Three lectures on mathematical fluid mechanics (Q2776595)

In these lectures the author discusses three open problems. The first lecture addresses the possibility of finite time singularity in three-dimensional incompressible Euler equations, which arises from smooth and compactly supported initial data. The author gives geometric necessary conditions for blow up, expressed in terms of the direction of vorticity, and describes a class of models -- ``active scalars'' -- which have a common mathematical structure and describe a large part of physically relevant phenomena in fluid dynamics.NEWLINENEWLINENEWLINEThe second lecture considers the inviscid limit for nonsmooth solutions of Euler equations. The open problem here is whether the Euler equations can describe adequately this limit. The author proves convergence results for solutions in appropriate Besov spaces, and demonstrates that the rate of convergence in these limits depends on detailed information about initial data, and likely degenerates in time.NEWLINENEWLINENEWLINEThe third lecture addresses the long-time behavior of solutions of two-dimensional Navier-Stokes equations. The open problem here is whether this long-time behavior can be adequately described by finitely many ordinary differential equations. The author constructs a weakly dense set of initial data for which the Euler equations can describe the temporal behavior at large energies.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00028].