| Summary: | The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Ω</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></mrow></semantics></math></inline-formula> with the axis of symmetry. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mn>1</mn></msub></semantics></math></inline-formula> is the boundary of the cylinder parallel to the axis of symmetry and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mn>2</mn></msub></semantics></math></inline-formula> is perpendicular to it. We have two parts of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mn>2</mn></msub></semantics></math></inline-formula>. For simplicity, we assume the periodic boundary conditions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mn>2</mn></msub></semantics></math></inline-formula>. On <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mn>1</mn></msub></semantics></math></inline-formula>, we impose the vanishing of the normal component of velocity, the angular component of velocity, and the angular component of vorticity. We prove the existence of global regular solutions. To prove this, it is necessary that the coordinate of velocity along the axis of symmetry vanishes on it. We have to emphasize that the technique of weighted spaces applied to the stream function plays a crucial role in the proof of global regular axially symmetric solutions. The weighted spaces used are such that the stream function divided by the radius must vanish on the axis of symmetry. Currently, we do not know how to relax this restriction. In part 2 of this topic, the periodic boundary conditions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mn>2</mn></msub></semantics></math></inline-formula> are replaced by the conditions that both the normal component of velocity and the angular component of vorticity must vanish. Moreover, it is assumed that the normal derivative of the angular component of velocity also vanishes on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mn>2</mn></msub></semantics></math></inline-formula>. A transformation from part 1 to part 2 is not trivial because it needs new boundary value problems, so new estimates must be derived.
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