Graduate Thesis Or Dissertation
 

On the Existence of Solutions to the Incompressible Navier-Stokes Equations with Constant Energy and Enstrophy

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  • In this dissertation, we use Fourier-analytic and spectral theory methods to analyze the behavior of solutions of the incompressible Navier-Stokes equations in 2D and 3D (with an eye towards better understanding turbulence). In particular, we investigate the possible existence of so-called ghost solutions to the Navier-Stokes Equations. Such solutions, if they exist, would be dynamic in time and yet have constant energy and enstrophy profiles. First, we completely analyze the case of ghost solutions in the simpler Stokes system, and use results from that to construct nonstationary constant-energy solutions (as well as solutions with constant higher order norms) when the spatial domain is the 3D torus. We then explore the properties of finite-mode solutions on the 2D torus, providing a constraining relationship between the spectral structure of a finite-mode solution and that of any force that might generate it. As a consequence of the main result regarding the spectral structure of finite-mode solutions we disprove the existence of so-called chained ghost solutions, which have been investigated in several recent papers in this area.
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  • The research in this dissertation was partially funded by the National Science Foundation, DMS grant 1211413.
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