Further Properties of a Continuum of Model Equations with Globally Defined Flux

To develop an understanding of singularity formation in vortex sheets, we consider model equations that exhibit shared characteristics with the vortex sheet equation but are slightly easier to analyze. A model equation is obtained by replacing the flux term in Burgers’ equation by alternatives that contain contributions depending globally on the solution. We consider the continuum of partial differential equations utsu H u.u.xq 1yu. u.uxqnux x, 0FuF1, n G0, where H u. is the Hilbert transform of u. We show that when u s1r2, for n )0, the solution of the equation exists for all time and is unique. We also show with a combination of analytical and numerical means that the solution when u s1r2 and n )0 is analytic. Using a pseudo-spectral method in space and the Adams]Moulton fourth-order predictor-corrector in time, we compute the numerical solution of the equation with u s1r2 for various viscosities. The results confirm that for n )0, the solution is well behaved and analytic. The numerical results also confirm that for n )0 and u s1r2, the solution becomes singular in finite time and finite viscosity prevents singularity formation. We also present, for a certain class of initial conditions, solutions of the equation, with 0-u -1r3 and u s1, that become infinite for n G0 in finite time.