On helical flows with little regularity: global<br/><br/>existence and vanishing viscosity

Monday, February 22, 2010 - 3:40pm - 4:25pm
EE/CS 3-180
Helena Nussenzveig Lopes (State University of Campinas (UNICAMP))
Keywords: Helical flow, swirl, vortex stretching, energy method, Delort type symmetrization

Abstract: Helical flows are flows which are covariant with respect to translation along helices. In recent work, B. Ettinger and E. Titi established global existence and uniqueness of helical flow solutions of the incompressible 3D Euler equations with bounded vorticity and no helical swirl (the component of velocity along helices). In earlier work, A. Mahalov, E. Titi and S. Leibovich had established well-posedness, in H1, of viscous helical flows (solutions of the Navier-Stokes equations) with no restriction on helical swirl. Absence of helical swirl prevents vortex stretching, but, although a conserved quantity for Euler, it is not conserved by the Navier-Stokes evolution. In both sets of results, the fluid domain was a bounded, smooth, helical subset of R3.

In this talk we will review these results and discuss the problem of taking the vanishing viscosity limit of helical flows with small, with respect to viscosity, helical swirl. We work with bounded domain flows, assuming Navier boundary conditions, but we discuss the full-space problem as well. Our results assume that the flow has finite enstrophy. We also discuss an alternative way to obtain global existence of helical solutions of the 3D Euler equations, without helical swirl, with vorticity only in Lp, p>3/2.

The collaborators involved are Anne Bronzi, Quansen Jiu, Milton Lopes Filho and Dongjuan Niu.

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