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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

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 ��3 := {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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