Regularization of ill-posed problems via the level set approach

We introduce a new formulation for the motion of curves in R-2 (easily extendable to the motion of surfaces in R-3), when the original motion generally corresponds to an ill-posed problem such as the Cauchy-Riemann equations. This is, in part, a generalization of our earlier work in [6], where we applied similar ideas to compute flows with highly concentrated vorticity, such as vortex sheets or dipoles, for incompressible Euler equations. Our new formulation involves extending the level set method of [12] to problems in which the normal velocity is not intrinsic. We obtain a coupled system of two equations, one of which is a level surface equation. This yields a fixed-grid, Eulerian method which regularizes the ill-posed problem in a topological fashion. We also present an analysis of curvature regularizations and some other theoretical justification. Finally, we present numerical results showing the stability properties of our approach and the novel nature of the regularization, including the development of bubbles for curves evolving under Cauchy-Riemann flow.