<jats:p>We point out some criteria that imply regularity of axisymmetric solutions to Navier–Stokes equations. We show that boundedness of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Vert {v_{r}}/{\sqrt{r^3}}\Vert _{L_2({\mathbb {R}}^3\times (0,T))}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>‖</mml:mo></mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:msqrt><mml:msup><mml:mi>r</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:msqrt><mml:msub><mml:mrow><mml:mo>‖</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msub></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> as well as boundedness of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Vert {\omega _{\varphi }}/{\sqrt{r}} \Vert _{L_2({\mathbb {R}}^3\times (0,T))}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>‖</mml:mo></mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mi>φ</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:msqrt><mml:mi>r</mml:mi></mml:msqrt><mml:msub><mml:mrow><mml:mo>‖</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msub></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$v_r$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>v</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:math></jats:alternatives></jats:inline-formula> is the radial component of velocity and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\omega _{\varphi }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ω</mml:mi><mml:mi>φ</mml:mi></mml:msub></mml:math></jats:alternatives></jats:inline-formula> is the angular component of vorticity, imply regularity of weak solutions.
</jats:p>
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