ANALYTIC STRUCTURE OF 2 1D-TRANSPORT EQUATIONS WITH NONLOCAL FLUXES

We replace the flux term in Burger's equation by two simple alternates that contain contributions depending globally on the solution. In one case, the term is in the form of a hyperbolic equation where the char acteristic speed is nonlocal, and in the other the term is in conserva tion form. In both cases, the nonanalytic is due to the presence of th e Hilbert transform. The equations have a loose analogy to the motion of vortex sheets. In particular, they both form singularities in finit e time in the absence of viscous effects, Our motivation then is to st udy the influence of viscosity. In one case, viscosity does not preven t singularity formation. In the other, we can prove solutions exist fo r all time, and determine the likely weak solution as viscosity vanish es. An interesting aspect of our work is that singularity formation ca n be viewed as the motion of singularities in the complex physical pla ne that reach the real axis in finite time. In one case, the singulari ty is a pole and causes the solution to blow up when it reaches the re al axis. In the other, numerical solutions and an asymptotic analysis suggest that the weak solution contains a square root singularity that reaches the real axis in finite time, and then propagates along it. W e hope our results will spur further interest in the role of singulari ties in the complex spatial plane in solutions to transport equations.