Nonnegative measures belonging to $H^{-1}(\mathbb{R}^2)$

摘要: Radon measures belonging to the negative Sobolev space $H^{-1}(\mathbb{R}^2)$ are important from point of view fluid mechanics as they model vorticity vortex-sheet solutions incompressible Euler equations. In this note we discuss regularity conditions sufficient for nonnegative supported on a line be in $H^{-1}(\mathbb{R}^2)$. Applying obtained results, derive consequences $\mathbb{R}^2$ with arbitrary support and prove elementarily, among other things, that may set Hausdorff dimension $0$. We comment possible numerical applications.