project involves a training component, where junior researchers at all levels will be mentored as scholars and educators and will work on research projects under the guidance of the PI. The first main focus of the proposal is on studying solutions of the classical equations of fluid mechanics, such as incompressible Euler, Navier-Stokes and 2D Boussinesq system. Recent numerical simulations of Hou and Luo suggest a new scenario for singularity formation for solutions of the 3D Euler equation. The scenario is axi-symmetric, and singularity formation happens at the boundary. The scenario also applies for the 2D inviscid Boussinesq system. Inspired by the geometry of the scenario, the PI (jointly with Vladimir Sverak) has constructed examples of solutions of 2D Euler equation with double exponential growth in vorticity gradient. Such growth is known to be sharp. The PI intends to use new insights obtained in the construction to approach more complex and long open questions for 2D Boussinesq system and 3D Euler equation. Methods employed will include functional analysis, Fourier analysis, PDE techniques and novel comparison principles. The second direction concerns efficiency of mixing in fluids given constraints on fluid velocity. Bounds on the mixing rates under various types of constraints and boundary conditions as well as insight into the nature of best mixing flows will be sought. This area lies at the intersection of dynamical systems, PDE and Fourier analysis. The third direction involves effects of chemical attraction between reacting species on the rates of reaction. Chemical attraction will be modeled by the Keller-Segel and related nonlinear terms. The PI proposes to obtain qualitative results about the behavior of solutions in such systems, and to derive precise bounds describing the strength of the effect and its dependence on various key parameters. Sharp bounds on convergence to equlibirum for Fokker-Planck operators with certain natural classes of potentials will also be obtained.