Finite time singularities in a class of hydrodynamic models - art. no. 056306

Models of inviscid incompressible fluid are considered, with the kinetic en ergy (i.e., the Lagrangian functional) taking the form L-integralk(alpha)\v (k)\(2)dk in 3D Fourier representation, where ct is a constant, 0< <alpha>< 1. Unlike the case <alpha>=0 (the usual Eulerian hydrodynamics), a finite v alue of alpha results in a finite energy for a singular, frozen-in vortex f ilament. This property allows us to study the dynamics of such filaments wi thout the necessity of a regularization procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solut ion is performed for a pair of antiparallel 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 syst em. Self-similar solutions. of the corresponding equations are found analyt ically. They describe the formation of a finite time singularity, with all length scales decreasing like (t* -t)(1/(2-alpha)), where t* is the singula rity time.