Analysis of Models in Fluid Dynamics

RUSIN DMS-1613831项目中涉及的问题涉及跨学科性质的数学。对描述流体流动的方程中的非线性现象进行分析研究，对于从天气预报、气候和环境研究、湍流和燃烧(例如发动机中的湍流和燃烧)到宇宙中质量分布的宇宙学问题等领域都是至关重要的。该项目的目的是开发适用的数学理论，从而在理解流体方面取得进展，并促进高效的计算工具。该项目的主要目的是研究流体力学中的偏微分方程解的适定性。主要的重点放在大初始数据背景下的不可压缩三维Navier-Stokes方程。特别地，分析了具有轴对称和时间周期解的解的存在性和性质。研究扩展到相关的模型，如地球物理流体动力学中出现的一类活动标量方程。在这里，主要目的是研究非线性问题的病态/适定现象，其中非线性是基于偶数或奇数符号，并附加一些固有的各向异性。该项目还涉及海洋原始方程解的唯一性问题。最后一个目标是欧拉方程的局域模型及其分析，特别是与三维欧拉方程中的奇异极限和涡度伸展有关的问题。

RusinDMS-1613831 The problems addressed in the project involve mathematics of an interdisciplinary character. Analytical study of the nonlinear phenomena in equations describing flows of fluids is essential to areas that range from weather prediction, climate and environmental studies, turbulent and combustion (as for instance in engines) to the cosmological question of mass distribution in the universe. The aim of the project is to develop applicable mathematics theory that yields progress in understanding of fluids and that facilitates efficient computational tools. The main objective of the project is the investigation of well-posedness of partial differential equations arising in fluid dynamics. The primary emphasis is placed on the incompressible three-dimensional Navier-Stokes equations in the context of large initial data. In particular, the analysis focuses on the existence and properties of solutions with axial symmetry and time-periodic solutions. The research extends to related models such as a class of active scalar equations that arises in the geophysical fluid dynamics. Here, the main aim is to investigate ill/well-posedness phenomena of nonlinear problems, where the nonlinearity is based on even or singular symbols, additionally equipped with some intrinsic anisotropy. The project concerns furthermore the problem of uniqueness of solutions to the primitive equation of the ocean. The last objective addresses a local model of the Euler equations and its analysis, in particular related to singular limits and vorticity stretching in the three-dimensional Euler equations.