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Small Scale Creation in Active Scalars

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Progress in Mathematical Fluid Dynamics

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2272))

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Abstract

The focus of the course is on small scale formation in solutions of the incompressible Euler equation of fluid dynamics and associated models. We first review the regularity results and examples of small scale growth in two dimensions. Then we discuss a specific singular scenario for the three-dimensional Euler equation discovered by Hou and Luo, and analyze some associated models. Finally, we will also talk about the surface quasi-geostrophic (SQG) equation, and construct an example of singularity formation in the modified SQG patch solutions as well as examples of unbounded growth of derivatives for the smooth solutions.

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Acknowledgements

The author acknowledges partial support of the NSF-DMS grant 1848790 and 2006372, and thanks the organizers of the 2019 CIME summer school on fluid mechanics for the kind invitation to give lectures and excellent organization.

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Authors and Affiliations

  1. Department of Mathematics, Duke University, Durham, NC, USA

    Alexander A. Kiselev

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  1. Alexander A. Kiselev
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Correspondence to Alexander A. Kiselev .

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  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Luigi C. Berselli

  2. Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Freiburg, Germany

    Michael Růžička

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Kiselev, A.A. (2020). Small Scale Creation in Active Scalars. In: Berselli, L.C., Růžička, M. (eds) Progress in Mathematical Fluid Dynamics. Lecture Notes in Mathematics(), vol 2272. Springer, Cham. https://doi.org/10.1007/978-3-030-54899-5_4

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