涡斑方法作为二级非牛顿流体
The vortex blob method as a second-grade non-Newtonian fluid
AI总结：
本文证明一类二维涡斑近似可严格视为零粘度二级非牛顿流体方程的解，给出全局弱解存在性及收敛性，为涡方法提供理论支撑。
摘要：
我们证明，二维理想流体力学中某一类涡斑近似可以被严格理解为具有零粘度的二级非牛顿流体方程的解，其初始数据位于Radon测度空间${\mathcal M}({\mathbb R}^2)$中。该正则化PDE（也称为平均Euler或Euler-$\alpha$方程）的解，是关于一个新的弱右不变度量下体积保持微分同胚群上的测地线。我们证明了对于初始涡量在${\mathcal M}({\mathbb R}^2)$中（如点涡数据）的情形，存在唯一的全局弱解（测地线），并表明相关的余伴随轨道由该流保持。此外，当初始数据在测度意义下弱逼近且逼近的总变差也收敛时，该特定涡斑方法的解收敛到具有有界初始涡量的Euler方程的解。特别地，这包括了通常用于涡方法的基于网格的逼近格式。
英文摘要：
We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in the space of Radon measures ${\mathcal M}({\mathbb R}^2)$. The solutions of this regularized PDE, also known as the averaged Euler or Euler-$α$ equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in ${\mathcal M}({\mathbb R}^2)$ such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes of the type that are usually used for vortex methods.
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