Ill-posedness of the Complex Euler Equations

Well-posedness is among the most important basic properties one should seek to establish for any PDE problem, especially for applications. For suitable initial/boundary data, do solutions of the problem exist, are the solutions unique, and do they depend continuously on the data? In this talk we introduce a complex-valued generalization of the Euler equations for inviscid, incompressible fluid flow. We show that, surprisingly, these equations are not well-posed in Sobolev spaces and there exist analytic initial data from which singularities form in finite time. This is in contrast to the situation for the real Euler equations in dimension 2, where smooth initial data produce solutions which remain smooth for all time.