Abstract
This chapter presents the existence, uniqueness, and regularity theory of solutions to the Navier–Stokes equations when they are formulated in vorticity form. It also discusses the large-time asymptotic behavior of solutions for sufficiently small initial data. In fact, the three-dimensional case has hardly been studied and therefore, it concentrates on the two-dimensional case. From the point of view of hydrodynamic phenomena an interesting case is the evolution of vorticity and its associated velocity field when it is initially given by isolated vortices, vortex filaments, or sheets. In the zero viscosity limit (i.e., v = 0, leading to the Euler equations) the circulation is preserved (Kelvin's theorem) and the use of vorticity in numerical methods has become very popular. In vortex methods even smooth initial data are replaced by a distribution of singular vortical objects.
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