Nonnegative measures belonging to H⁻¹(R²)

الأصل · EN

Radon measures belonging to the negative Sobolev space H⁻¹(R²) are important from the point of view of fluid mechanics as they model vorticity of vortex-sheet solutions of incompressible Euler equations. In this note we discuss regularity conditions sufficient for nonnegative Radon measures supported on a line to be in H⁻¹(R²). Applying the obtained results, we derive consequences for measures on R² with arbitrary support and prove elementarily, among other things, that measures belonging to H⁻¹(R²) may be supported on a set of Hausdorff dimension 0. We comment on possible numerical applications.