Upper bounds on the maximum growth of the total enstrophy or palinstrophy of a fluid flow have far-reaching implications for the regularity of solutions of the incompressible Navier-Stokes equations. However, they are notoriously difficult to obtain even for simpler models. This talk will describe a general approach to bounding the maximum value of a given objective function $\Phi$ (e.g., the total enstrophy of a fluid flow) along trajectories of a dynamical system that start from a prescribed set of initial conditions. Specifically, bounds are obtained by constructing a feasible solution to a convex optimization problem over continuously differentiable Lyapunov-type functions. It will be demonstrated that many of the results obtained so far for the Navier-Stokes equations, as well as for the simpler viscous Burgers equation, amount to particular applications of this approach. It will also be shown that, for systems governed by well posed ODEs, arbitrarily sharp bounds on the largest value of $\Phi$ along trajectories can be computed numerically using algorithms for sum-of-squares (SOS) programming. This will be demonstrated for a low-dimensional model of shear flow. Finally, the remaining challenges for the application of SOS programming to full PDE systems, including the Burgers and Navier-Stokes equations, will be discussed.