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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 partialŽ . where H u is the Hilbert transform of u. We show that when s 1r2, 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 s 1r2 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 s 1r2 for various viscosities. The results confirm that for ) 0, the solution is well behaved and analytic. The numerical results also confirm that for ) 0 and s 1r2, 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 --1r3 and s 1, that become infinite for G 0 in finite time. ᮊ

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