On helical flows: vanishing viscosity limit and global existence for ideal fluids
by Helena Nussenzveig Lopes (IMECC-UNICAMP, Brasil)
Helical flows are 3D flows which are covariant with respect to helical symmetry, in which simultaneous rotation and translation along the rotation axis occur. For incompressible helical flows it has been proved that there exist global strong solutions to the Navier-Stokes equations. Well-posedness for helical flows in the inviscid case has also been established under the assumption that the vorticity be bounded and that the helical swirl, the component of velocity along the helices, vanish. In this talk we will examine the vanishing viscosity limit of helical flows with finite enstrophy; we show, in several contexts, that the solutions converge to a solution of the inviscid problem, if the initial data has finite enstrophy and vanishingly small helical swirl. We also prove the existence of a weak helical flow, solution to the inviscid equations, with vorticity pth-power integrable, for some p>4/3, by using a different approximation.
Finally, we comment on the two-dimensional limits of helical flows.