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A Regularity Criterion of Weak Solutions to the 3D Boussinesq Equations

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Abstract

The paper deals with the regularity criteria for the weak solutions to the 3D Boussinesq equations in terms of the partial derivatives in Besov spaces. It is proved that the weak solution \((u,\theta )\) becomes regular provided that

$$\begin{aligned} (\nabla _{h}{\widetilde{u}},\nabla _{h}\theta )\in L^{1}(0,T;\overset{\cdot }{B }_{\infty ,\infty }^{0}({\mathbb {R}}^{3})) \end{aligned}$$

Our results improve and extend the well-known result of Dong and Zhang (Nonlinear Anal 11:2415–2421, 2010) for the Navier–Stokes equations.

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References

  • Abidi, H., Hmidi, T.: On the global well-posedness for the Boussinesq system. J. Differ. Equ. 233, 199–220 (2007)

    Article  MathSciNet  Google Scholar 

  • Alghamdi, A.M., Gala, S., Ragusa, M.A.: A regularity criterion of weak solutions to the 3D Boussinesq equations. AIMS Math. 2, 451–457 (2017)

    Article  Google Scholar 

  • Berselli, L.C.: On a regularity criterion for the solutions to 3D Navier–Stokes equations. Differ. Integral Equ. 15, 1129–1137 (2002)

    MathSciNet  MATH  Google Scholar 

  • Brandolese, L., Schonbek, M.E.: Large time decay and growth for solutions of a viscous Boussinesq system. Trans. Am. Math. Soc. 364, 5057–5090 (2012)

    Article  MathSciNet  Google Scholar 

  • Cannon, J.R., Dibenedetto, E.: The initial problem for the Boussinesq equations with data in \(L^{p}\). In: Rautmann, R. (ed.) Approximation Methods for Navier-Stokes Problems. Lecture Notes in Mathematics, vol. 771, pp. 129–144. Springer, Berlin (1980)

    Chapter  Google Scholar 

  • Cao, C., Wu, J.: Two regularity criteria for the 3D MHD equations. J. Differ. Equ. 248, 2263–2274 (2010)

    Article  MathSciNet  Google Scholar 

  • Chae, D., Nam, H.S.: Local existence and blow-up criterion for the Boussinesq equations. Proc. R. Soc. Edinb. Sect. A 127, 935–946 (1997)

    Article  MathSciNet  Google Scholar 

  • Chae, D., Kim, S.-K., Nam, H.-S.: Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations. Nagoya Math. J. 155, 55–80 (1999)

    Article  MathSciNet  Google Scholar 

  • Chen, W., Gala, S.: A regularity criterion for the Navier–Stokes equations in terms of the horizontal derivatives of two velocity components. Electron. J. Differ. Equ. 2011(06), 1–7 (2011)

    MathSciNet  MATH  Google Scholar 

  • Dong, B.Q., Zhang, Z.: The BKM criterion for the 3D Navier–Stokes equations via two velocity components. Nonlinear Anal. RWA 11, 2415–2421 (2010)

    Article  MathSciNet  Google Scholar 

  • Dong, B., Gala, S., Chen, Z.: On the regularity criteria of the 3D Navier–Stokes equations in critical spaces. Acta Math. Sci. Ser. B 31, 591–600 (2011)

    Article  MathSciNet  Google Scholar 

  • Dong, B., Jia, Y., Zhang, X.: Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Commun. Pure Appl. Anal. 12, 923–937 (2012)

    Article  MathSciNet  Google Scholar 

  • Fan, J., Ozawa, T.: Regularity criterion for 3D density-dependent Boussinesq equations. Nonlinearity 22, 553–568 (2009)

    Article  MathSciNet  Google Scholar 

  • Fan, J., Zhou, Y.: A note on regularity criterion for the 3D Boussinesq systems with partial viscosity. Appl. Math. Lett. 22, 802–805 (2009)

    Article  MathSciNet  Google Scholar 

  • Gala, S.: A remark on the regularity for the 3D Navier–Stokes equations in terms of the two components of the velocity. Electron. J. Differ. Equ. 2009(148), 1–6 (2009)

    MathSciNet  MATH  Google Scholar 

  • Gala, S.: On the regularity criterion of strong solutions to the 3D Boussinesq equations. Appl. Anal. 90, 1829–1835 (2011)

    Article  MathSciNet  Google Scholar 

  • Gala, S., Ragusa, M.A.: A new regularity criterion for the Navier–Stokes equations in terms of two components of the velocity. Electron. J. Qual. Theory Differ. Equ. 26, 1–9 (2016)

    Article  MathSciNet  Google Scholar 

  • Gala, S., Ragusa, M.A.: Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices. Appl. Anal. 95, 1271–1279 (2016)

    Article  MathSciNet  Google Scholar 

  • Gala, S., Guo, Z., Ragusa, M.A.: A remark on the regularity criterion of Boussinesq equations with zero heat conductivity. Appl. Math. Lett. 27, 70–73 (2014)

    Article  MathSciNet  Google Scholar 

  • Gala, S., Mechdene, M., Ragusa, M.A.: Logarithmically improved regularity criteria for the Boussinesq equations. AIMS Math. 2, 336–347 (2017)

    Article  Google Scholar 

  • Guo, Z., Gala, S.: Remarks on logarithmical regularity criteria for the Navier–Stokes equations. J. Math. Phys. 52, 063503 (2011)

    Article  MathSciNet  Google Scholar 

  • Guo, Z., Gala, S.: Regularity criterion of the Newton–Boussinesq equations in R3. Commun. Pure Appl. Anal. 11, 443–451 (2012)

    Article  MathSciNet  Google Scholar 

  • Hou, T.Y., Li, C.: Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12, 1–12 (2005)

    Article  MathSciNet  Google Scholar 

  • Ishimura, N., Morimoto, H.: Remarks on the blow-up criterion for the 3-D Boussinesq equations. Math. Model. Methods Appl. Sci. 9, 1323–1332 (1999)

    Article  MathSciNet  Google Scholar 

  • Kukavica, I., Zinae, M.: Navier–Stokes equation with regularity in one direction. J. Math. Phys. 48, 065203 (2007). 10 pp

    Article  MathSciNet  Google Scholar 

  • Majda, A.: Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, no. 9, AMS/CIMS (2003)

  • Mechdene, M., Gala, S., Guo, Z., Ragusa, A.M.: Logarithmical regularity criterion of the three-dimensional Boussinesq equations in terms of the pressure. Z. Angew. Math. Phys. 67, 1–10 (2016)

    Article  MathSciNet  Google Scholar 

  • Meyer, Y., Gerard, P., Oru, F.: Inégalités de Sobolev précisées; in Séminaire sur les Équations aux Dérivé es Partielles, 1996–1997, Exp. IV, 11 pp, École Polytech., Palaiseau

  • Pedlosky, J.: Geophysical Fluid Dynsmics. Springer, New York (1987)

    Book  Google Scholar 

  • Skalák, Z.: Criteria for the regularity of the solutions to the Navier–Stokes equations based on the velocity gradient. Nonlinear Anal. 118, 1–21 (2015)

    Article  MathSciNet  Google Scholar 

  • Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)

    Book  Google Scholar 

  • Zhou, Y.: A new regularity criteria for weak solutions to the Navier–Stokes equations. J. Math. Pures Appl. 84, 1496–1514 (2005)

    Article  MathSciNet  Google Scholar 

  • Zhou, Y., Pokorny, M.: On the regularity of the solutions of the Navier–Stokes equations via one velocity component. Nonlinearity 23, 1097–1107 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was done while Sadek Gala was visiting the Catania University in Italy. He would like to thank the hospitality and support of the University, where this work was completed. This research is partially supported by Piano della Ricerca 2016-2018 - Linea di intervento 2: “Metodi variazionali ed equazioni differenziali”. Maria Alessandra Ragusa wish to thank the support of “RUDN University Program 5-100”. The authors wish to express their thanks to the referee for his/her very careful reading of the paper, giving valuable comments and helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, ENS of Mostaganem, Box 227, 27000, Mostaganem, Algeria

    Sadek Gala

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

    Sadek Gala & Maria Alessandra Ragusa

  3. RUDN University, 6 Miklukho-Maklay St, Moscow, 117198, Russia

    Maria Alessandra Ragusa

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  1. Sadek Gala
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  2. Maria Alessandra Ragusa
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Correspondence to Sadek Gala.

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Gala, S., Ragusa, M.A. A Regularity Criterion of Weak Solutions to the 3D Boussinesq Equations. Bull Braz Math Soc, New Series 51, 513–525 (2020). https://doi.org/10.1007/s00574-019-00162-z

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  • DOI: https://doi.org/10.1007/s00574-019-00162-z

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