American Institute of Mathematical Sciences

Advanced
November  2010, 27(4): 1571-1586. doi: 10.3934/dcds.2010.27.1571

Group foliation of equations in geophysical fluid dynamics

Jeffrey J. Early 1, , Juha Pohjanpelto 2, and  Roger M. Samelson 1,

1. 

College of Oceanic and Atmospheric Sciences, 104 COAS Admin Bldg, Oregon State University, Corvallis, OR 97331-5503, United States, United States
2. 

Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605, United States

Received  September 2009 Revised  February 2010 Published  March 2010

The method of group foliation can be used to construct solutions to a system of partial differential equations that, as opposed to Lie's method of symmetry reduction, are not invariant under any symmetry of the equations. The classical approach is based on foliating the space of solutions into orbits of the given symmetry group action, resulting in rewriting the equations as a pair of systems, the so-called automorphic and resolvent systems, involving the differential invariants of the symmetry group, while a more modern approach utilizes a reduction process for an exterior differential system associated with the equations. In each method solutions to the reduced equations are then used to reconstruct solutions to the original equations. We present an application of the two techniques to the one-dimensional Korteweg-de Vries equation and the two-dimensional Flierl-Petviashvili (FP) equation. An exact analytical solution is found for the radial FP equation, although it does not appear to be of direct geophysical interest.
Keywords: exterior differential system, geophysical fluid dynamics., group foliation, Symmetry.
Mathematics Subject Classification: Primary: 34A26, 58J70; Secondary: 58A1.
Citation: Jeffrey J. Early, Juha Pohjanpelto, Roger M. Samelson. Group foliation of equations in geophysical fluid dynamics. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1571-1586. doi: 10.3934/dcds.2010.27.1571
[1]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic and Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025
[2]

Maoli Chen, Yicheng Liu, Xiao Wang. Delay-dependent flocking dynamics of a two-group coupling system. Discrete and Continuous Dynamical Systems - B, 2023, 28 (1) : 808-832. doi: 10.3934/dcdsb.2022099
[3]

L. Bakker. A reducible representation of the generalized symmetry group of a quasiperiodic flow. Conference Publications, 2003, 2003 (Special) : 68-77. doi: 10.3934/proc.2003.2003.68
[4]

Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2155-2185. doi: 10.3934/cpaa.2021062
[5]

Santiago Capriotti. Dirac constraints in field theory and exterior differential systems. Journal of Geometric Mechanics, 2010, 2 (1) : 1-50. doi: 10.3934/jgm.2010.2.1
[6]

Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345
[7]

Eugenii Shustin. Dynamics of oscillations in a multi-dimensional delay differential system. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 557-576. doi: 10.3934/dcds.2004.11.557
[8]

Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925
[9]

Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121
[10]

Olof Heden, Fabio Pasticci, Thomas Westerbäck. On the existence of extended perfect binary codes with trivial symmetry group. Advances in Mathematics of Communications, 2009, 3 (3) : 295-309. doi: 10.3934/amc.2009.3.295
[11]

Zhi-Ying Sun, Lan Huang, Xin-Guang Yang. Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry. Electronic Research Archive, 2020, 28 (2) : 861-878. doi: 10.3934/era.2020045
[12]

Lan Huang, Zhiying Sun, Xin-Guang Yang, Alain Miranville. Global behavior for the classical solution of compressible viscous micropolar fluid with cylinder symmetry. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1595-1620. doi: 10.3934/cpaa.2022033
[13]

Xiaotao Huang, Lihe Wang. Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1121-1134. doi: 10.3934/cpaa.2017054
[14]

Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161
[15]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure and Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[16]

M. Jotz. The leaf space of a multiplicative foliation. Journal of Geometric Mechanics, 2012, 4 (3) : 313-332. doi: 10.3934/jgm.2012.4.313
[17]

Mahesh Nerurkar. Forced linear oscillators and the dynamics of Euclidean group extensions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1201-1234. doi: 10.3934/dcdss.2016049
[18]

Laurent Bourgeois, Jérémi Dardé. The "exterior approach" to solve the inverse obstacle problem for the Stokes system. Inverse Problems and Imaging, 2014, 8 (1) : 23-51. doi: 10.3934/ipi.2014.8.23
[19]

Alain Miranville, Mazen Saad, Raafat Talhouk. Preface: Workshop in fluid mechanics and population dynamics. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : i-i. doi: 10.3934/dcdss.2014.7.2i
[20]

A. V. Borisov, I.S. Mamaev, S. M. Ramodanov. Dynamics of two interacting circular cylinders in perfect fluid. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 235-253. doi: 10.3934/dcds.2007.19.235

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy