Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On unstable and neutral spectra of incompressible inviscid and viscid fluids on the $ 2$D torus

Author: Vincent Xiaosong Liu
Journal: Quart. Appl. Math. 53 (1995), 465-486
MSC: Primary 76D05; Secondary 35Q30, 76E05, 76N10
DOI: https://doi.org/10.1090/qam/1343462
MathSciNet review: MR1343462
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We completely determine all the unstable modes and neutral modes for the Kolmogorov flow $ \left( \sin \left( 2{x_2} \right), 0 \right)$ on the two-dimensional torus $ \left[ 0, 2\pi \right] \times \left[ 0, 2\pi \right]$ for both the Navier-Stokes equations and the Euler equations. We also give comments on ``vorticity generation".

References [Enhancements On Off] (What's this?)

  • [1] P. Constantin and C. Foias, Navier-Stokes Equations, the University of Chicago Press, Chicago, 1988
  • [2] P. Constantin, C. Foias, and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Physica D. 30, 284-296 (1988)
  • [3] S. Friedlander and M. M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett. 66, no. 17
  • [4] W. Gautschi, Computational aspects of three-term recurrence relations, SIAM Review 9 (Jan., 1967)
  • [5] W. B. Jones and W. J. Thron, Continued fractions, analytic theory and applications, Encyclopedia of Math. and Its Appl. 11 (1980)
  • [6] C. C. Lin, The Theory of Hydrodynamic Stability, Cambridge University Press, Cambridge, 1964
  • [7] V. X. Liu, An example of instability for the Navier-Stokes equations on the 2-dimensional torus, Comm. Partial Differential Equations (to appear)
  • [8] V. X. Liu, Instability for the Navier-Stokes equations on the 2-dimensional torus and a lower bound for the Hausdorff dimension of their global attractors, Comm. Math. Phys. 147, 217-230 (1992)
  • [9] V. X. Liu, On bifurcations of the Navier-Stokes equations on the 2D torus, in preparation
  • [10] C. Marchioro, An example of absence of turbulence for any Reynolds number, Comm. Math. Phys. 105, 99-106 (1986)
  • [11] L. D. Meshalkin and Ya. G. Sinai, Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid, J. Appl. Math. Mech. 25 (1961)
  • [12] L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan, London, 1933
  • [13] O. Perron, Uber linear Differenzengleichungen, Acta Math. 34, 109-137 (1911)
  • [14] H. Poincaré, Sur les équations linéaires aux différentielles ordinaires et aux differences finies, Amer. J. Math. 7, 203-258 (1885)
  • [15] D. H. Sattinger, The mathematical problem of hydrodynamic stability, J. Math. Mech. 19 (1970)
  • [16] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Math. 41, SIAM, Philadelphia, 1983
  • [17] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam, New York, 1984
  • [18] M. M. Vishik and S. Friedlander, Dynamo theory methods for hydrodynamic stability, J. Math. Pure Appl. (to appear)
  • [19] V. I. Yudovich, Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid, J. Appl. Math. Mech. 29 (1965)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 76D05, 35Q30, 76E05, 76N10

Retrieve articles in all journals with MSC: 76D05, 35Q30, 76E05, 76N10

Additional Information

DOI: https://doi.org/10.1090/qam/1343462
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society