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The pullback attractor Read chapter Read first chapter

2013 | OriginalPaper | Chapter

11. The Navier–Stokes equations with non-autonomous forcing

Authors : Alexandre N. Carvalho, José A. Langa, James C. Robinson

Published in: Attractors for infinite-dimensional non-autonomous dynamical systems

Publisher: Springer New York

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Abstract

The two-dimensional incompressible Navier–Stokes equations provide one of the canonical examples of an infinite-dimensional dynamical system. In this chapter we illustrate the results of Chaps. 2 and 4 by driving the dynamics with a non-autonomous forcing term. With such an equation, which has no clear underlying structure (like a Lyapunov function, for example), the application of the more ‘global’ results of these two chapters (existence of a finite-dimensional pullback attractor) is essentially as far as one can currently proceed.

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previous chapter Delay differential equations
next chapter Applications to parabolic problems
Footnotes
1
If we keep this term and integrate from s to t then, using (11.7), we obtain the following additional estimate, which will be useful later:
$$\|w{(t)\|}^{2} + \nu {\int \nolimits \nolimits }_{s}^{t}\|{A}^{1/2}w{(r)\|}^{2}\,\mathrm{d}r \leq \| w{(s)\|}^{2}\left [1 + D(t,s){\mathrm{e}}^{D(t,s)}\right ],$$
where \(D(t,s) = \frac{c} {\nu }{\int \nolimits \nolimits }_{s}^{t}\|{A}^{1/2}u{(r)\|}^{2}\,\mathrm{d}r\).
 
2
If we were to take f( ⋅) uniformly bounded in D(A  − 1 ∕ 2), in line with the assumptions in the existence result of Theorem 11.1, then we would have to show the asymptotic compactness of S( ⋅,  ⋅) to obtain a pullback attractor; see Rosa (1998) or García-Luengo et al. (2012a,b).
 
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Metadata
Title
The Navier–Stokes equations with non-autonomous forcing
Authors
Alexandre N. Carvalho
José A. Langa
James C. Robinson
Copyright Year
2013
Publisher
Springer New York
Book
Attractors for infinite-dimensional non-autonomous dynamical systems

Print ISBN: 978-1-4614-4580-7

Electronic ISBN: 978-1-4614-4581-4

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-4581-4_11

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