Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.752521
Title: Energy conservation for the Euler equations with boundaries
Author: Skipper, Jack
ISNI:       0000 0004 7425 6519
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2018
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Abstract:
In this thesis we study energy conservation for the incompressible Euler equations that model non-viscous fluids. This has been a topic of interest since Onsager conjectured regularity conditions for solutions to conserve energy in 1949. Very recently the full conjecture has been resolved in the case without boundaries. We first perform a study of the different conditions used to ensure energy conservation for domains without boundaries. Results are presented in Chapter 2, as well as an analysis of the similarities between the weakest of these conditions and the conditions we use later with a boundary. We then study the time regularity in Chapter 3 and present a detailed proof for energy conservation without boundaries imposing the conditions ꭒ Ꞓ2 LꝪ (0; T ; L3 ) and Lim │y│→0 1│y│ ∫ ꓔ 0 ∫ │ꭒ(x + y) −u(x│3 dx dt = 0: In Chapters 4 and 5 we consider the easiest case of a at finite boundary corresponding to the domain T2x R+. In Chapter 4 we use an extension argument and impose a condition of continuity at the boundary to prove energy conservation under the conditions that ꭒ Ꞓ2 LꝪ (0; T ; L3 (T2x R+)), Lim │y│∫T0 T2∫∞ │u(x + y) . u(x)j3 dx3dx2dx1dt = 0; ꭒ ꞒL3 (0; T ; L∞ (T2 x[0;ẟ )) and u is continuous at the boundary. We then improve this result further by making it a local method in Chapter 5 and use a different definition of a weak solution where there is no pressure term involved. Chapter 6 considers various different definitions of weak solutions for the incompressible Euler equations on a bounded domain. We study the relations between these varying definitions with and without pressure terms. We then use the recent work of Bardos & Titi (2018), who showed energy conservation with pressure terms included, to get a condition for energy conservation when we consider a weak solution without reference to the pressure term.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.752521  DOI: Not available
Keywords: QA Mathematics
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