Generalization of some regularity criteria for 3D Boussinesq equations in homogeneous Besov and Triebel–Lizorkin spaces

Abstract In this paper, we generalize some regularity criteria for weak solutions of three-dimensional (3D) Boussinesq equations in homogeneous Besov spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mover accent="true"> <m:mi>B</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mmultiscripts> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mprescripts/> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mi>s</m:mi> </m:mmultiscripts> </m:mrow> </m:math> {\dot{{B}}{{}_{p,q}^{s}}(\mathbb{R}^{3})} and homogeneous Triebel–Lizorkin spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mover accent="true"> <m:mi>F</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mmultiscripts> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mprescripts/> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mi>s</m:mi> </m:mmultiscripts> </m:mrow> </m:math> {\dot{{F}}{{}_{p,q}^{s}}(\mathbb{R}^{3})} . We also deduce them for the homogeneous Sobolev spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mover accent="true"> <m:mi>W</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mi>p</m:mi> <m:mi>m</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\dot{W}^{m}_{p}(\mathbb{R}^{3})} in a certain sense. We show that the weak solution <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mi>ϕ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(\omega,\phi)} is regular on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">]</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> {]0,T]} for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>T</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {T&gt;0} if the velocity ω satisfies <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ω</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>L</m:mi> <m:mi>β</m:mi> </m:msup> <m:mrow> <m:mo maxsize="120%" minsize="120%">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo>;</m:mo> <m:mover accent="true"> <m:mi>B</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mmultiscripts> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mprescripts/> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mi>s</m:mi> </m:mmultiscripts> <m:mo maxsize="120%" minsize="120%">)</m:mo> </m:mrow> <m:mo rspace="12.5pt">,</m:mo> <m:mi>ω</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>L</m:mi> <m:mi>β</m:mi> </m:msup> <m:mrow> <m:mo maxsize="120%" minsize="120%">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo>