Beltrami fields and knotted vortex structures in incompressible fluid flows

This paper gives a survey on recent results about the existence of knotted vortex structures in incompressible fluids. includes proof Lord Kelvin's conjecture tubes steady Euler flows and new probabilistic approach to address Arnold's speculation that typical Beltrami fields should exhibit lines arbitrary topological complexity. We review key tools establish these results: global approximation properties fields, inverse localization principle spectral theory, theory Gaussian random which permits introduction considerations picture. Finally, we include some applications techniques other problems, such as reconnections for smooth solutions 3D Navier–Stokes equation evolution local hot spots under heat equation.