Global regular axially symmetric solutions to the Navier–Stokes equations. Part 3

The series of papers denoted by Part 1, Part 2, and Part 3 is devoted to motions of axially symmetric solutions to the Navier–Stokes equations in a cylindrical domain. Part 1 and Part 2 deal with motions such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi 1 equals psi slash r"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mi> ψ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\psi _1=\psi /r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi"> <mml:semantics> <mml:mi> ψ </mml:mi> <mml:annotation encoding="application/x-tex">\psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the stream function and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the radius, vanishes on the axis of symmetry. The boundary of the cylinder equals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 1 union upper S 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> ∪ </mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">S_1\cup S_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 1"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is parallel to the axis of symmetry and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 2"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is perpendicular to it. It is clear that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 2"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has two parts. In Part 1 it was assumed that the normal component of velocity, angular components of velocity, and vorticity vanish on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 1"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the periodic boundary conditions were assumed on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 2"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In Part 2 the same boundary conditions were imposed on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 1"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as in Part 1 and it was assumed that the normal component of velocity, angular component of vorticity, and the normal derivative of the angular component of velocity vanish on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 2"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper, Part 3, the same boundary conditions as in Part 2 are imposed but the restriction <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi 1 equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\psi _1=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the axis of symmetry is dropped. This is done by looking for solutions to the problem under study in the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v equals v prime plus v Overscript 1 Endscripts plus v Overscript 2 Endscripts"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>v</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml: