**Title:** Existence of knotted vortex tubes in steady Euler flows.

**Author(s):** Alberto Enciso (CSIC-ICMAT), Daniel Peralta-Salas (CSIC-ICMAT).

**Source:** Acta Mathematica. Volume 214, Issue 1, pp 61-134.

**Date of publication:** March, 2015.

**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.

**Link:** http://link.springer.com/article/10.1007/s11511-015-0123-z

** ICMAT Authors**

**Daniel Peralta.** Daniel Peralta is a member of the ICMAT mainly interested in the interactions between dynamical systems, partial differential equations and differential geometry. He got his PhD at Universidad Complutense in 2006. He has published over 50 papers in some of the best mathematical journals like Annals of Mathematics, ActaMathematica, Journal of Differential Geometry, American Journal of Mathematics, Advances in Mathematics and Communications in Mathematical Physics. He was a plenary speaker at the European Congress of Mathematics that will be held in Berlin in 2016. In 2013 he was awarded a Starting Grant from the European Research Council.

**Alberto Enciso.** Alberto Enciso is an ERC Researcher at the Instituto de CienciasMatemáticas – ICMAT. His research interests lie at the boundary between analysis and geometry, and includes several topics in partial differential equations, geometric analysis, dynamical systems and mathematical physics. In particular, recently he has been considering problems in fluid mechanics, elliptic equations, spectral theory and wave equations on manifolds. He has coauthored around 50 research papers, including articles in the Annals of Mathematics, ActaMathematica, the Journal of Differential Geometry, Advances in Mathematics and Communications in Mathematical Physics. Alberto has been awarded the José Luis Rubio de Francia Prize of the Royal Mathematical Society of Spain in 2011, the Antonio Valle Prize of the Spanish Society for Applied Mathematics in 2013, the Prince of Girona Prize for Scientific Research in 2014, and the Barcelona Dynamical Systems Prize in 2015. In 2014 he was also awarded a Starting Grant from the European Research Council.