59 research outputs found
Derivation and analysis of a new 2D Green-Naghdi system
We derive here a variant of the 2D Green-Naghdi equations that model the
propagation of two-directional, nonlinear dispersive waves in shallow water.
This new model has the same accuracy as the standard Green-Naghdi
equations. Its mathematical interest is that it allows a control of the
rotational part of the (vertically averaged) horizontal velocity, which is not
the case for the usual Green-Naghdi equations. Using this property, we show
that the solution of these new equations can be constructed by a standard
Picard iterative scheme so that there is no loss of regularity of the solution
with respect to the initial condition. Finally, we prove that the new
Green-Naghdi equations conserve the almost irrotationality of the vertically
averaged horizontal component of the velocity
A new class of two-layer Green-Naghdi systems with improved frequency dispersion
We introduce a new class of Green-Naghdi type models for the propagation of
internal waves between two (1+1)-dimensional layers of homogeneous, immiscible,
ideal, incompressible, irrotational fluids, vertically delimited by a flat
bottom and a rigid lid. These models are tailored to improve the frequency
dispersion of the original bi-layer Green-Naghdi model, and in particular to
manage high-frequency Kelvin-Helmholtz instabilities, while maintaining its
precision in the sense of consistency. Our models preserve the Hamiltonian
structure, symmetry groups and conserved quantities of the original model. We
provide a rigorous justification of a class of our models thanks to
consistency, well-posedness and stability results. These results apply in
particular to the original Green-Naghdi model as well as to the Saint-Venant
(hydrostatic shallow-water) system with surface tension.Comment: to appear in Stud. Appl. Mat
An improved result for the full justification of asymptotic models for the propagation of internal waves
We consider here asymptotic models that describe the propagation of
one-dimensional internal waves at the interface between two layers of
immiscible fluids of different densities, under the rigid lid assumption and
with uneven bottoms. The aim of this paper is to show that the full
justification result of the model obtained by Duch\^ene, Israwi and Talhouk
[{\em SIAM J. Math. Anal.}, 47(1), 240--290], in the sense that it is
consistent, well-posed, and that its solutions remain close to exact solutions
of the full Euler system with corresponding initial data, can be improved in
two directions. The first direction is taking into account medium amplitude
topography variations and the second direction is allowing strong nonlinearity
using a new pseudo-symmetrizer, thus canceling out the smallness assumptions of
the Camassa-Holm regime for the well-posedness and stability results.Comment: arXiv admin note: substantial text overlap with arXiv:1304.4554; text
overlap with arXiv:1208.6394 by other author
Large Time existence For 1D Green-Naghdi equations
We consider here the Green-Naghdi equations that are commonly used in
coastal oceanography to describe the propagation of large amplitude surface
waves. We show that the solution of the Green-Naghdi equations can be
constructed by a standard Picard iterative scheme so that there is no loss of
regularity of the solution with respect to the initial condition
Equations for small amplitude shallow water waves over small bathymetric variations
A generalized version of the -Boussinesq class of systems is derived to
accommodate variable bottom topography in two-dimensional space. This extension
allows for the conservation of suitable energy functionals in some cases and
enables the description of water waves in closed basins with well-justified
slip-wall boundary conditions. The derived systems possess a form that ensures
their solutions adhere to important principles of physics and mathematics. By
demonstrating their consistency with the Euler equations and estimating their
approximation error, we establish the validity of these new systems. Their
derivation is based on the assumption of small bathymetric variations. With
practical applications in mind, we assess the effectiveness of some of these
new systems through comparisons with standard benchmarks. The results indicate
that the assumptions made during the derivation are not overly restrictive. The
applications of the new systems encompass a wide range of scenarios, including
the study of tsunamis, tidal waves, and waves in ports and lakes
A mathematical justification of the momentum density function associated to the KdV equation
Consideration is given to the KdV equation as an approximate model for long waves of small amplitude at the free surface of an inviscid fluid. It is shown that there is an approximate momentum density associated to the KdV equation, and the difference between this density and the physical momentum density derived in the context of the full Euler equations can be estimated in terms of the long-wave parameter.publishedVersio
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