We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable Hölder space and the vorticity may not have a distinguished sign. Our solutions are obtained by means of convex integration; they are smooth outside a “turbulence” zone which grows linearly in time around the vortex sheet. As a by-product, this approach shows how the growth of the turbulence zone is controlled by the local energy inequality and measures the maximal initial dissipation rate in terms of the vortex sheet strength. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

中文翻译：

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无可分辨符号的涡片初始数据的耗散欧拉流

我们为涡流片初始数据构造了无限多的二维不可压缩欧拉方程的可容许弱解。我们的初始数据的涡量集中在合适的 Hölder 空间中的一条简单闭合曲线上，并且涡量可能没有明显的符号。我们的解决方案是通过凸积分获得的；它们在“湍流”区域之外是光滑的，“湍流”区域在涡流片周围随时间线性增长。作为副产品，这种方法显示了湍流区的增长如何受局部能量不平等的控制，并根据涡流片强度测量最大初始耗散率。© 2021 作者。Communications on Pure and Applied Mathematics由 Wiley Periodicals LLC 出版。