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Title: On some alternative formulations of the Euler and Navier-Stokes equations
Author: Pooley, Benjamin C.
ISNI:       0000 0004 6062 0105
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2016
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In this thesis we study well-posedness problems for certain reformulations and models of the Euler equations and the Navier{Stokes equations. We also prove several global well-posedness results for the diffusive Burgers equations. We discuss the Eulerian-Lagrangian formulation of the incompressible Euler equations considered by Constantin (2000). Using this formulation we give a new proof that the Euler equations are locally well-posed in Hs (Td ) for s > d/2 + 1. Constantin proved a local well-posedness result for this system in the Hӧlder spaces C1; for μ> 0, but an analysis in Sobolev spaces is perhaps more natural. After suggesting a possible Eulerian-Lagrangian formulation for the incompressible Navier{Stokes equations in which the back-to-labels map is not di used, we obtain the formulation written in terms of the so-called magnetization variables, as studied by Montgomery-Smith and Pokornẏ (2001). We give a rigorous analysis of the equivalence between this formulation and the classical one, in the context of weak solutions. Noting certain similarities between this formulation and the diffusive Burgers equations we begin a study of the latter. We prove that the diffusive Burgers equations are globally well-posed in Lp ∩ L2 (Ω ) for certain domains Rd , p > d, and d = 2 or d = 3. Moreover, we prove a global well-posedness result in H1= 2 (T3 ). Lastly, we consider a new model of the Navier{Stokes equations, obtained by modifying one of the nonlinear terms in the magnetization variables formulation. This new system admits a maximum principle and we prove a global well-posedness result in H1=2 (T3 ) following our analysis of the Burgers equations.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics