The incompressible Euler and Navier-Stokes equations have been central models in hydrodynamics for centuries;
however, no satisfactory results about the propagation of regularity of solutions for general initial data have been
obtained so far. The object of this research project will be a generalized vorticity model equation that has been
introduced by the authors. This model equation includes, via an artificial parameter, other model equations arising
in hydrodynamics. Special emphasis will be laid on the rôle of the convection term; in particular, we will
investigate how it interacts with the stretching term to cause of prevent global existence in time or finite-time
blow-up of the corresponding solutions.
Furthermore, for the viscous extension of the generalized vorticity model equation, we propose the introduction of
another parameter acting as the exponent of the fractional Laplacian.
We expect that the results of our research project will take us a further substantial step towards a deeper
understanding of the smoothness (or lack thereof) of solutions to the hydrodynamic equations mentioned above.