通过竞争运输标记对多相二维欧拉方程进行正则化
Regularization for Multi-Phase 2D Euler Equations via Competing Transport Markers
AI总结：
本文提出了一种基于竞争运输标记的正则化方法，用于多相二维欧拉方程，通过标记选择机制实现涡旋场的精确保持和收敛。
摘要：
我们介绍了一种新的正则化框架，用于二维不可压缩欧拉方程，该框架能够精确保持多相涡旋场的传输结构。关键步骤是将多相涡旋斑块数据重新表述为有限个被动传播的标量标记函数：在每个点，局部涡旋由来自这些标记之间的竞争中产生的光滑、点wise选择规则确定。该方案不引入空间扩散或 mollification；所有正则化都仅来源于标记选择机制。当尖锐参数β→∞时，我们证明了传输标记函数在有限时间区间内的均匀收敛。此外，在底层欧拉界面网络上满足几何非退化条件的情况下，我们建立了演化的界面结构的Hausdorff收敛，并建立了正则化涡旋在非tie集附近以指数形式收敛到相应的多相涡旋斑块解。最后，我们显示了点wise收敛的丧失恰好与欧拉界面动力学中的几何退化现象相吻合。
英文摘要：
We introduce a novel regularization framework for the two-dimensional incompressible Euler equation that exactly preserves the transport structure of multi-phase vorticity fields. The key step is a reformulation of multi-phase vortex patch data in terms of a finite family of passively advected scalar marker functions: at each point, the local vorticity is determined by a smooth, pointwise selection rule arising from competition among these markers. The scheme introduces no spatial diffusion or mollification; all regularization originates solely from the marker selection mechanism. As the sharpness parameter $β\to\infty$, we prove uniform convergence of the transported marker functions on finite time intervals. Moreover, under a geometric nondegeneracy condition on the underlying Euler interface network, we establish Hausdorff convergence of the evolving interfacial structures and exponential-in-$β$ pointwise convergence of the regularized vorticity away from the tie sets to the corresponding sharp multi-phase vortex patch solution. Finally, we show that the loss of pointwise convergence coincides precisely with the onset of geometric degeneracy in the Euler interface dynamics.
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