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Dynamical Systems and the Two-Dimensional Navier-Stokes Equations

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Dynamics of Partial Differential Equations

Part of the book series: Frontiers in Applied Dynamical Systems: Reviews and Tutorials ((FIADS,volume 3))

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Abstract

The focus of this chapter is on the application of dynamical systems ideas to the study of dissipative partial differential equations with a particular focus on the two-dimensional Navier-Stokes equations. The notion of dissipativity arises in physics where it is generally thought of as a dissipation of some “energy” associated with the system and such systems are contrasted with energy conserving systems like Hamiltonian systems.

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Notes

  1. 1.

    The situation is very different if one considers the equation on a bounded domain with periodic boundary conditions - see the discussion of the work of Foias and Saut [FS84a] in the following section.

  2. 2.

    The link between the attractor dimension and the Lyapunov exponents was first proposed by Kaplan and Yorke, [KY79], and is often known as the Kaplan-Yorke formula.

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Acknowledgements

The support of the author’s research by the National Science Foundation grants, DMS-0908093 and DMS-1311553 is gratefully acknowledged. For those parts of this survey which describe my own research it is a pleasure to thank my collaborators - Thierry Gallay and Margaret Beck for the theoretical results in Sections 2 and 3, and Alethea Barbaro, Ray Nagem, Guido Sandri, and David Uminsky for the numerical methods described in Section 2.

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  1. Department of Mathematics and Statistics, Boston University, Cummington Hall 111, Boston, MA, 02215, USA

    C. Eugene Wayne

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Wayne, C.E. (2015). Dynamical Systems and the Two-Dimensional Navier-Stokes Equations. In: Dynamics of Partial Differential Equations. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-19935-1_1

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