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Weak Solutions to the Navier–Stokes Equations with Data in \(\mathbb {L}(3,\infty )\)

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Mathematics for Nonlinear Phenomena — Analysis and Computation (MNP2015 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 215))

Abstract

The paper concerns the existence of weak solutions to the 3d-Navier–Stokes initial boundary value problem in exterior domains. The problem is considered with an initial data belonging to \(\mathbb L(3,\infty )\) which is a special subspace of the Lorentz’s space \(L(3,\infty )\). The nature of the domain and the initial data in \(L(3,\infty )\) make the result of existence not comparable with the usual Leray-Hopf theory of weak solutions. However, we are able to prove both that the weak solutions enjoy the partial regularity in the sense of Leray’s structure theorem and the asymptotic limit of \(|u(t)|_{3\infty }\).

Dedicated to Professor Y. Giga on his 60th birthday

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Notes

  1. 1.

    The results of this note were communicated in some meetings: http://www.math.sci.hokudai.ac.jp/sympo/sapporo/program150819_en.html, http://perso-math.univ-mlv.fr/users/danchin.raphael/Porq15/abstract15.pdf, https://www.maths.ox.ac.uk/events/past/2895.

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Acknowledgements

- This research was partly supported by GNFM-INdAM, and by MIUR via the PRIN 2012 “Nonlinear Hyperbolic Partial Differential Equations, Dispersive and Transport Equations: Theoretical and Applicative Aspects”.

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Seconda Università degli Studi di Napoli, via Vivaldi 43, 81100, Caserta, Italy

    P. Maremonti

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  1. P. Maremonti
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Correspondence to P. Maremonti .

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Editors and Affiliations

  1. Mathematical Institute, Tohoku University, Sendai, Miyagi, Japan

    Yasunori Maekawa

  2. Department of Mathematics, Hokkaido University, Sapporo, Hokkaido, Japan

    Shuichi Jimbo

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Maremonti, P. (2017). Weak Solutions to the Navier–Stokes Equations with Data in \(\mathbb {L}(3,\infty )\) . In: Maekawa, Y., Jimbo, S. (eds) Mathematics for Nonlinear Phenomena — Analysis and Computation. MNP2015 2015. Springer Proceedings in Mathematics & Statistics, vol 215. Springer, Cham. https://doi.org/10.1007/978-3-319-66764-5_8

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