Under a critical smallness condition on initial data, axisymmetric Navier-Stokes solutions with NHL boundary conditions in a finite cylinder remain globally regular.
Leray,Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Mathematica 63(1), 193–248 (1934)
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Global Regularity of Axisymmetric Navier-Stokes Equations with NHL Boundary Conditions under a Critical Smallness Condition
Under a critical smallness condition on initial data, axisymmetric Navier-Stokes solutions with NHL boundary conditions in a finite cylinder remain globally regular.