# Mathematical investigation of models of shallow water with a varying bottom

http://hdl.handle.net/10150/191198
Title:
Mathematical investigation of models of shallow water with a varying bottom
Author:
Oliver, Marcel,1963-
Issue Date:
1996
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
This dissertation is a mathematical investigation of the so-called lake and the great lake equations, which are shallow water equations that describe the long-time motion of an inviscid, incompressible fluid contained in a shallow basin with a slowly spatially varying bottom, a free upper surface and vertical side walls, under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude. It is shown that these equations are globally well-posed, i.e. that they possess unique global weak solutions that depend continuously on the initial data and on the bottom topography. Provided the initial data is in a class of sufficiently differentiable functions, it remains a member of that class for all times. In other words, the lake and great lake equations have global classical solutions. Moreover, if the equations are posed on a space-periodic domain and the initial data is real analytic, the solution remains real analytic for all times. The proof is based on a characterization of Gevrey classes in terms of decay of Fourier coefficients. Finally, a partial mathematical justification of the formal derivation of the lake equations is given. It is shown that solutions of the lake equation stay close to solutions of the rigid lid equations—the three dimensional Euler equations in the limit of small surface wave amplitude—in the following sense: For every error bound 6 there exists a time T = T(ε) such that for all times t ∈ [0, T] the difference between a solution to the lake equations and the solution to the rigid lid equation corresponding to the same initial data is less than E in a suitably chosen norm. Moreover, T tends to infinity as the aspect ratio of the basin tends to zero.
Type:
Dissertation-Reproduction (electronic); text
Keywords:
Hydrology.; Water levels -- Mathematical models.; Equations.
Degree Name:
Ph. D.
Degree Level:
doctoral
Degree Program:
Degree Grantor:
University of Arizona
Committee Chair:
Levermore, C. David

DC FieldValue Language
dc.language.isoenen_US
dc.titleMathematical investigation of models of shallow water with a varying bottomen_US
dc.creatorOliver, Marcel,1963-en_US
dc.contributor.authorOliver, Marcel,1963-en_US
dc.date.issued1996en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThis dissertation is a mathematical investigation of the so-called lake and the great lake equations, which are shallow water equations that describe the long-time motion of an inviscid, incompressible fluid contained in a shallow basin with a slowly spatially varying bottom, a free upper surface and vertical side walls, under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude. It is shown that these equations are globally well-posed, i.e. that they possess unique global weak solutions that depend continuously on the initial data and on the bottom topography. Provided the initial data is in a class of sufficiently differentiable functions, it remains a member of that class for all times. In other words, the lake and great lake equations have global classical solutions. Moreover, if the equations are posed on a space-periodic domain and the initial data is real analytic, the solution remains real analytic for all times. The proof is based on a characterization of Gevrey classes in terms of decay of Fourier coefficients. Finally, a partial mathematical justification of the formal derivation of the lake equations is given. It is shown that solutions of the lake equation stay close to solutions of the rigid lid equations—the three dimensional Euler equations in the limit of small surface wave amplitude—in the following sense: For every error bound 6 there exists a time T = T(ε) such that for all times t ∈ [0, T] the difference between a solution to the lake equations and the solution to the rigid lid equation corresponding to the same initial data is less than E in a suitably chosen norm. Moreover, T tends to infinity as the aspect ratio of the basin tends to zero.en_US
dc.description.notehydrology collectionen_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.typetexten_US
dc.subjectHydrology.en_US
dc.subjectWater levels -- Mathematical models.en_US
dc.subjectEquations.en_US
thesis.degree.namePh. D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US