Finite time singularities in a class of hydro dynamic models

摘要

Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form L ～∫κ~α|v_k|~2d~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 α 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.