Applied Math Seminar
Winter Quarter 2008
3:15 p.m.
Sloan Mathematics Corner
Building 380, Room 380-C

Friday, March 21, 2008

Walter Pauls
Max Planck Institute for Dynamics and Self-Organization
Gottingen, Germany

Complex singularities of solutions of hydrodynamic equations: numerical and analytical study


Abstract:

It is still an open question whether solutions of the 3D Euler equation blow up in a finite time. However, it is known that for solutions with analytic initial conditions real singularities are necessarily proceeded by singularities located in the complex domain. Therefore, our approach to a solution of the blow-up problem consists in analysing the nature and the dynamical characteristics of the complex singularities, especially their role in the depletion of non-linearity. 

We use formal expansion methods and high-precision calculations to obtain detailed information on the analytic properties of solutions of the Euler equation in two dimensions for some simple complex-valued initial conditions. It has been found numerically that such solutions  have scaling properties that depend on the initial conditions. In particular, the scaling exponent varies continuously between two simple values, one of which can be well described by a variant of the linearised Euler equation, the other one corresponding to the 2D Burgers equation.