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On Steady Water Waves and Flows with Vorticity in Three Dimensions

Svensson Seth, Douglas LU (2021)
Abstract
In this thesis we study the steady Euler equations in three dimensions where the solution is assumed to have nonvanishing vorticity. The thesis is based on three research papers. In the first and the third we study the steady Euler equations in the context of the water wave problem, which means we are solving a free boundary problem, while in the second paper we study the equations in a fixed cylinder-like domain. In the first paper we prove the existence of small amplitude doubly periodic waves when the velocity of the water is assumed to be a Beltrami field. Divergence free Beltrami fields are special solutions to the steady Euler equations where the velocity and vorticity are parallel. In the second paper we prove the existence of... (More)
In this thesis we study the steady Euler equations in three dimensions where the solution is assumed to have nonvanishing vorticity. The thesis is based on three research papers. In the first and the third we study the steady Euler equations in the context of the water wave problem, which means we are solving a free boundary problem, while in the second paper we study the equations in a fixed cylinder-like domain. In the first paper we prove the existence of small amplitude doubly periodic waves when the velocity of the water is assumed to be a Beltrami field. Divergence free Beltrami fields are special solutions to the steady Euler equations where the velocity and vorticity are parallel. In the second paper we prove the existence of solutions of the steady Euler equations in cylinder-like domains, where the fluid flows through the domain, like water through a pipe. Here the vorticity is specified by two boundary conditions on the part of the surface where the fluid flows into the domain. In the third paper we also prove the existence of small amplitude doubly periodic waves, but with a different assumption on the vorticity. This assumption is more technical in nature and comes from magnetohydrodynamics. This theory is applicable because the governing equations for the magnetic field in a magnetohydrostatic equilibrium is equivalent to the steady Euler equations. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor Ambrose, David, Drexel University, USA
organization
publishing date
type
Thesis
publication status
published
subject
keywords
The steady Euler equations, the water wave problem, fluid dynamics, partial differential equations
pages
225 pages
publisher
Lund University, Faculty of Science, Centre for Mathematical Sciences, Mathematics
defense location
MH:H. Join via zoom: https://lu-se.zoom.us/j/68353800257
defense date
2021-06-04 13:15:00
ISBN
978-91-7895-838-2
978-91-7895-837-5
project
Mathematical aspects of three-dimensional water waves with vorticity
language
English
LU publication?
yes
id
5a2ca814-827b-4855-9082-e65473b33893
date added to LUP
2021-05-10 22:42:30
date last changed
2025-04-04 14:36:13
@phdthesis{5a2ca814-827b-4855-9082-e65473b33893,
  abstract     = {{In this thesis we study the steady Euler equations in three dimensions where the solution is assumed to have nonvanishing vorticity. The thesis is based on three research papers. In the first and the third we study the steady Euler equations in the context of the water wave problem, which means we are solving a free boundary problem, while in the second paper we study the equations in a fixed cylinder-like domain. In the first paper we prove the existence of small amplitude doubly periodic waves when the velocity of the water is assumed to be a Beltrami field. Divergence free Beltrami fields are special solutions to the steady Euler equations where the velocity and vorticity are parallel. In the second paper we prove the existence of solutions of the steady Euler equations in cylinder-like domains, where the fluid flows through the domain, like water through a pipe. Here the vorticity is specified by two boundary conditions on the part of the surface where the fluid flows into the domain. In the third paper we also prove the existence of small amplitude doubly periodic waves, but with a different assumption on the vorticity. This assumption is more technical in nature and comes from magnetohydrodynamics. This theory is applicable because the governing equations for the magnetic field in a magnetohydrostatic equilibrium is equivalent to the steady Euler equations.}},
  author       = {{Svensson Seth, Douglas}},
  isbn         = {{978-91-7895-838-2}},
  keywords     = {{The steady Euler equations; the water wave problem; fluid dynamics; partial differential equations}},
  language     = {{eng}},
  publisher    = {{Lund University, Faculty of Science, Centre for Mathematical Sciences, Mathematics}},
  school       = {{Lund University}},
  title        = {{On Steady Water Waves and Flows with Vorticity in Three Dimensions}},
  url          = {{https://lup.lub.lu.se/search/files/97646792/Kappa.pdf}},
  year         = {{2021}},
}