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A shallow water model with eddy viscosity for basins with varying bottom topography
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In this article, the authors introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model to derive a two-dimensional shallow water model and prove the global regularity of the resulting model.Abstract:
The motion of an incompressible fluid confined to a shallow basin with a varying bottom topography is considered. We introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model to derive a two-dimensional shallow water model. The global regularity of the resulting model is proved. The anisotropic form of the stress tensor in our three-dimensional eddy viscosity model plays a critical role in ensuring that the resulting shallow water model dissipates energy.read more
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On Some Compressible Fluid Models: Korteweg, Lubrication, and Shallow Water Systems
TL;DR: In this article, the authors give some mathematical results for an isothermal model of capillary compressible fluids derived by Dunn and Serrin in 1985, which can be used as a phase transition model.
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Existence of Global Weak Solutions for a 2D Viscous Shallow Water Equations and Convergence to the Quasi-Geostrophic Model
Didier Bresch,Benoît Desjardins +1 more
TL;DR: In this article, the authors consider a two dimensional viscous shallow water model with friction term and prove the existence of global weak solutions and convergence to the strong solution of the viscous quasi-geostrophic equation with free surface term.
Journal ArticleDOI
Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects
TL;DR: In this article, the authors derived a two-dimensional viscous shallow water model in rotating framework, with irregular topography, linear and quadratic bottom friction terms and capillary effects.
Journal ArticleDOI
A new two-dimensional shallow water model including pressure effects and slow varying bottom topography
Stefania Ferrari,Fausto Saleri +1 more
TL;DR: In this article, the authors introduced appropriate scalings into a three-dimensional anisotropic eddy viscosity model; after averaging on the vertical direction and considering some asymptotic assumptions, they obtained a two-dimensional model, which approximates the threedimensional model at the second order with respect to the ratio between the vertical scale and the longitudinal scale.
Recent mathematical results and open problems about shallow water equations. Analysis and simulation of fluid dynamic
TL;DR: The purpose of this work is to present recent mathematical results about the shallow water model and to mention related open problems of high mathematical interest.
References
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Journal ArticleDOI
Long-time effects of bottom topography in shallow water
TL;DR: In this article, the authors present and discuss new shallow water equations that provide an estimate of the long-time asymptotic effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid with a free surface which is moving under the force of gravity.
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Long-time shallow-water equations with a varying bottom
TL;DR: In this article, new shallow-water equations were proposed to model the long-time effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid possessing a free surface and moving under the force of gravity.
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Global well-posedness for the lake equations
TL;DR: In this article, the authors prove global well-posedness for the Euler lake equations in a basin with a free upper surface and a spatially varying bottom topography, where the bottom topology is assumed to have a non-degenerate bottom surface.
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Classical Solutions for a Generalized Euler Equation in Two Dimensions
TL;DR: In this paper, the Euler equations in two spatial dimensions have global classical solutions, and a new proof which is analytic rather than geometric is provided which is set in an abstract framework that applies to the so-called lake and the great lake equations describing weakly non-hydrostatic effects of bottom topography on the motion of shallow water.
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Justification of the Shallow-Water Limit for a Rigid-Lid Flow with Bottom Topography
TL;DR: In this article, it was shown that any solution to the lake equations remains close to some solution of the rigid-lid equations for an interval of time that can be made arbitrarily large by choosing the aspect ratio of the basin small.