Research article

A regularity criterion of weak solutions to the 3D Boussinesq equations

  • Received: 02 June 2017 Accepted: 08 August 2017 Published: 25 August 2017
  • In this note, a regularity criterion of weak solutions to the 3D-Boussinesq equations with respect to Serrin type condition under the framework of Besov space $\dot B_{\infty, \infty.}^r$. It is shown that the weak solution $(u, \theta)$ is regular on $% (0, T] $ if $u$ satisfies $ \int\limits_0^T {\left\| {u( \cdot ,t)} \right\|_{\dot B_{\infty ,\infty .}^r}^{\frac{2}{{1 + r}}}} dt \lt \infty , $ for $ 0 \lt r \lt 1 $. This result improves some previous works.

    Citation: Ahmad Mohammed Alghamdi, Sadek Gala, Maria Alessandra Ragusa. A regularity criterion of weak solutions to the 3D Boussinesq equations[J]. AIMS Mathematics, 2017, 2(3): 451-457. doi: 10.3934/Math.2017.2.451

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  • In this note, a regularity criterion of weak solutions to the 3D-Boussinesq equations with respect to Serrin type condition under the framework of Besov space $\dot B_{\infty, \infty.}^r$. It is shown that the weak solution $(u, \theta)$ is regular on $% (0, T] $ if $u$ satisfies $ \int\limits_0^T {\left\| {u( \cdot ,t)} \right\|_{\dot B_{\infty ,\infty .}^r}^{\frac{2}{{1 + r}}}} dt \lt \infty , $ for $ 0 \lt r \lt 1 $. This result improves some previous works.


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    [1] J. R. Cannon and E. Dibenedetto, The initial problem for the Boussinesq equation with data in Lp, Lecture Notes in Mathematics, Springer, Berlin, 771 (1980), 129-144.
    [2] L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the navierstokesequations Comm. Pure Appl. Math., 35 (1982), 771-831.
    [3] D. Chae and H.-S. Nam, Local existence and blow-up criterion for the Boussinesq equations, Proc.Roy. Soc. Edinburgh, Sect. A, 127 (1997), 935-946.
    [4] B. Q. Dong, J. Song, and W. Zhang, Blow-up criterion via pressure of three-dimensional Boussinesqequations with partial viscosity (in Chinese), Sci. Sin. Math., 40 (2010), 1225-1236.
    [5] J. Fan and Y. Zhou, A note on regularity criterion for the 3D Boussinesq system with partial viscosity, Appl. Math. Lett., 22 (2009), 802-805.
    [6] J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity, 22 (2009), 553-568.
    [7] S. Gala, On the regularity criterion of strong solutions to the 3D Boussinesq equations, ApplicableAnalysis, 90 (2011), 1829-1835.
    [8] S. Gala and M.A. Ragusa, Logarithmically improved regularity criterion for the Boussinesq equationsin Besov spaces with negative indices, Applicable Analysis, 95 (2016), 1271-1279.
    [9] S. Gala, Z. Guo, and M. A. Ragusa, A remark on the regularity criterion of Boussinesq equationswith zero heat conductivity, Appl. Math. Lett., 27 (2014), 70-73.
    [10] Z. Guo and S. Gala, Regularity criterion of the Newton-Boussinesq equations in $\mathbb{R}^3$, Commun. PureAppl. Anal., 11 (2012), 443-451.
    [11] J. Geng and J. Fan, A note on regularity criterion for the 3D Boussinesq system with zero thermalconductivity, Appl. Math. Lett., 25 (2012), 63-66.
    [12] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamichen Grundgleichungen, Math. Nach. , 4 (1950/1951), 213-231.
    [13] Y. Jia, X. Zhang, and B. Dong, Remarks on the blow-up criterion for smooth solutions of theBoussinesq equations with zero diffusion, C.P.A.A., 12 (2013), 923-937.
    [14] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Commun.Pure Appl. Math., 41 (1988), 891-907.
    [15] C. Kenig, G. Ponce, and L. Vega, Well-posedness of the initial value problem for the Korteweg-de-Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.
    [16] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta. Math., 63 (1934), 183-248.
    [17] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notesin Mathematics, AMS/CIMS, 9 (2003).
    [18] M. Mechdene, S. Gala, Z. Guo, and M.A. Ragusa, Logarithmical regularity criterion of the threedimensionalBoussinesq equations in terms of the pressure, Z. Angew. Math. Phys., 67 (2016), 67-120.
    [19] G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.
    [20] J. Serrin, The initial value problem for the Navier-Stokes equations, In Nonlinear Problems (Proc. Sympos. , Madison, Wis. ), Univ. of Wisconsin Press, Madison, Wis. , 1963, 69-98.
    [21] M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458.
    [22] N. Ishimura and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations, Math. Meth. Appl. Sci., 9 (1999), 1323-1332.
    [23] H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983.
    [24] H. Qiu, Y. Du, and Z. Yao, Blow-up criteria for 3D Boussinesq equations in the multiplier space, Comm. Nonlinear Sci. Num. Simulation, 16 (2011), 1820-1824.
    [25] H. Qiu, Y. Du, and Z. Yao, A blow-up criterion for 3D Boussinesq equations in Besov spaces, Nonlinear Analysis TMA, 73 (2010), 806-815.
    [26] Z. Xiang, The regularity criterion of the weak solution to the 3D viscous Boussinesq equations inBesov spaces, Math. Methods Appl. Sci., 34 (2011), 360-372.
    [27] 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.
    [28] Z. Ye, A Logarithmically improved regularity criterion of smooth solutions for the 3D Boussinesqequations, Osaka J. Math., 53 (2016), 417-423.
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