The goal of this numerical study is to get insight into singular solutions of the two-dimensional (2D) Euler equations for non-smooth initial data, in particular for vortex sheets. To this end high resolution computations of vortex layers in 2D incompressible Euler flows are performed using the characteristic mapping method (CMM). This semi-Lagrangian method evolves the flow map using the gradient-augmented level set method (GALS). The semi-group structure of the flow map allows its decomposition into sub-maps (each over a finite time interval), and thus the precision can be controlled by choosing appropriate remapping times. Composing the flow map yields exponential resolution in linear time, a unique feature of CMM, and thus fine scale flow structures can be resolved in great detail. Here the roll-up process of vortex layers is studied varying the thickness of the layer showing its impact on the growth of palinstrophy and possible blow up of absolute vorticity. The curvature of the vortex sheet shows a singular-like behavior. The self-similar structure of the vortex core is investigated in the vanishing thickness limit. Conclusions on the non-uniqueness of weak solutions of 2D Euler for non-smooth initial data are drawn and the presence of flow singularities is revealed tracking them in the complex plane.

翻译：本数值研究旨在深入理解二维欧拉方程在非光滑初值（特别是涡面）条件下的奇异解特性。为此，采用特征映射方法（CMM）对二维不可压缩欧拉流中的涡层进行高分辨率计算。该半拉格朗日方法通过梯度增强水平集方法（GALS）演化流映射。流映射的半群结构允许将其分解为子映射（每个子映射对应有限时间区间），从而可通过选择适当的重映射时间控制计算精度。流映射的复合运算可在线性时间内实现指数级分辨率（这是CMM的独特优势），因此能够高精度解析精细尺度流动结构。本研究系统考察了不同厚度涡层的卷起过程，揭示了厚度参数对涡量梯度增长及绝对涡度可能爆发现象的影响规律。涡面曲率呈现类奇异性行为，并在厚度趋近于零的极限情形下研究了涡核的自相似结构。基于复平面追踪奇点轨迹，得出了二维欧拉方程在非光滑初值条件下弱解非唯一性的结论，并揭示了流动奇点的存在性。