Three-dimensional Navier-Stokes equations for singular filament initial data
A slightly altered form of the traditional Navier-Stokes equations can be written in which, instead of the velocity vector \(\vec u\), the vorticity \({\vec \omega}\) (\({\vec \omega}=curl \vec u)\) is treated as unknown. This transformation is particularly useful when the vortical motions in a viscous fluid must be studied. In particular, it may happen that vortices are initially active on an element of a curve, and consequently, the system of equations of Navier-Stokes type describing this motion possesses a singular initial datum, analytically characterized by a measure located on the initial filament. The central assumptions to treating the problem is that, at any subsequent time, the instantaneous filament is diffeomorphic to the initial filament. This assumption being made, the so called mollified problem is considered, in which, instead of the originary quantities \(\vec u\), \({\vec \omega}\), their time convolutions, denoted by \(\vec u^{\epsilon}\), \({\vec \omega}^{\epsilon}\), are introduced. The system in terms of the new unknowns admits a unique classical solution, for which an a priori bound, independent of \(\epsilon\), can be given. It follows that, by applying compactness arguments, each \(\vec u^{\epsilon}\) weakly converges to \(\vec u\) in \(L^{\infty}\). Moreover, the further assumption that the \(\vec u\) norm decreases allows the uniqueness of the solution to be proven.
scientific article; zbMATH DE number 4106495
scientific article