NAVIER-STOKES INSTABILITY, MULTI-VORTEX, AND ATTRACTORS

A fundamentally new method for obtaining lower bounds for the dimension of attractors in equations related to hydrodynamics is proposed. The method is not based on Kolmogorov flows and is applicable to the classical two-dimensional Navier-Stokes equations in a bounded domain with the Dirichlet condition, as well as to the Navier-Stokes equations with Ekman friction in any plane. For a bounded domain, lower bounds are proved that are similar to the known bound for a torus and a sphere, and for the entire plane, our estimate is exact. Note that no lower bounds for these two cases were previously known. The method is based on the use of a so-called multi-vortex, consisting of well-separated Vishik vortices (i.e., spectrally unstable spatially localized flows constructed by M.M. Vishik) as an analog of Kolmogorov flows. It should also be noted that this method reproduces a well-known result on a torus and is applicable to many other hydrodynamic equations. (Results of joint research with S.V. ZELIK)