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Well-posedness of the Kolmogorov two-equation model of turbulence in optimal Sobolev spaces

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Abstract

In this paper, we study the well-posedness of the Kolmogorov two-equation model of turbulence in a periodic domain \(\mathbb {T}^d\), for space dimensions \(d=2,3\). We admit the average turbulent kinetic energy k to vanish in part of the domain, i.e. we consider the case \(k \ge 0\); in this situation, the parabolic structure of the equations becomes degenerate. For this system, we prove a local well-posedness result in Sobolev spaces \(H^s\), for any \(s>1+d/2\). We expect this regularity to be optimal, due to the degeneracy of the system when \(k \approx 0\). We also prove a continuation criterion and provide a lower bound for the lifespan of the solutions. The proof of the results is based on Littlewood-Paley analysis and paradifferential calculus on the torus, together with a precise commutator decomposition of the nonlinear terms involved in the computations.

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Notes

  1. Throughout we agree that f(D) stands for the pseudo-differential operator \(u\mapsto \mathcal {F}^{-1}(f\,\mathcal {F}u)\), where \(\mathcal F^{-1}\) is the inverse Fourier transform.

  2. Throughout this text, we note by [s] the integer part of a real number \(s\in \mathbb {R}\), namely the biggest integer which is lower than, or equal to, s.

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Acknowledgements

The work of the second author has been partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissement d’Avenir” (ANR-11-IDEX-0007), and by the projects SingFlows (ANR-18-CE40-0027) and CRISIS (ANR-20-CE40-0020-01), all operated by the French National Research Agency (ANR). The work of the third author has been partially supported by the grant PRE2018-083984, funded by MCIN/AEI/ 10.13039/501100011033, by the ERC through the Starting Grant H2020-EU.1.1.-639227, by MICINN through the grants EUR2020-112271 and PID2020- 114703GB-I00 and by Junta de Andalucía through the grant P20-00566.

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Author notes

  1. Ophélie Cuvillier

    Present address: Lycée Joffre, 150, Allée de la Citadelle, 34060, Montpellier cedex, France

Authors and Affiliations

  1. Université de Lyon, Université Claude Bernard Lyon 1, Institut Camille Jordan – UMR 5208, 43 blvd. du 11 novembre 1918, 69622, Villeurbanne cedex, France

    Ophélie Cuvillier & Francesco Fanelli

  2. ICMAT & Universidad de Sevilla, Avda. Reina Mercedes s/n, CITIUS II, 41012, Sevilla, Spain

    Elena Salguero

  3. Max Planck Institute for Mathematics in the Sciences, Inselstrae 22, 04103, Leipzig, Germany

    Elena Salguero

  4. Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Spain

    Francesco Fanelli

  5. Ikerbasque, Basque Foundation for Science, Plaza Euskadi 5, 48009, Bilbao, Spain

    Francesco Fanelli

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  1. Ophélie Cuvillier
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  2. Francesco Fanelli
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Correspondence to Elena Salguero.

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Cuvillier, O., Fanelli, F. & Salguero, E. Well-posedness of the Kolmogorov two-equation model of turbulence in optimal Sobolev spaces. J. Evol. Equ. 23, 68 (2023). https://doi.org/10.1007/s00028-023-00914-x

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