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Geometric problems in PDEs with applications to fluid mechanics

Total Cost €


EC-Contrib. €






Project "GEOFLUIDS" data sheet

The following table provides information about the project.


Organization address
address: CALLE SERRANO 117
city: MADRID
postcode: 28006

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country Spain [ES]
 Project website
 Total cost 1˙256˙375 €
 EC max contribution 1˙256˙375 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2014-STG
 Funding Scheme ERC-STG
 Starting year 2015
 Duration (year-month-day) from 2015-03-01   to  2021-02-28


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 


 Project objective

There are many high-profile problems in PDEs that ultimately boil down to assertions of a strongly geometric or topological nature. One feature that makes these problems both very difficult and extremely appealing is that there is not a standard set of techniques that one can routinely resort to in order to attack them. Indeed, the very nature of these questions makes them strongly interdisciplinary, so successful approaches require finely tailored combinations of ideas and techniques coming from different branches of mathematics (analysis, geometry and topology), often interspersed with some physical intuition. In this project I aim at going significantly beyond the state of the art in a wide class of geometric questions in PDEs, with an emphasis on problems in fluid mechanics and encompassing long-standing questions that can be traced back to leading analysts and geometers such as Arnold, De Giorgi and Yau. The project is divided in three interrelated blocks, respectively devoted to the study of Beltrami fields in steady incompressible fluids, to geometric evolution problems and to global approximation theorems. Key to the proposal is a versatile new approach to a number of geometric problems in PDEs that I have pioneered and applied in several seemingly unrelated contexts. The power of this technique is laid bare by my recent proofs of a well-known conjecture on knotted vortex lines in topological fluid mechanics that was popularized by Arnold and Moffatt in the 1960s and of a long-standing conjecture on the existence of thin vortex tubes in steady solutions to the Euler equation that dates back to Lord Kelvin in 1875. The award of a Starting Grant will enable me to establish a top-level research group on these topics.


year authors and title journal last update
List of publications.
2019 Miguel Domínguez-Vázquez, Alberto Enciso, Daniel Peralta-Salas
Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities
published pages: 718-760, ISSN: 0001-8708, DOI: 10.1016/j.aim.2019.05.017
Advances in Mathematics 351 2019-10-28
2018 Alberto Enciso, David Hartley, Daniel Peralta-Salas
A problem of Berry and knotted zeros in the eigenfunctions of the harmonic oscillator
published pages: 301-314, ISSN: 1435-9855, DOI: 10.4171/JEMS/767
Journal of the European Mathematical Society 20/2 2019-09-02
2018 Alberto Enciso, Daniel Peralta-Salas
A problem of Ulam about magnetic fields generated by knotted wires
published pages: , ISSN: 0143-3857, DOI:
Ergodic Theory & Dynamical Systems in press 2019-09-02
2018 A. Enciso, M.A. Garcia-Ferrero, D. Peralta-Salas
Minimal graphs with micro-oscillations
published pages: in press, ISSN: 0022-0396, DOI:
Journal of Differential Equations 2019-09-02
2018 Alberto Enciso, M. Ángeles García-Ferrero, Daniel Peralta-Salas
The Biot-Savart operator of a bounded domain
published pages: , ISSN: 0021-7824, DOI:
Journal de Mathématiques Pures et Appliquées in press 2019-09-02
2016 Alberto Enciso, Daniel Peralta-Salas, Stefan Steinerberger
Prescribing the nodal set of the first eigenfunction in each conformal class
published pages: , ISSN: 1073-7928, DOI:
International Mathematics Research Notices 2019-09-02
2017 Alberto Enciso, María del Mar González, Bruno Vergara
Fractional powers of the wave operator via Dirichlet-to-Neumann maps in anti-de Sitter spaces
published pages: 2144-2166, ISSN: 0022-1236, DOI: 10.1016/j.jfa.2017.06.009
Journal of Functional Analysis 273/6 2019-09-02
2016 Diego Córdoba, Alberto Enciso, Nastasia Grubic
On the existence of stationary splash singularities for the Euler equations
published pages: 922-941, ISSN: 0001-8708, DOI: 10.1016/j.aim.2015.09.035
Advances in Mathematics 288 2019-09-02
2016 Alberto Enciso, David Hartley, Daniel Peralta-Salas
Laplace operators with eigenfunctions whose nodal set is a knot
published pages: 182-200, ISSN: 0022-1236, DOI: 10.1016/j.jfa.2016.04.016
Journal of Functional Analysis 271/1 2019-09-02
2017 Alberto Enciso, Renato Lucà, Daniel Peralta-Salas
Vortex reconnection in the three dimensional Navier–Stokes equations
published pages: 452-486, ISSN: 0001-8708, DOI: 10.1016/j.aim.2017.01.025
Advances in Mathematics 309 2019-09-02
2018 Alberto Enciso, David Hartley, Daniel Peralta-Salas
Dislocations of arbitrary topology in Coulomb eigenfunctions
published pages: , ISSN: 0213-2230, DOI:
Revista Matemática Iberoamericana in press 2019-09-02
2016 Alberto Enciso, Daniel Peralta-Salas, Francisco Torres de Lizaur
Helicity is the only integral invariant of volume-preserving transformations
published pages: 2035-2040, ISSN: 0027-8424, DOI: 10.1073/pnas.1516213113
Proceedings of the National Academy of Sciences 113/8 2019-09-02
2016 Alberto Enciso, Daniel Peralta-Salas
Beltrami Fields with a Nonconstant Proportionality Factor are Rare
published pages: 243-260, ISSN: 0003-9527, DOI: 10.1007/s00205-015-0931-5
Archive for Rational Mechanics and Analysis 220/1 2019-09-02
2018 Alberto Enciso, David Poyato, Juan Soler
Stability Results, Almost Global Generalized Beltrami Fields and Applications to Vortex Structures in the Euler Equations
published pages: 197-269, ISSN: 0010-3616, DOI: 10.1007/s00220-017-3063-y
Communications in Mathematical Physics 360/1 2019-09-02
2017 Alberto Enciso, Daniel Peralta-Salas, Francisco Torres de Lizaur
Knotted structures in high-energy Beltrami fields on the torus and the sphere
published pages: 995-1016, ISSN: 0012-9593, DOI:
Annales Scientifiques de l\'École Normale Supérieure 50 2019-09-02
2016 Alberto Enciso, Daniel Peralta-Salas
Bounded solutions to the Allen–Cahn equation with level sets of any compact topology
published pages: 1433-1446, ISSN: 2157-5045, DOI: 10.2140/apde.2016.9.1433
Analysis & PDE 9/6 2019-09-02
2018 A. Enciso, P. Tempesta
Uniqueness and characterization theorems for generalized entropies
published pages: , ISSN: 1742-5468, DOI:
Journal of Statistical Mechanics 2019-09-02
2018 Enciso, Alberto; Kamran, Niky
Lorentzian Einstein metrics with prescribed conformal infinity
published pages: , ISSN: 0022-040X, DOI:
Journal of Differential Geometry in press 2019-09-02

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