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A note on the regularity criterion of the Boussinesq equations with zero heat conductivity

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Abstract

This note provides a new simple proof of a result obtained in 2009 by Fan and Ozawa on the regularity criterion for a 3D Boussinesq equations with zero heat conductivity.

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References

  1. Chae, D.: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203, 497–513 (2006)

    Article  MathSciNet  Google Scholar 

  2. Chae, D., Lee, J.: Local existence and blow-up criterion for the inhomogeneous Euler equations. J. Math. Fluid Mech. 5, 144–165 (2003)

    Article  MathSciNet  Google Scholar 

  3. Cannon, J.R., Dibenedetto, E.: The Initial Problem for the Boussinesq Equation with Data in \(L^{p}\). Lecture Notes in Mathematics, vol. 771, pp. 129–144. Springer, Berlin (1980)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. 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 

  8. Geng, J., Fan, J.: A note on regularity criterion for the 3D Boussinesq system with zero thermal conductivity. Appl. Math. Lett. 25, 63–66 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  10. Jia, Y., Zhang, X., Dong, B.: Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Commun. Pure Appl. Anal. 12, 923–937 (2013)

    Article  MathSciNet  Google Scholar 

  11. Koch, H., Tataru, D.: Well-posedness for the Navier–Stokes equations. Adv. Math. 157, 22–35 (2001)

    Article  MathSciNet  Google Scholar 

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

  13. Ogawa, T., Shimizu, S.: End-point maximal regularity and its application to two-dimensional Keller-Segel system. Math. Z. 264, 601–628 (2010)

    Article  MathSciNet  Google Scholar 

  14. Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Walter de Gruyter, Berlin (1996)

    Book  Google Scholar 

  15. Stein, E.M.: Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    Google Scholar 

  16. Strichartz, R.S.: Boundard mean oscillations and Sobolev spaces. Indiana Univ. Math. J. 29, 539–558 (1980)

    Article  MathSciNet  Google Scholar 

  17. Triebel, H.: Interpolation theory, function spaces, differential operators. North-Holland Mathematical Library 18, North-Holland, Amsterdam- New York (1978)

  18. Youssfi, A.: Regularity properties of commutators and BMO-Triebel-Lizorkin spaces. Ann. Inst. Fourier (Grenoble) 45, 795–807 (1995)

    Article  MathSciNet  Google Scholar 

  19. Zhou, Y., Fan, J.: On the Cauchy problems for certain Boussinesq-\(\alpha \) equations. Proc. R. Soc. Edinb. Sect. A 140(2), 319–327 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank the referee for his careful reading of the work and his many helpful suggestions.

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Correspondence to S. Gala.

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Gala, S. A note on the regularity criterion of the Boussinesq equations with zero heat conductivity. Ricerche mat (2024). https://doi.org/10.1007/s11587-024-00860-x

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