Numerical Study on the Asymptotic Equation for the Deformation of Vortex Lattice

We study a coarse-grained asymptotic equation which describes deformation of vortex lattices [Smirnov & Chukbar 2001]. It reads φt=φxxφyy–φ2xy, where φ denotes displacement of vortex locations. This equation is valid for a lattice with short-ranged interaction, e.g. geostrophic vortices with a screened potential. New self-similar blow-up solutions with infinite total energy are found. We ask whether or not finite-time blow-up can take place developing from smooth initial data with finite energy. The numerical simulations show finite-time blow-up in such a way that φ∈H3 but φ∉H4.