American Institute of Mathematical Sciences

Advanced
October  2005, 13(5): 1153-1186. doi: 10.3934/dcds.2005.13.1153

Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation

Thomas Y. Hou 1, , Danping Yang 2, and  Hongyu Ran 3,

1. 

Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States
2. 

School of Mathematics and System Science, Shandong University, Jinan, 250100, China
3. 

Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States

Received  November 2004 Revised  March 2005 Published  September 2005

We perform a systematic multiscale analysis for the 2-D incompressible Euler equation with rapidly oscillating initial data using a Lagrangian approach. The Lagrangian formulation enables us to capture the propagation of the multiscale solution in a natural way. By making an appropriate multiscale expansion in the vorticity-stream function formulation, we derive a well-posed homogenized equation for the Euler equation. Based on the multiscale analysis in the Lagrangian formulation, we also derive the corresponding multiscale analysis in the Eulerian formulation. Moreover, our multiscale analysis reveals some interesting structure for the Reynolds stress term, which provides a theoretical base for establishing systematic multiscale modeling of 2-D incompressible flow.
Keywords: Euler equation, microstructures., multiscale analysis.
Mathematics Subject Classification: 76F25, 76B47, 35L4.
Citation: Thomas Y. Hou, Danping Yang, Hongyu Ran. Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1153-1186. doi: 10.3934/dcds.2005.13.1153
[1]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052
[2]

Carlos Jerez-Hanckes, Irina Pettersson, Volodymyr Rybalko. Derivation of cable equation by multiscale analysis for a model of myelinated axons. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 815-839. doi: 10.3934/dcdsb.2019191
[3]

Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281
[4]

David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319
[5]

Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511
[6]

Annalisa Malusa, Matteo Novaga. Crystalline evolutions in chessboard-like microstructures. Networks & Heterogeneous Media, 2018, 13 (3) : 493-513. doi: 10.3934/nhm.2018022
[7]

Terence Tao. On the universality of the incompressible Euler equation on compact manifolds. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1553-1565. doi: 10.3934/dcds.2018064
[8]

S. Huff, G. Olumolode, N. Pennington, A. Peterson. Oscillation of an Euler-Cauchy dynamic equation. Conference Publications, 2003, 2003 (Special) : 423-431. doi: 10.3934/proc.2003.2003.423
[9]

Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449
[10]

Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations & Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297
[11]

M. Petcu. Euler equation in a channel in space dimension 2 and 3. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 755-778. doi: 10.3934/dcds.2005.13.755
[12]

Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4375-4405. doi: 10.3934/dcdsb.2020292
[13]

Denis Mercier. Spectrum analysis of a serially connected Euler-Bernoulli beams problem. Networks & Heterogeneous Media, 2009, 4 (4) : 709-730. doi: 10.3934/nhm.2009.4.709
[14]

In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012
[15]

Juan Calvo. On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1341-1347. doi: 10.3934/cpaa.2013.12.1341
[16]

Adnan H. Sabuwala, Doreen De Leon. Particular solution to the Euler-Cauchy equation with polynomial non-homegeneities. Conference Publications, 2011, 2011 (Special) : 1271-1278. doi: 10.3934/proc.2011.2011.1271
[17]

Flavia Antonacci, Marco Degiovanni. On the Euler equation for minimal geodesics on Riemannian manifoldshaving discontinuous metrics. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 833-842. doi: 10.3934/dcds.2006.15.833
[18]

David González-Sánchez, Onésimo Hernández-Lerma. On the Euler equation approach to discrete--time nonstationary optimal control problems. Journal of Dynamics & Games, 2014, 1 (1) : 57-78. doi: 10.3934/jdg.2014.1.57
[19]

Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013
[20]

Sergey A. Denisov. Infinite superlinear growth of the gradient for the two-dimensional Euler equation. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 755-764. doi: 10.3934/dcds.2009.23.755

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy