Monge–Ampère structures and the geometry of incompressible flows

Bertrand Banos, Vladimir Roubtsov, Ian Roulstone

Type: Article

Publication Date: 2016-05-11

Citations: 0

DOI: https://doi.org/10.1088/1751-8113/49/24/244003

View Chat PDF

Abstract

We show how a symmetry reduction of the equations for incompressible hydrodynamics in three dimensions leads naturally to a Monge-Amp\`ere structure, and Burgers'-type vortices are a canonical class of solutions associated with this structure. The mapping of such solutions, which are characterised by a linear dependence of the third component of the velocity on the coordinate defining the axis of rotation, to solutions of the incompressible equations in two dimensions is also shown to be an example of a symmetry reduction The Monge-Amp\`ere structure for incompressible flow in two dimensions is shown to be hypersymplectic.

Locations

arXiv (Cornell University) - View - PDF
HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF
Surrey Open Research repository (University of Surrey) - View - PDF
DataCite API - View
Journal of Physics A Mathematical and Theoretical - View

Similar Works

Action	Title	Year	Authors
+ PDF Chat	Monge-Ampère Structures and the Geometry of Incompressible Flows	2015	Bertrand Banos Vladimir Roubtsov Ian Roulstone
+ PDF Chat	Monge–Ampère geometry and vortices	2024	Lewis Napper Ian Roulstone Vladimir Rubtsov Martin Wolf
+	Monge-Ampere Geometry and Vortices	2023	Lewis Napper Ian Roulstone Vladimir Rubtsov Martin Wolf
+	The Monge-Ampere equation and two-dimensional incompressible flows	1990	Michèle Larchevêque
+	K"ahler Geometry and the Navier-Stokes Equations	2005	Ian Roulstone Bertrand Banos John Gibbon Vladimir Roubtsov
+	Monge–Ampère equations on two-dimensional manifolds and geometric structures	2010	Alexei Kushner Valentin Lychagin Vladimir Rubtsov
+	Symmetries and contact transformations of Monge–Ampère equations	2010	Alexei Kushner Valentin Lychagin Vladimir Rubtsov
+	On the role of continuous symmetries in the solution of the three-dimensional Euler fluid equations and related models	2022	Miguel D. Bustamante
+	Attractors and symmetries of vector fields: The inverse problem	2007	F. González-Gascón Daniel Peralta‐Salas
+ PDF Chat	Kahler geometry and Burgers' vortices	2009	Ian Roulstone Bertrand Banos John Gibbon Vladimir Roubtsov
+	On the role of continuous symmetries in the solution of the 3D Euler fluid equations and related models	2021	Miguel D. Bustamante
+	Kinematical symmetries of three-dimensional incompressible flows	2000	Hasan Gümral
+	Geometry of Monge–Ampère Structures	2019	V. Rubtsov
+	Chaos from Symmetry: Navier Stokes Equations, Beltrami Fields and the Universal Classifying Crystallographic Group	2023	Pietro G. Fré Mario Trigiante
+ PDF Chat	Special Issue on Symmetry and Fluid Mechanics	2020	R. Ellahi
+	Chaos from Symmetry: Navier Stokes equations, Beltrami fields and the Universal Classifying Crystallographic Group	2022	Pietro G. Fré Mario Trigiante
+ PDF Chat	Auxiliary Monge–Ampère equations in geometric analysis	2023	Bin Guo D. H. Phong
+	A Geometric approximation to the euler equations: the Vlasov?Monge?Amp�re system	2004	Yann Brenier Grégoire Loeper
+	Symmetries, morse polynomials and applications to bifurcation problems	1990	Zhi-Qiang Wang
+ PDF Chat	Kármán vortex street in incompressible fluid models	2020	Claudia García

Cited by (0)

Action	Title	Year	Authors

Citing (14)

Action	Title	Year	Authors
+ PDF Chat	The Geometry of Three-Forms in Six Dimensions	2000	Nigel Hitchin
+	Examples of quaternionic and Kähler structures in Hamiltonian models of nearly geostrophic flow	1997	V. Rubtsov Ian Roulstone
+	Monge–Ampère equations and generalized complex geometry— The two-dimensional case	2006	Bertrand Banos
+	The three-dimensional Euler equations: Where do we stand?	2007	John Gibbon
+ PDF Chat	Numerical study of singularity formation in a class of Euler and Navier–Stokes flows	2000	Koji Ohkitani John Gibbon
+ PDF Chat	Holomorphic structures in hydrodynamical models of nearly geostrophic flow	2001	Vladimir Roubtsov Ian Roulstone
+	Prescribing invariants of Lagrangian surfaces	1992	Marek Kossowski
+ PDF Chat	A classification of Monge-Ampère equations	1993	Valentin Lychagin V. Rubtsov I. V. Chekalov
+ PDF Chat	U(1)-invariant special Lagrangian 3-folds. I. Nonsingular solutions	2004	Dominic Joyce
+	ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS	1986	Neil White
+	Lectures on Mechanics	1992	Jerrold E. Marsden
+ PDF Chat	Compatible Structures on Lie Algebroids and Monge-Ampère Operators	2009	Yvette Kosmann–Schwarzbach Vladimir Rubtsov
+ PDF Chat	Hyper-Kähler geometry and semi-geostrophic theory	2009	Sylvain Delahaies Ian Roulstone
+	None	2002	Bertrand Banos