solutions of incompressible navier-stokes equations can exhibit a wide spectrum of different types of behavior. in various regimes, the equations contain as special limiting cases for example the classical heat equation, the non-linear schroedinger equation, various other dispersive equations with strange dispersion relations, various non-trivial finite-dimensional dynamical systems, some classical geometric semilinear elliptic equations, etc. in addition, when thinking about realistic fluid flows and applications, ideas from statistical mechanics enter the picture. in the lecture i will explain (a limited number of) some pde aspects of these equations.