Blowup of smooth solutions and conditional regularity in mathematical fluid mechanics

• Applicant: Dr. Vu Hoang
• Subject Area: Mathematics
• Term: from 2013 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 233177374

The three-dimensional Euler equations for the motion of incompressible, ideal fluids are one of the most fundamental models in mathematical fluid mechanics. An outstanding unsolved mathematical problem is to decide whether solutions remain regular globally in time, or whether smooth solutions can develop singularities in finite time. The analogous problem for the Navier-Stokes equations is one of the celebrated Millennium problems of the Clay Mathematics Institute. Recently interesting breakthroughs have been made in the field of Euler equations. In a groundbreaking paper T.Hou and G. Luo (California Institute of Technology) found strong numerical evidence for a possible blowup mechanism for axisymmetrical Euler equations. This discovery led to vivid research revolving around the so called hyperbolic flow scenario. In a hyperbolic flow the fluid is supposed to be compressed strongly in one direction. The idea is to create strong gradients by this scenario hoping to achieve blowup in finite time. Symmetries and suitable boundary conditions play a crucial role in this approach. Due to the complexity of the problem an intense investigation of simplified one- and two-dimensional problems is in progress. In my research program I study the hyperbolic flow scenario for a Boussinesq-System with simplified velocity law and the surface quasigeostrophic equations as well as some one-dimensional problems. My goal is either to rigorously show blowup in finite time or to deduce a conditional regularity result, i.e. to exclude certain blowup scenarios. The main obstacle for a blowup proof is the creation of a stable instability in the following sense: On the one hand it is desirable that certain quantities grow infinitely in finite time, on the other hand one has to remain sufficient control over the nonlocal and nonlinear evolution such that the blowup scenario does not break down. Proving a conditional regularity result, however, one assumes a certain blowup scenario and tries to exclude it, usually using the dynamics of the system. Here, too, the control over the evolution is crucial. For this purpose, in my project I develop new a-priori estimates which make use of the geometry of the flow in an optimal way.

• DFG Programme: Research Fellowships
• International Connection: USA
• Host: Professor Dr. Alexander Kiselev