Behavior of Solutions of Model Equations for Incompressible Fluid Flow

We study the behavior of solutions of model equations of inviscid incompressible fluid flow proposed by Constantin, Lax and Majda together with a viscous version studied by Schochet. A condition is found on initial data to guarantee that solution of viscous equation remains smooth when the inviscid solution blows up. We prove that global smooth solution of viscous model equation exists for a class of initial data, for which the explicit solution is not known. Singularities of solutions previously obtained are characterized here as distributions.