Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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THE 3D BOUSSINESQ EQUATIONS WITH REGULARITY IN THE HORIZONTAL COMPONENT OF THE VELOCITY

  • Liu, Qiao (Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Statistics Hunan Normal University)
  • Received : 2019.04.16
  • Accepted : 2019.11.06
  • Published : 2020.05.31
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Abstract

This paper proves a new regularity criterion for solutions to the Cauchy problem of the 3D Boussinesq equations via one directional derivative of the horizontal component of the velocity field (i.e., (∂iu1; ∂ju2; 0) where i, j ∈ {1, 2, 3}) in the framework of the anisotropic Lebesgue spaces. More precisely, for 0 < T < ∞, if $$\large{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_o}^T}({\HUGE\left\|{\small{\parallel}{\partial}_iu_1(t){\parallel}_{L^{\alpha}_{x_i}}}\right\|}{\small^{\gamma}_{L^{\beta}_{x_{\hat{i}}x_{\bar{i}}}}+}{\HUGE\left\|{\small{\parallel}{\partial}_iu_2(t){\parallel}_{L^{\alpha}_{x_j}}}\right\|}{\small^{\gamma}_{L^{\beta}_{x_{\hat{i}}x_{\bar{i}}}}})dt<{{\infty}},$$ where ${\frac{2}{{\gamma}}}+{\frac{1}{{\alpha}}}+{\frac{2}{{\beta}}}=m{\in}[1,{\frac{3}{2}})$ and ${\frac{3}{m}}{\leq}{\alpha}{\leq}{\beta}<{\frac{1}{m-1}}$, then the corresponding solution (u, θ) to the 3D Boussinesq equations is regular on [0, T]. Here, (i, ${\hat{i}}$, ${\tilde{i}}$) and (j, ${\hat{j}}$, ${\tilde{j}}$) belong to the permutation group on the set 𝕊3 := {1, 2, 3}. This result reveals that the horizontal component of the velocity field plays a dominant role in regularity theory of the Boussinesq equations.

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References

  1. H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system, J. Differential Equations 233 (2007), no. 1, 199-220. https://doi.org/10.1016/j.jde.2006.10.008
  2. H.-O. Bae and H. J. Choe, $L^{\infty}$-bound of weak solutions to Navier-Stokes equations, in Proceedings of the Korea-Japan Partial Differential Equations Conference (Taejon, 1996), 13 pp, Lecture Notes Ser., 39, Seoul Nat. Univ., Seoul, 1997.
  3. H.-O. Bae and H. J. Choe, A regularity criterion for the Navier-Stokes equations, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1173-1187. https://doi.org/10.1080/03605300701257500
  4. H. Beirao da Veiga, A new regularity class for the Navier-Stokes equations in ${\mathbf{R}}^n$, Chinese Ann. Math. Ser. B 16 (1995), no. 4, 407-412.
  5. C. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal. 202 (2011), no. 3, 919-932. https://doi.org/10.1007/s00205-011-0439-6
  6. C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations 248 (2010), no. 9, 2263-2274. https://doi.org/10.1016/j.jde.2009.09.020
  7. D. Chae and H.-J. Choe, Regularity of solutions to the Navier-Stokes equation, Electron. J. Differential Equations 1999 (1999), No. 05, 7 pp.
  8. L. Iskauriaza, G. A. Seregin, and V. Shverak, $L^{3,{\infty}}$ solutions to the Navier-Stokes equations and backward uniqueness, Russian Math. Surveys 58 (2003), no. 2, 211-250; translated from Uspekhi Mat. Nauk 58 (2003), no. 2(350), 3-44. https://doi.org/10.1070/RM2003v058n02ABEH000609
  9. J. Fan, S. Jiang, G. Nakamura, and Y. Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech. 13 (2011), no. 4, 557-571. https://doi.org/10.1007/s00021-010-0039-5
  10. S. Gala, On the regularity criterion of strong solutions to the 3D Boussinesq equations, Appl. Anal. 90 (2011), no. 12, 1829-1835. https://doi.org/10.1080/00036811.2010.530261
  11. S. Gala and M. A. Ragusa, Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices, Appl. Anal. 95 (2016), no. 6, 1271-1279. https://doi.org/10.1080/00036811.2015.1061122
  12. S. Gala and M. A. Ragusa, A logarithmic regularity criterion for the two-dimensional MHD equations, J. Math. Anal. Appl. 444 (2016), no. 2, 1752-1758. https://doi.org/10.1016/j.jmaa.2016.07.001
  13. J. Geng and J. Fan, A note on regularity criterion for the 3D Boussinesq system with zero thermal conductivity, Appl. Math. Lett. 25 (2012), no. 1, 63-66. https://doi.org/10.1016/j.aml.2011.07.008
  14. Y. Jia, X. Zhang, and B.-Q. Dong, Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion, Commun. Pure Appl. Anal. 12 (2013), no. 2, 923-937. https://doi.org/10.3934/cpaa.2013.12.923
  15. A. Majda, Introduction to PDEs and waves for the atmosphere and ocean, Courant Lecture Notes in Mathematics, 9, New York University, Courant Institute of Mathematical Sciences, New York, 2003. https://doi.org/10.1090/cln/009
  16. J. Neustupa, A. Novotny, and P. Penel, An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity, in Topics in mathematical fluid mechanics, 163-183, Quad. Mat., 10, Dept. Math., Seconda Univ. Napoli, Caserta, 2002.
  17. P. Penel and M. Pokorny, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math. 49 (2004), no. 5, 483-493. https://doi.org/10.1023/B:APOM.0000048124.64244.7e
  18. G. Prodi, Un teorema di unicita per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl. (4) 48 (1959), 173-182. https://doi.org/10.1007/BF02410664
  19. C. Qian, A generalized regularity criterion for 3D Navier-Stokes equations in terms of one velocity component, J. Differential Equations 260 (2016), no. 4, 3477-3494. https://doi.org/10.1016/j.jde.2015.10.037
  20. H. Qiu, Y. Du, and Z. Yao, Serrin-type blow-up criteria for 3D Boussinesq equations, Appl. Anal. 89 (2010), no. 10, 1603-1613. https://doi.org/10.1080/00036811.2010.492505
  21. H. Qiu, Y. Du, and Z. Yao, Blow-up criteria for 3D Boussinesq equations in the multiplier space, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 4, 1820-1824. https://doi.org/10.1016/j.cnsns.2010.08.036
  22. J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187-195. https://doi.org/10.1007/BF00253344
  23. F. Xu, Q. Zhang, and X. Zheng, Regularity criteria of the 3D Boussinesq equations in the Morrey-Campanato space, Acta Appl. Math. 121 (2012), 231-240. https://doi.org/10.1007/s10440-012-9705-3
  24. Z. Zhang, A logarithmically improved regularity criterion for the 3D Boussinesq equations via the pressure, Acta Appl. Math. 131 (2014), 213-219. https://doi.org/10.1007/s10440-013-9855-y
  25. Z. Zhang, Global regularity criteria for the n-dimensional Boussinesq equations with fractional dissipation, Electron. J. Differential Equations 2016 (2016), Paper No. 99, 5 pp.
  26. Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in ${\mathbb{R}}^3$, Proc. Amer. Math. Soc. 134 (2006), no. 1, 149-156. https://doi.org/10.1090/S0002-9939-05-08312-7
  27. Y. Zhou and M. Pokorny, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity 23 (2010), no. 5, 1097-1107. https://doi.org/10.1088/0951-7715/23/5/004