Topics in Applied PDE

该提案集中在三个方向：流体流动中的强烈结构，流体流动的混合，以及化学吸引力在生物学，生态学和医学中的作用。流体流动表现出高度的复杂性，并且可以容易地形成强烈的结构。更好地理解流体流动中强烈特征的形成机制对于工程、天气预报和其他领域的许多应用是非常重要的。该提案的第一个方向旨在研究流体力学的经典方程，重点是强烈的流体运动可以自发发展的情况。PI最近在这个方向上获得的新结果为理解这些复杂而重要的现象提供了重要的进展。在第二个方向上，提出了流体流动中最有效的混合方式的研究。有效的混合在许多应用中至关重要，从发动机燃烧到生态学。在第三个方向，PI将研究化学传感和化学吸引力在生物学反应增强中的作用。其中一种相关的情况涉及身体的愈合，感染或受伤的组织释放特殊的化合物，吸引免疫系统细胞对抗感染。 化学吸引力也可能是一种不良影响：已知一些肿瘤依赖于这种机制来生长。PI计划开发新的数学工具来分析这些更先进、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辅导，并在PI的指导下开展研究项目。该提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Euler方程，Navier-Stokes方程和二维Boussinesq方程组。最近的数值模拟侯和罗提出了一个新的方案奇点形成的三维欧拉方程的解决方案。该方案是轴对称的，奇点形成发生在边界处。该方案也适用于二维无粘Boussinesq系统。受该方案几何形状的启发，PI（与弗拉基米尔Sverak共同）构建了涡度梯度双指数增长的二维欧拉方程的解的示例。众所周知，这种增长是急剧的。PI打算使用在构建过程中获得的新见解来解决2D Boussinesq系统和3D Euler方程的更复杂和长期未决问题。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的比较原则。第二个方向涉及在流体速度受到约束的情况下流体的混合效率。将寻求在各种类型的约束和边界条件下的混合率的界限，以及对最佳混合流的性质的洞察。这一领域位于动力系统，偏微分方程和傅立叶分析的交叉点。第三个方向涉及反应物种之间的化学吸引力对反应速率的影响。化学吸引力将通过Keller-Segel和相关的非线性项来建模。PI建议获得有关此类系统中解决方案行为的定性结果，并推导出描述效果强度及其对各种关键参数的依赖性的精确界限。也将得到具有某些自然类势的Fokker-Planck算子收敛到平衡点的精确界。

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.