Existence, uniqueness, regularity and long time behavior of hydrodynamic evolution equations

In these lecture notes we present some recent results regarding existence, uniqueness and regularity for two well-known mathematical problems arising from hydrodynamics. Although the problems are very different and therefore need a specic analysis, it turns out that in both problems at least one of the tools needed comes fromharmonic analysis, for instance the study of fractional Laplace operators and singular integrals. The first part deals with a class of problems which can be described by systems of non-linear partial differential equations consisting of incompressible Navier-Stokes or Stokes equations coupled to other field equations. The most important examples in this class are the ideal magneto-hydrodynamics or complex fluids of Oldroyd-B type. For these problems we present some results concerning existence and uniqueness obtained using a combined Lagrangian-Eulerian approach and suitable commutator estimates. In the second partwe summarize some recent developments concerning the regularity of the critical surface quasi-geostrophic equation (SQG) on the two-dimensional torus. After recalling some classical a-priori estimates and a non-linear lower bound for fractional Laplacians, we prove global (in time) regularity for the critical dissipative SQG by combining the De Giorgiiteration technique with the (above mentioned) non-linear lower bound. In particular, this result shows the existence of a compact absorbing set in H¹(T²) which attracts uniformly all the dynamics.