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Aspects of PDEs Related to Fluid Flows

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Vector-Valued Partial Differential Equations and Applications

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2179))

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Abstract

In the first part of the notes we discuss the problem of long-time behavior of some infinite-dimensional Hamiltonian system from the point of view of statistical mechanics. In the second part we discuss the issue of well-posedness of the Leray-Hopf weak solutions in the energy space in the context of recent developments concerning scale-invariant solutions.

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Notes

  1. 1.

    We should exclude the completely integrable systems, for example.

  2. 2.

    This may not be the case for infinite-energy solutions which may have enough energy to fill all the available phase-space, even though it is infinite-dimensional.

  3. 3.

    In the context of this example these were drawn to the author’s attention by Jalal Shatah.

  4. 4.

    We avoid the more logical but unwieldy notation \(\mathcal{O}_{\omega _{ 0}^{(N)}}^{(N)}\).

  5. 5.

    Consider for example n = 2, m = 1 and f(x) = x 1 x 2. The reader can check that the measure δ( f(x)) is well-defined in R 2 ∖{0}, but not in R 2 .

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Acknowledgements

The author thanks Sergei Kuksin, Geordie Richards and Ofer Zeitouni for very helpful discussions. The research of was supported in part by grants DMS 1159376 and DMS 1362467 from the National Science Foundation.

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  1. School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN, 5545, USA

    Vladimír Šverák

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  1. Mathematical Institute, University of Oxford , Oxford, United Kingdom

    John Ball

  2. Dipartimento di Matematica, Università di Firenze, Firenze, Italy

    Paolo Marcellini

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Šverák, V. (2017). Aspects of PDEs Related to Fluid Flows. In: Ball, J., Marcellini, P. (eds) Vector-Valued Partial Differential Equations and Applications. Lecture Notes in Mathematics(), vol 2179. Springer, Cham. https://doi.org/10.1007/978-3-319-54514-1_4

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