Dissipation caused by a vorticity field and generation of singularities in Madelung fluid

We consider a generalization of Madelung fluid equations, whichwas derived in the 1980s by means of a pathwise stochastic calculus of variationswith the classical action functional. At variance with the original ones, the newequations allowus to consider velocity fields with vorticity. Such a vorticity causesdissipation of energy and it may concentrate, asymptotically, in the zeros of thedensity of the fluid. We study, by means of numerical methods, some Cauchyproblems for the bidimensional symmetric harmonic oscillator and observe thegeneration of zeros of the density and concentration of the vorticity close tocentral lines and cylindrical sheets. Moreover, keeping the same initial data, weperturb the harmonic potential by a term proportional to the density of the fluid,thus obtaining an extension with vorticity of the Gross–Pitaevskii equation, andobserve analogous behaviours.