Symmetry, Singularity, and Stability in Fluids and Plasmas

该项目旨在研究流体和等离子体中具有数学意义以及物理和工程重要性的非线性现象。它们包括在可压缩和不可压缩流体中在表面张力、粘度和电场的影响下的空化、射流夹断和液滴形成。该项目还涉及奇异热羽流与多尺度流的相互作用。具有复杂结构对称性的不可压缩流体方程的多尺度平衡附近的动力学以及准对称等离子体平衡的构造和限制特性是该项目的第二个组成部分。这些问题与等离子体聚变约束有关。该项目的主要研究领域之一是两相流体的拓扑变化和流体界面奇点的形成。物理实验和数值计算表明，表面张力和电场力会导致细长射流的不稳定性和有限时间夹断。表面张力可以使连接嵌入另一种流体中的流体细胞的细线断裂。非局部和非线性分析的进展，加上几何分析将被开发来研究这些奇异事件。热羽流在湍流对流中起着重要的作用，并且与地球物理学有关。热羽流相互作用的动力学将需要使用涉及奇异积分和拉格朗日观点相互作用的分析方法。具有非平凡准对称性的静磁平衡的构造涉及几何和非线性偏微分方程方法。对这些平衡的限制特性的研究涉及到对具有非常奇异系数的非均匀动力学模型的研究。该奖项反映了NSF的法定使命，并被认为值得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来支持。

The project is aimed at the study of nonlinear phenomena in fluids and plasmas that are of mathematical interest and of physical and engineering importance. They include cavitation, jet pinch-off and drop formation under the influence of surface tension, viscosity, and electrical fields, in compressible and incompressible fluids. The project is also concerned with singular thermal plume interactions with multiscale flows. Dynamics near multiscale equilibria of incompressible fluid equations with complex structured symmetry and the construction and confinement properties of quasisymmetric plasma equilibria are a second component of the project. These issues are relevant to plasma fusion confinement. The project will also provide opportunities for the involvement of graduate students in the research.One of the main areas of the project concerns topological change in two phase fluids and singularity formation in fluid interfaces. Physical experimental and numerical evidence show that surface tension and electrical forces can produce instability and finite time pinch-off of slender fluid jets. Surface tension may rupture thin threads connecting fluid cells embedded in another fluid. Advances in nonlocal and nonlinear analysis, coupled with geometric analysis are going to be developed to study these singular events. Thermal plumes play an important role in turbulent convection and are relevant to geophysics. The dynamics of thermal plume interactions will require the use of methods of analysis involving the interplay of singular integral and Lagrangian points of view. The construction of magnetostatic equilibria with nontrivial quasisymmetry involves geometrical and nonlinear PDE methods. The investigation of the confinement properties of these equilibria involves studies of inhomogeneous kinetic models with very singular coefficients.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.