Finite time singularities in a class of hydro dynamic models

Abstract

Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form L∼∫k ^α|v_k|² d ³k in 3D Fourier representation, where α is a constant, 0“;α<1. Unlike the case α=0 (the usual Eulerian hydrodynamics), a finite value of & ga results in a finite energy for a singular, frozen-in vortex filament. This property allows us to study the dynamics of such filaments without the necessity of a regularisation procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution is performed for a pair of anti-parallel vortex filaments and an analog of the Crow instability is found at small wave-numbers. A local approximate Hamiltonian is obtained for the nonlinear long-scale dynamics of this system. Self-similar solutions of the corresponding equations are found analytically. They describe the formation of a finite time singularity, with all length scales decreasing like (t ^*−t)^1/(2−α) where t* is the singularity time.