Invariant helical subspaces for the Navier-Stokes equations. (English) Zbl 0708.76044

Summary: Three-dimensional solutions with helical symmetry are shown to form an invariant subspace for the Navier-Stokes equation. Uniqueness of weak helical solutions in the sense of Leray is proved, and these weak solutions are shown to be regular (strong) solutions existing for arbitrary time \(t\). The global universal attractor for the infinite- dimensional dynamical system generated by the corresponding semigroup of helical flows is shown to be compact and finite-dimensional. The Hausdorff and fractal dimensions of the global attractors are estimated in terms of the governing physical parameters and in terms of the helical parameters for several problems in the class, with the most detailed results obtained for rotating Hagen-Poiseuille (pipe) flow. In this case, the dimension, either Hausdorff or fractal, up to an absolute constant is bounded from above by \(\mathrm{Re}/\sqrt{\alpha^ 2+n^ 2}\), where \(\alpha\) is the axial wavenumber, \(n\) is the azimuthal wavenumber and Re is the Reynolds number based on the radius of the pipe. These upper bounds are independent of the rotation rate.


76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
Full Text: DOI


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