Book contents
- Frontmatter
- Contents
- Preface
- 1 The equations of motion
- 2 Dimensionless parameters and stability
- 3 Turbulence
- 4 Degrees of freedom, dynamical systems, and attractors
- 5 On the existence, uniqueness, and regularity of solutions
- 6 Ladder results for the Navier-Stokes equations
- 7 Regularity and length scales for the 2d and 3d Navier-Stokes equations
- 8 Exponential decay of the Fourier power spectrum
- 9 The attractor dimension for the Navier-Stokes equations
- 10 Energy dissipation rate estimates for boundary-driven flows
- Appendix A Inequalities
- References
- Index
9 - The attractor dimension for the Navier-Stokes equations
Published online by Cambridge University Press: 02 February 2010
- Frontmatter
- Contents
- Preface
- 1 The equations of motion
- 2 Dimensionless parameters and stability
- 3 Turbulence
- 4 Degrees of freedom, dynamical systems, and attractors
- 5 On the existence, uniqueness, and regularity of solutions
- 6 Ladder results for the Navier-Stokes equations
- 7 Regularity and length scales for the 2d and 3d Navier-Stokes equations
- 8 Exponential decay of the Fourier power spectrum
- 9 The attractor dimension for the Navier-Stokes equations
- 10 Energy dissipation rate estimates for boundary-driven flows
- Appendix A Inequalities
- References
- Index
Summary
Introduction
In this chapter we show how the dimension of the global attractor ℕ can be estimated for the Navier-Stokes equations. The approach is an extension of that developed in Chapter 4 for ordinary differential equations where it was shown that if N-dimensional volume elements in the system phase space contract to zero, then the attractor dimension dL(ℕ) must be bounded by N. For partial differential equations the technical chore remains the same; namely, to derive estimates on the spectrum of the linearized evolution operator, linearized around solutions on the attractor, and to perform this operation in some function space instead of an a priori finite dimensional phase space. As we saw in Chapter 4 in the context of the Lorenz equations, this requires some knowledge of the location of the attractor, i.e., a priori estimates on the solutions. This approach is pursued in section 9.2 which deals with the 2d Navier-Stokes equations. It was shown in Chapter 7 that a global attractor si exists in this case, and we have good control of the solutions on the attractor. It turns out that the result for periodic boundary conditions is quite sharp, within logarithms of both the conventional heuristic estimate for the number of degrees of freedom in a 2d turbulent flow and rigorous lower bounds.
The 3d Navier-Stokes equations on a periodic domain are the concern of section 9.3. The lack of a regularity proof for this case results in some uncertainty concerning the very existence of a compact attractor. To achieve any formal estimate of the attractor dimension it is necessary to assume that H1 remains bounded for all t.
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- Applied Analysis of the Navier-Stokes Equations , pp. 169 - 180Publisher: Cambridge University PressPrint publication year: 1995