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Applications of optimization and optimal control to some fundamental problems in mathematical fluid dynamics

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Optimization and optimal dynamical control are used to investigate the accuracy of analytical estimates for solutions of some basic nonlinear partial differential equations of mathematical hydrodynamics. Even though many mathematical estimates are demonstrably sharp, the result of a sequence of applications of such estimates need not be sharp leaving uncertainty in the ultimate result of the analysis. We examine the classical analysis for enstrophy and palinstrophy amplification in Burgers’ and the Navier-Stokes equations and discover that the best known instantaneous growth rates estimates are indeed sharp. Integrating the estimates in time, however, may (as in the 2D Navier-Stokes case) but does not always (as in the Burgers case) produce sharp estimates in which case optimal control techniques must be brought to bear to determine the actual extreme behavior of the nonlinear dynamics. The question of 3D Navier-Stokes regularity remains unresolved although work is in progress to apply these tools to the challenge.

This talk is part of the Fluid Mechanics (DAMTP) series.

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