Inertial manifolds of incompressible, nonlinear bipolar viscous fluids
Authors:
Frederick Bloom and Wenge Hao
Journal:
Quart. Appl. Math. 54 (1996), 501-539
MSC:
Primary 76A05; Secondary 35Q35, 76D99
DOI:
https://doi.org/10.1090/qam/1402407
MathSciNet review:
MR1402407
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The existence of an inertial manifold is established for the nonlinear system of equations describing the motion of a bipolar incompressible viscous fluid. In this paper we consider only the case of a spatially periodic velocity field. Existence of an inertial manifold for the model complements earlier work on the existence of compact global attractors for bipolar viscous fluids and serves to further highlight the differences between the bipolar model and the usual model based on the linear Stokes constitutive relation.
A. E. Green and R. S. Rivlin, Simple force and stress multipoles, Arch. Rational Mech. Anal. 16, 325–353 (1964)
A. E. Green and R. S. Rivlin, Multipolar continuum mechanics, Arch. Rational Mech. Anal. 17, 113–147 (1964)
J. L. Bleustein and A. E. Green, Dipolar fluids, Internat. J. Engrg. Sci. 5, 323–340 (1967)
J. Nečas and M. Šilhavý, Multipolar viscous fluids, Quart. Appl. Math. 49, 247–265 (1991)
H. Bellout, F. Bloom, and J. Nečas, Phenomenological behavior of multipolar viscous fluids, Quart. Appl. Math. 50, 559–583 (1992)
M. Shinbrot, Lectures on Fluid Mechanics, Gordon and Breach, New York, 1993
D. Ruelle, The turbulent fluid as a dynamical system, in New Perspectives in Turbulence, L. Sirovich, ed., Springer-Verlag, Berlin, 1991, pp. 123–138
O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969
O. Ladyzhenskaya, New equations for the description of viscous incompressible fluids and solvability in the large of the boundary value problems for them, Proc. Steklov Inst. Math. 102, 95–118 (1970)
S. Kaniel, On the initial-value problem for an incompressible fluid with nonlinear viscosity, J. Math. Mech. 19, 681–706 (1970)
Q. Du and M. D. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Anal. Appl. 155, 21–45 (1991)
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969
Y.-R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations. I, Quart. Appl. Math. 49, 651–685 (1991)
Y.-R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations. II, Quart. Appl. Math. 49, 687–728 (1991)
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988
R. A. Toupin, Theories of elasticity with couple stress, Arch. Rational Mech. Anal. 17, 85–112 (1964)
H. Bellout, F. Bloom, and J. Nečas, Existence, uniqueness, and stability of solutions to the initial boundary value problem for bipolar viscous fluids, Differential Integral Equations 8, 453–464 (1995)
H. Bellout, F. Bloom, and J. Nečas, Young measure-valued solutions for non-Newtonian incompressible fluids, Comm. Partial Differential Equations 19, 1763–1803 (1994)
H. Bellout, F. Bloom, and J. Nečas, Solutions for incompressible non-Newtonian fluids, C. R. Acad. Sci. Paris Sér. I Math. 317, 795–800 (1993)
J. Nečas, Sur les normes équivalentes dans $W_p^{\left ( k \right )}\left ( \Omega \right )$ et sur la coercivité des formes formellement positives, Équations aux Dérivées Partielles, Presses Univ. Montréal, Montréal, 1965
H. Bellout and F. Bloom, Existence and asymptotic stability of time-dependent Poiseuille flows of isothermal bipolar fluids, Applicable Anal. 50, 115–130 (1993)
H. Bellout and F. Bloom, On the uniqueness of plane Poiseuille solutions of the equations of incompressible bipolar fluids, Internat. J. Engrg. Sci. 31, 1535–1549 (1993)
H. Bellout and F. Bloom, Steady plane Poiseuille flows of incompressible multipolar fluids, Internat. J. Non-Linear Mech. 23, 503–518 (1993)
F. Bloom and W. Hao, Steady flows of nonlinear bipolar viscous fluids between rotating cylinders, Quart. Appl. Math. 53, 143–171 (1995)
H. Bellout, F. Bloom, and J. Nečas, Bounds for the dimensions of the attractors of nonlinear bipolar viscous fluids, Asymptotic Anal. 11, 131–167 (1995)
F. Bloom, Attractors of bipolar and non-Newtonian viscous fluids, First World Congr. Nonlinear Analysts (Tampa, FL, August 1992), Vol. I, V. Lakshmikantham, ed., Walter de Gruyter Publishers, 1996, pp. 583–596
F. Bloom, Attractors of non-Newtonian fluids, J. Dynamics and Differential Equations 7, 109–140 (1995)
P. Constantin, C. Foiaş, and R. Temam, Attractors Representing Turbulent Flows, Mem. Amer. Math. Soc. No. 314 (1985)
P. Constantin, C. Foiaş, and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Phys. D. 30, 284–296 (1988)
J. Malek and J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, J. Differential Equations, in press
C. Foiaş and G. R. Sell, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73, 309–353 (1988)
P. Constantin, C. Foiaş, B. Nicolaenko, and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, New York, 1989
P. Constantin and C. Foiaş, Navier-Stoker Equations, University of Chicago Press, Chicago, 1988
M. W. Smiley, Global attractors and approximate inertial manifolds for nonautonomous dissipative equations, Applicable Anal. 50, 217–241 (1993)
G. R. Sell and Y. You, Inertial manifolds : the nonselfadjoint case, J. Differential Equations 96, 203–255 (1992)
M. Kwak, Finite-dimensional inertial forms for the 2D Navier-Stokes equations, Indiana Univ. Math. J. 41, 927–981 (1993)
G. R. Sell and J. Mallet-Paret, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Amer. Math. Soc. 1, 805–866 (1988)
M. Luskin and G. R. Sell, Parabolic regularization and inertial manifolds, IMA Preprint, University of Minnesota, Minneapolis, 1989
P. Constantin, C. Foiaş, B. Nicolaenko, and R. Temam, Spectral barriers and inertial manifolds for dissipative partial differential equations, J. Dynamics and Differential Equations 1, 45–73 (1989)
C. Foiaş, G. R. Sell, and E. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dynamics and Differential Equations 1, 199–244 (1989)
M. Marion, Approximate inertial manifolds for reaction-diffusion equations in higher space dimensions, J. Dynamics and Differential Equations 1, 245–267 (1989)
M. Luskin and G. R. Sell, The construction of inertial manifolds for reaction-diffusion equations by elliptic regularization, IMA Preprint, University of Minnesota, Minneapolis, 1989
E. Titi, On approximate inertial manifolds to the Navier-Stokes equations, J. Math. Anal. Appl. 149, 540–557 (1990)
P. Constantin, A construction of inertial manifolds, in The Connection Between Infinite-Dimensional and Finite-Dimensional Dynamical Systems (Boulder, CO, 1987), Contemp. Math., vol. 99, Amer. Math. Soc., Providence, RI, 1989, pp. 27–62
F. Demenge and J. M. Ghidaglia, Some remarks on the smoothness of inertial manifolds, Nonlinear Anal. 16, 79–87 (1991)
A. Eden, C. Foiaş, B. Nicolaenko, and R. Temam, Inertial sets for dissipative evolution equations, IMA Preprint, University of Minnesota, Minneapolis, 1990; cf. Ensembles inertiels pour des équations d’évolution dissipatives, C. R. Acad. Sci. Paris Sér. I Math. 310, 559–562 (1990)
M. Marion, Inertial manifolds associated to partly dissipative reaction-diffusion systems, J. Math. Anal. Appl. 143, 295–326 (1989)
R. Temam, Induced trajectories and approximate inertial manifolds, RAIRO Modél. Math. Anal. Numér. 23, 541–561 (1989)
J. Heywood and R. Rannacher, On the question of turbulence modelling by approximate inertial manifolds and the nonlinear Galerkin method, SIAM J. Numer. Anal. 30, 1603–1621 (1993)
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992
A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations and estimates of their dimension, Russian Math. Surveys 38, no. 4, 151–213 (1983)
A. V. Babin and M. I. Vishik, Attractors for the Navier-Stokes system and for parabolic equations and estimates of their dimension, J. Soviet Math. 28, 619–627 (1983)
O. A. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, J. Soviet Math. 3, 458–479 (1975)
C. Foiaş and R. Temam, The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory, in Directions in Partial Differential Equations, Academic Press, NY, 1987, pp. 55–73
A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl. 62, 441–491 (1983)
A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A 116, 221–243 (1990)
A. V. Babin, The attractor of a Navier-Stokes system in an unbounded channel-like domain, J. Dynamics and Differential Equations 4, 555–584 (1992)
P. Constantin, C. Foiaş, O. Manky, and R. Temam, Determining modes and fractal dimension of turbulent flows, J. Fluid Mech. 150, 427–440 (1985)
F. Abergel, Attractor for a Navier-Stokes flow in an unbounded domain, RAIRO Modél. Math. Anal. Numér. 23, 359–370 (1989)
F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations 83, 85–108 (1990)
J. M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures et Appl. 66, 273–319 (1987)
Ian Richards, On the gap between numbers which are the sum of two squares, Adv. Math. 46, 1–2 (1982)
A. E. Green and R. S. Rivlin, Simple force and stress multipoles, Arch. Rational Mech. Anal. 16, 325–353 (1964)
A. E. Green and R. S. Rivlin, Multipolar continuum mechanics, Arch. Rational Mech. Anal. 17, 113–147 (1964)
J. L. Bleustein and A. E. Green, Dipolar fluids, Internat. J. Engrg. Sci. 5, 323–340 (1967)
J. Nečas and M. Šilhavý, Multipolar viscous fluids, Quart. Appl. Math. 49, 247–265 (1991)
H. Bellout, F. Bloom, and J. Nečas, Phenomenological behavior of multipolar viscous fluids, Quart. Appl. Math. 50, 559–583 (1992)
M. Shinbrot, Lectures on Fluid Mechanics, Gordon and Breach, New York, 1993
D. Ruelle, The turbulent fluid as a dynamical system, in New Perspectives in Turbulence, L. Sirovich, ed., Springer-Verlag, Berlin, 1991, pp. 123–138
O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969
O. Ladyzhenskaya, New equations for the description of viscous incompressible fluids and solvability in the large of the boundary value problems for them, Proc. Steklov Inst. Math. 102, 95–118 (1970)
S. Kaniel, On the initial-value problem for an incompressible fluid with nonlinear viscosity, J. Math. Mech. 19, 681–706 (1970)
Q. Du and M. D. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Anal. Appl. 155, 21–45 (1991)
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969
Y.-R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations. I, Quart. Appl. Math. 49, 651–685 (1991)
Y.-R. Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations. II, Quart. Appl. Math. 49, 687–728 (1991)
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988
R. A. Toupin, Theories of elasticity with couple stress, Arch. Rational Mech. Anal. 17, 85–112 (1964)
H. Bellout, F. Bloom, and J. Nečas, Existence, uniqueness, and stability of solutions to the initial boundary value problem for bipolar viscous fluids, Differential Integral Equations 8, 453–464 (1995)
H. Bellout, F. Bloom, and J. Nečas, Young measure-valued solutions for non-Newtonian incompressible fluids, Comm. Partial Differential Equations 19, 1763–1803 (1994)
H. Bellout, F. Bloom, and J. Nečas, Solutions for incompressible non-Newtonian fluids, C. R. Acad. Sci. Paris Sér. I Math. 317, 795–800 (1993)
J. Nečas, Sur les normes équivalentes dans $W_p^{\left ( k \right )}\left ( \Omega \right )$ et sur la coercivité des formes formellement positives, Équations aux Dérivées Partielles, Presses Univ. Montréal, Montréal, 1965
H. Bellout and F. Bloom, Existence and asymptotic stability of time-dependent Poiseuille flows of isothermal bipolar fluids, Applicable Anal. 50, 115–130 (1993)
H. Bellout and F. Bloom, On the uniqueness of plane Poiseuille solutions of the equations of incompressible bipolar fluids, Internat. J. Engrg. Sci. 31, 1535–1549 (1993)
H. Bellout and F. Bloom, Steady plane Poiseuille flows of incompressible multipolar fluids, Internat. J. Non-Linear Mech. 23, 503–518 (1993)
F. Bloom and W. Hao, Steady flows of nonlinear bipolar viscous fluids between rotating cylinders, Quart. Appl. Math. 53, 143–171 (1995)
H. Bellout, F. Bloom, and J. Nečas, Bounds for the dimensions of the attractors of nonlinear bipolar viscous fluids, Asymptotic Anal. 11, 131–167 (1995)
F. Bloom, Attractors of bipolar and non-Newtonian viscous fluids, First World Congr. Nonlinear Analysts (Tampa, FL, August 1992), Vol. I, V. Lakshmikantham, ed., Walter de Gruyter Publishers, 1996, pp. 583–596
F. Bloom, Attractors of non-Newtonian fluids, J. Dynamics and Differential Equations 7, 109–140 (1995)
P. Constantin, C. Foiaş, and R. Temam, Attractors Representing Turbulent Flows, Mem. Amer. Math. Soc. No. 314 (1985)
P. Constantin, C. Foiaş, and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Phys. D. 30, 284–296 (1988)
J. Malek and J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, J. Differential Equations, in press
C. Foiaş and G. R. Sell, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73, 309–353 (1988)
P. Constantin, C. Foiaş, B. Nicolaenko, and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, New York, 1989
P. Constantin and C. Foiaş, Navier-Stoker Equations, University of Chicago Press, Chicago, 1988
M. W. Smiley, Global attractors and approximate inertial manifolds for nonautonomous dissipative equations, Applicable Anal. 50, 217–241 (1993)
G. R. Sell and Y. You, Inertial manifolds : the nonselfadjoint case, J. Differential Equations 96, 203–255 (1992)
M. Kwak, Finite-dimensional inertial forms for the 2D Navier-Stokes equations, Indiana Univ. Math. J. 41, 927–981 (1993)
G. R. Sell and J. Mallet-Paret, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Amer. Math. Soc. 1, 805–866 (1988)
M. Luskin and G. R. Sell, Parabolic regularization and inertial manifolds, IMA Preprint, University of Minnesota, Minneapolis, 1989
P. Constantin, C. Foiaş, B. Nicolaenko, and R. Temam, Spectral barriers and inertial manifolds for dissipative partial differential equations, J. Dynamics and Differential Equations 1, 45–73 (1989)
C. Foiaş, G. R. Sell, and E. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dynamics and Differential Equations 1, 199–244 (1989)
M. Marion, Approximate inertial manifolds for reaction-diffusion equations in higher space dimensions, J. Dynamics and Differential Equations 1, 245–267 (1989)
M. Luskin and G. R. Sell, The construction of inertial manifolds for reaction-diffusion equations by elliptic regularization, IMA Preprint, University of Minnesota, Minneapolis, 1989
E. Titi, On approximate inertial manifolds to the Navier-Stokes equations, J. Math. Anal. Appl. 149, 540–557 (1990)
P. Constantin, A construction of inertial manifolds, in The Connection Between Infinite-Dimensional and Finite-Dimensional Dynamical Systems (Boulder, CO, 1987), Contemp. Math., vol. 99, Amer. Math. Soc., Providence, RI, 1989, pp. 27–62
F. Demenge and J. M. Ghidaglia, Some remarks on the smoothness of inertial manifolds, Nonlinear Anal. 16, 79–87 (1991)
A. Eden, C. Foiaş, B. Nicolaenko, and R. Temam, Inertial sets for dissipative evolution equations, IMA Preprint, University of Minnesota, Minneapolis, 1990; cf. Ensembles inertiels pour des équations d’évolution dissipatives, C. R. Acad. Sci. Paris Sér. I Math. 310, 559–562 (1990)
M. Marion, Inertial manifolds associated to partly dissipative reaction-diffusion systems, J. Math. Anal. Appl. 143, 295–326 (1989)
R. Temam, Induced trajectories and approximate inertial manifolds, RAIRO Modél. Math. Anal. Numér. 23, 541–561 (1989)
J. Heywood and R. Rannacher, On the question of turbulence modelling by approximate inertial manifolds and the nonlinear Galerkin method, SIAM J. Numer. Anal. 30, 1603–1621 (1993)
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992
A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations and estimates of their dimension, Russian Math. Surveys 38, no. 4, 151–213 (1983)
A. V. Babin and M. I. Vishik, Attractors for the Navier-Stokes system and for parabolic equations and estimates of their dimension, J. Soviet Math. 28, 619–627 (1983)
O. A. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, J. Soviet Math. 3, 458–479 (1975)
C. Foiaş and R. Temam, The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory, in Directions in Partial Differential Equations, Academic Press, NY, 1987, pp. 55–73
A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl. 62, 441–491 (1983)
A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A 116, 221–243 (1990)
A. V. Babin, The attractor of a Navier-Stokes system in an unbounded channel-like domain, J. Dynamics and Differential Equations 4, 555–584 (1992)
P. Constantin, C. Foiaş, O. Manky, and R. Temam, Determining modes and fractal dimension of turbulent flows, J. Fluid Mech. 150, 427–440 (1985)
F. Abergel, Attractor for a Navier-Stokes flow in an unbounded domain, RAIRO Modél. Math. Anal. Numér. 23, 359–370 (1989)
F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations 83, 85–108 (1990)
J. M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures et Appl. 66, 273–319 (1987)
Ian Richards, On the gap between numbers which are the sum of two squares, Adv. Math. 46, 1–2 (1982)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
76A05,
35Q35,
76D99
Retrieve articles in all journals
with MSC:
76A05,
35Q35,
76D99
Additional Information
Article copyright:
© Copyright 1996
American Mathematical Society
