Weak Singularities and Transport for Incompressible Flows of an Ideal Fluid

首席研究员将研究流体运动的数学问题。用Littlewood-Paley理论和小波方法研究理想流体不可压缩流动的弱奇异性。欧拉方程的基本非定常问题将在基本上具有无界涡度的函数类中进行研究。他将通过非规则的流动来解决线性媒介运输的问题。输运理论的这些方面直接适用于欧拉方程的存在和唯一性问题。流体运动的数学理论在气象学、地球物理学、天体物理学和数学本身的应用中都是基本的。自然界中的大多数流动，如洋流、飓风或星系中物质的运动，都是高度不规则的。奇异结构的起源和演化，如流动的快速振荡，目前还不是很清楚。此外，不规则流动的数值模拟由于流动不断产生的小尺度结构而受到不稳定因素的限制。PI将调查这一问题的两个相关方面。首先，具有奇点的流的基本数学性质。第二，太阳光球层中不规则流动对磁场等量的传输。

9876947VishikThe principal investigator will study the mathematical problems of fluid motion. Methods of Littlewood-Paley theory and wavelets will be used to study weak singularities for the incompressible flows of an ideal fluid. The basic nonstationary problem for the Euler equations will be investigated in function classes with, generally, essentially unbounded vorticity. He will address the problem of linear vector transport by flows that are less than regular. These aspects of transport theory are directly applicable to questions of existence and uniqueness for the Euler equations.The mathematical theory of fluid motion is fundamental both for applications in meteorology, geophysics, astrophysics and within mathematics itself. Most of the flows in nature such as ocean currents, or hurricanes, or the motion of matter in a galaxy, are highly irregular. The origin and evolution of singular structures such as rapid oscillations of a flow are still not well understood. In addition to this, numerical modelling of irregular flows is limited by instabilities due to the small scale structures constantly produced by the flow. The PI will investigate two related aspects of this subject. First, the basic mathematical nature of the flows with singularities. Second, transport of quantities such as the magnetic field by irregular flows in the solar photosphere.