We consider the two-dimensional Navier-Stokes equations subject to the Dirichlet boundary condition in a half plane for initial vorticity with finite measures. We study local well-posedness of the associated vorticity equations for measures with a small pure point part and global well-posedness for measures with a small total variation. Our construction is based on an $L^1$-estimate of a solution operator for the vorticity equations associated with the Stokes equations.

中文翻译：

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以测度为初始数据的半平面涡量方程

我们考虑二维 Navier-Stokes 方程在半平面中服从 Dirichlet 边界条件，以获得有限测度的初始涡度。我们研究了具有小的纯点部分的测量的相关涡度方程的局部适定性和具有小的总变化的测量的全局适定性。我们的建设基于${升}^1$- 与斯托克斯方程相关的涡度方程的解算子的估计。