Further Properties of a Continuum of Model Equations

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 ux term in Burgers' equation by alternatives that contain contributions depending globally on the solution. We consider the continuum of partial diierential equations ut = (H(u)u)x + (1 ?)H(u)ux + uxx, 0 1, 0, where H(u) is the Hilbert transform of u. We show that when = 1=2, for > 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 = 1=2 and > 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 = 1=2 for various viscosities. The results connrm that for > 0, the solution is well behaved and analytic. The numerical results also connrm that for = 0 and = 1=2, the solution becomes singular in nite time and nite viscosity prevents singularity formation. We also present, for a certain class of initial conditions, solutions of the equation, with 0 < < 1=3 and = 1, that become innnite for 0 in nite time.