In this talk we have obtained explicit and accurate estimates of the sup-norm for
solutions of the Navier-Stokes Equations (NSE) in two space dimensions. By using
the best (so far) available estimates of the embedding constants which appear in
the classical functional interpolation inequalities used in the study of solutions
of dissipative partial differential equations, we have evaluated in an explicit manner
the values of the sup-norm of the solutions of the NSE. In addition we have calculated
the so-called time-averaged dissipative length scale associated to the above solutions.