Characterization and Regularity for Axisymmetric Solenoidal Vector Fields with Application to Navier-Stokes Equation

Abstract

We consider the vorticity-stream formulation of axisymmetric incompressible flows and its equivalence with the primitive formulation. It is shown that, to characterize the regularity of a divergence free axisymmetric vector field in terms of the swirling components, an extra set of pole condition is necessary to give a full description of the regularity. In addition, smooth solutions up to the axis of rotation gives rise to smooth solutions of primitive formulation in the case of Navier-Stokes equations, but not the Euler equations. We also establish proper weak formulations and show its equivalence to Leray’s solutions.

DOI: 10.1137/080739744

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Cite this paper

@article{Liu2009CharacterizationAR, title={Characterization and Regularity for Axisymmetric Solenoidal Vector Fields with Application to Navier-Stokes Equation}, author={Jian-Guo Liu and Wei-Cheng Wang}, journal={SIAM J. Math. Analysis}, year={2009}, volume={41}, pages={1825-1850} }