Abstract
Three-dimensional solutions with helical symmetry are shown to form an invariant subspace for the Navier-Stokes equations. Uniqueness of weak helical solutions in the sense of Leray is proved, and these weak solutions are shown to be regular (strong) solutions existing for arbitrary time t. The global universal attractor for the infinite-dimensional dynamical system generated by the corresponding semi-group of helical flows is shown to be compact and finite-dimensional. The Hausdorff and fractal dimensions of the global attractors are estimated in terms of the governing physical parameters and in terms of the helical parameters for several problems in the class, with the most detailed results obtained for rotating Hagen-Poiseuille (pipe) flow. In this case, the dimension, either Hausdorff or fractal, up to an absolute constant is bounded from above by \(\frac{{\operatorname{Re} }}{{\sqrt {\alpha ^2 + n^2 } }}\), where α is the axial wavenumber, n is the azimuthal wavenumber and Re is the Reynolds number based on the radius of the pipe. These upper bounds are independent of the rotation rate.
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References
M. Abramowitz & I. E. Stegun [1968], Handbook of Mathematical Functions, Dover, New York.
R. S. Adams [1975], Sobolev Spaces, Academic Press, New York.
S. Agmon [1965], Lectures on Elliptic Boundary Value Problems, Mathematical Studies, Van Nostrand, New York.
P. Constantin [1989], Remarks on the Navier-Stokes equations. Proceedings of the Newport Conference on Turbulence, June 1989 (to appear).
P. Constantin & C. Foias [1988], Navier-Stokes Equations, The University of Chicago Press, Chicago.
P. Constantin & C. Foias [1985], Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations, Comm. Pure Appl. Math. 38, p. 1–27.
P. Constantin, C. Foias & R. Témam [1985], Attractors representing turbulent flows, Memoir A.M.S., No. 314, 53.
P. Constantin, C. Foias, B. Nicolaenko & R. Témam [1988], Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Applied Math. Science Series, No. 70, Springer-Verlag, New York.
F. Cotton & H. Salwen [1981], Linear stability of rotating Hagen-Poiseuille flow, J. Fluid Mech. 108, p. 101–125.
C. Foias, G. R. Sell & R. Témam [1988a], Inertial manifolds for nonlinear evolutionary equations, J. Diff. Eq. 73, p. 309–353.
C. Foias, O. Manley & R. Témam [1988b], Modelling of the interaction of small and large eddies in two dimensional turbulent flows, Mathematical Modelling and Num. Anal. 22, No. 1, p. 93–118.
C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell & E. S. Titi [1988c], On the computation of inertial manifolds, Phys. Letters A 131, p. 433–436.
J.-M. Ghidaglia [1984], Régularité des solutions des certains problèmes aux limites liés aux équations d'Euler, Comm. in Part. Diff. Eq. 9, p. 1237–1264.
M. Graham, P. Steen & E. S. Titi [1990], Dynamics of Benard convection in porous media: computations with approximate inertial manifolds (in preparation).
J. K. Hale [1988], Asymptotic behavior of dissipative systems, Math. Surveys and Monographs 25 AMS, Providence, R.I..
D. B. Henry [1981], Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, No. 840, Springer-Verlag, New York.
J. G. Heywood [1980], The Navier-Stokes equations: On the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29, p. 639–681.
M. S. Jolly, I. G. Kevrekidis & E. S. Titi [1990a], Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Physica D. (in press).
M. S. Jolly, I. G. Kevrekidis & E. S. Titi [1990b], Preserving dissipation in approximate inertial forms, J. Dynamics & Diff. Eqs. (in press).
D. D. Joseph [1976], Stability of Fluid Motions 1, Springer Tracts in Natural Philosophy 27.
O. A. Ladyzhenskaya [1969], Mathematical Theory of Viscous Incompressible Flow, 2nd ed., Gordon and Breach, New York.
O. A. Ladyzhenskaya [1970], Unique solvability in the large of three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry. Seminar in Mathematics, V. A. Steklov Mathematical Institute, Leningrad, 7, Boundary value problems of Mathematical Physics and Related Aspects of Function Theory, Part 2, Edited by O. A. Ladyzhenskaya, p. 70–79.
M. J. Landman [1989], Time dependent helical waves in rotating pipe flow (submitted for publication).
E. H. Lieb & W. Thirring [1976], Inequalities for the moments of the eigenvalues of the Schrödinger equation and their relation to Sobolev spaces, in Studies in Mathematical Physics: Essays in honor of Valentin Bargman (E. H. Lieb, B. Simon & A. Wightman, eds.), Princeton Univ. Press, Princeton, New Jersey.
J.-L. Lions [1969], Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris.
J.-L. Lions & E. Magenes [1972], Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, New York.
A. Mahalov & S. Leibovich [1988], Amplitude expansion for viscous rotating pipe flow near a degenerate bifurcation point, Bulletin of the American Physical Society, 33, No. 10, p. 2247.
G. Métivier [1978], Valeurs propres d'opérateurs définis par la restriction de systèmes variationnels a des sous-espaces, J. Math, pures et appl., p. 133–156.
H. M. Nagib [1972], On instabilities and secondary motions in swirling flows through annuli, Illinois Institute of Technology, PhD Thesis.
M. Onoe [1958], Tables of modified quotients of Bessel functions of the first kind for real and imaginary arguments, Columbia University Press, New York.
M. Reed & B. Simon [1978], Methods of Modern Mathematical Physics. Vol. 4: Analysis of Operators, Academic Press.
C. Rosier [1989], Thesis, Université de Paris-Sud, Orsay.
H. Salwen & C. E. Grosch [1972], The stability of Poiseuille flow in a pipe of circular cross-section, J. Fluid Mech. 54, p. 93–112.
G. R. Sell [1989], Approximation dynamics: hyperbolic sets and inertial manifolds, University of Minnesota Supercomputer Institute, Preprint No. 89/39.
M. A. Shubin [1987], Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin and New York.
R. Témam [1983], Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS Regional Conference series, No. 41, SIAM, Philadelphia.
R. Témam [1988a], Infinite Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer-Verlag, New York.
R. Témam [1988b], Variétés inertielles approximatives pour les équations de NavierStokes bidimensionnelles, C. R. Acad. Sci. Paris, Série II, 306, p. 399–402.
R. Témam ([1988c], Induced trajectories and approximate inertial manifolds, in: Proc. Luminy Conference on infinite dimensional dynamical systems, Marseille, France (1987).
R. Témam [1988d], Dynamical systems, turbulence and numerical solutions of the Navier-Stokes equations, The Inst. for Appl. Math. & Sci. Comp., Indiana Univ., Reprint # 8809.
E. S. Titi [1988], Une variété approximante de l'attracteur universel des équations de Navier-Stokes, non linéaire, de dimension finie, C. R. Acad. Sci., Paris, 307, Serie I, p. 383–385.
E. S. Titi [1990], On approximate inertial manifolds to the Navier-Stokes equations, J. Math. Anal. & Appl. 149, p. 540–557.
N. Toplosky & T. R. Akylas [1988], Nonlinear spiral waves in rotating pipe flow, J. Fluid Mech. 190, p. 39–55.
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Mahalov, A., Titi, E.S. & Leibovich, S. Invariant helical subspaces for the Navier-Stokes equations. Arch. Rational Mech. Anal. 112, 193–222 (1990). https://doi.org/10.1007/BF00381234
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DOI: https://doi.org/10.1007/BF00381234