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
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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".
P. Constantin and C. Foias, Navier-Stokes Equations, the University of Chicago Press, Chicago, 1988
P. Constantin, C. Foias, and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Physica D. 30, 284–296 (1988)
S. Friedlander and M. M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett. 66, no. 17
W. Gautschi, Computational aspects of three-term recurrence relations, SIAM Review 9 (Jan., 1967)
W. B. Jones and W. J. Thron, Continued fractions, analytic theory and applications, Encyclopedia of Math. and Its Appl. 11 (1980)
C. C. Lin, The Theory of Hydrodynamic Stability, Cambridge University Press, Cambridge, 1964
V. X. Liu, An example of instability for the Navier-Stokes equations on the 2-dimensional torus, Comm. Partial Differential Equations (to appear)
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)
V. X. Liu, On bifurcations of the Navier-Stokes equations on the 2D torus, in preparation
C. Marchioro, An example of absence of turbulence for any Reynolds number, Comm. Math. Phys. 105, 99–106 (1986)
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)
L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan, London, 1933
O. Perron, Uber linear Differenzengleichungen, Acta Math. 34, 109–137 (1911)
H. Poincaré, Sur les équations linéaires aux différentielles ordinaires et aux differences finies, Amer. J. Math. 7, 203–258 (1885)
D. H. Sattinger, The mathematical problem of hydrodynamic stability, J. Math. Mech. 19 (1970)
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Math. 41, SIAM, Philadelphia, 1983
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam, New York, 1984
M. M. Vishik and S. Friedlander, Dynamo theory methods for hydrodynamic stability, J. Math. Pure Appl. (to appear)
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)
P. Constantin and C. Foias, Navier-Stokes Equations, the University of Chicago Press, Chicago, 1988
P. Constantin, C. Foias, and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Physica D. 30, 284–296 (1988)
S. Friedlander and M. M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett. 66, no. 17
W. Gautschi, Computational aspects of three-term recurrence relations, SIAM Review 9 (Jan., 1967)
W. B. Jones and W. J. Thron, Continued fractions, analytic theory and applications, Encyclopedia of Math. and Its Appl. 11 (1980)
C. C. Lin, The Theory of Hydrodynamic Stability, Cambridge University Press, Cambridge, 1964
V. X. Liu, An example of instability for the Navier-Stokes equations on the 2-dimensional torus, Comm. Partial Differential Equations (to appear)
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)
V. X. Liu, On bifurcations of the Navier-Stokes equations on the 2D torus, in preparation
C. Marchioro, An example of absence of turbulence for any Reynolds number, Comm. Math. Phys. 105, 99–106 (1986)
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)
L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan, London, 1933
O. Perron, Uber linear Differenzengleichungen, Acta Math. 34, 109–137 (1911)
H. Poincaré, Sur les équations linéaires aux différentielles ordinaires et aux differences finies, Amer. J. Math. 7, 203–258 (1885)
D. H. Sattinger, The mathematical problem of hydrodynamic stability, J. Math. Mech. 19 (1970)
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Math. 41, SIAM, Philadelphia, 1983
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam, New York, 1984
M. M. Vishik and S. Friedlander, Dynamo theory methods for hydrodynamic stability, J. Math. Pure Appl. (to appear)
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)
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Article copyright:
© Copyright 1995
American Mathematical Society
