Title: Uniqueness for the Calderón problem with Lipschitz conductivities.
Author(s): Pedro Caro (BCAM), Keith Rogers (CSIC-ICMAT).
Source: Forum of Mathematics, Pi, Volume 4. DOI: http://dx.doi.org/10.1017/fmp.2015.9
Date of publication: January 13, 2016.
Abstract: In this article, the authors prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in R3. More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in R3, they show that they can be transformed with a Cm-small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted thin vortex tubes can be traced back to Lord Kelvin.
Keith Rogers. Keith Rogers completed his degree in mathematics at the University of Edinburgh in 1999. After a year at Cambridge University and working in London for a period, he obtained his PhD in 2004 from the University of New South Wales in Australia. In 2011, he was awarded a Starting Grant from the European Research Council. His research interests revolve around harmonic analysis – the decomposition of signals into their frequencies. This fundamental tool is used in the analysis of various partial differential equations arising in mathematical physics. He has coauthored more than 30 research articles, including publications in the Journal of the European Mathematical Society, the American Journal of Mathematics, and the Proceedings of the National Academy of Sciences USA. His results have achieved considerable international visibility, such as his extension, in joint work with Javier Parcet, of the Fundamental Theorem of Calculus in three dimensions.