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On the Regularity of Weak Solutions of the Boussinesq Equations in Besov Spaces

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Abstract

The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov space \(\dot {B}_{\infty ,\infty }^{-1}(\mathbb {R}^{3})\), that, if the solution of the Boussinesq equation (1) below (starting with an initial data in H2) is such that \((\nabla u,\nabla \theta )\in L^{2}(0,T;\dot {B}_{\infty ,\infty }^{-1}(\mathbb {R}^{3}))\), then the solution remains smooth forever after T. In this contribution, we prove the same result for weak solutions just by assuming the condition on the velocity u and not on the temperature 𝜃.

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Acknowledgements

Research of A. Barbagallo is partially supported by PRIN 2017 Nonlinear Differential Problems via Variational, Topological and Set-valued Methods (Grant 2017AYM8XW).

Part of the work was carried out while S. Gala was a long-term visitor at the University of Catania. The hospitality of Catania University is graciously acknowledged.

Research of M.A. Ragusa is partially supported by the Ministry of Education and Science of the Russian Federation (5-100 program of the Russian Ministry of Education).

Research of M. Théra is supported by the Australian Research Council (ARC) grant DP160100854 and benefited from the support of the FMJH Program PGMO and from the support of EDF.

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Authors and Affiliations

  1. Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, via Cintia, 80126, Naples, Italy

    Annamaria Barbagallo

  2. Department of Sciences exactes, Ecole Normale Supérieure de Mostaganem, Box 227, Mostaganem, 27000, Algeria

    Sadek Gala

  3. Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6 95125, Catania, Italy

    Maria Alessandra Ragusa

  4. RUDN University, 6 Miklukho, Maklay St., Moscow, 117198, Russia

    Maria Alessandra Ragusa

  5. XLIM UMR-CNRS 7252 Université de Limoges, Limoges, France

    Michel Théra

  6. Centre for Informatics and Applied Optimisation, Federation University Australia, Ballarat, Australia

    Michel Théra

Authors
  1. Annamaria Barbagallo
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  2. Sadek Gala
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  3. Maria Alessandra Ragusa
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  4. Michel Théra
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Correspondence to Michel Théra.

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Barbagallo, A., Gala, S., Ragusa, M.A. et al. On the Regularity of Weak Solutions of the Boussinesq Equations in Besov Spaces. Vietnam J. Math. 49, 637–649 (2021). https://doi.org/10.1007/s10013-020-00420-4

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  • DOI: https://doi.org/10.1007/s10013-020-00420-4

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