Eulerian dynamics with a commutator forcing

Abstract

We study a general class of Euler equations driven by a forcing with a commutator structure of the form [L, u]() = L(u) -L()u, where u is the velocity field and L is the 'action' which belongs to a rather general class of translation invariant operators. Such systems arise, for example, as the hydrodynamic description of velocity alignment, where action involves convolutions with bounded, positive influence kernels, L (f ) = * f . Our interest lies with a much larger class of L's which are neither bounded nor positive.