Finite time singularities in a class of hydrodynamic 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}dk 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, frozenin vortex filament. This property allows us to study the dynamics of such filaments without the necessity of a regularization procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution 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 longscale dynamics of this system. Selfsimilar 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/(2α)}, where t^{*} is the singularity time.
 Publication:

Physical Review E
 Pub Date:
 May 2001
 DOI:
 10.1103/PhysRevE.63.056306
 arXiv:
 arXiv:physics/0012007
 Bibcode:
 2001PhRvE..63e6306R
 Keywords:

 47.15.Ki;
 47.32.Cc;
 Physics  Fluid Dynamics
 EPrint:
 LaTeX, 17 pages, 3 eps figures. This version is close to the journal paper