65 research outputs found
The Camassa-Holm equation as a geodesic flow on the diffeomorphism group
Misiolek has shown that the Camassa-Holm (CH) equation is a geodesic flow on
the Bott-Virasoro group. In this paper it is shown that the Camassa-Holm
equation for the case is the geodesic spray of the weak Riemannian
metric on the diffeomorphism group of the line or the circle obtained by right
translating the inner product over the entire group. This paper uses the
right-trivialisation technique to rigorously verify that the Euler-Poincar\'{e}
theory for Lie groups can be applied to diffeomorphism groups. The observation
made in this paper has led to physically meaningful generalizations of the
CH-equation to higher dimensional manifolds (see Refs. \cite{HMR} and
\cite{SH}).Comment: 10 single-spaced pages, Geometric Methods in Fluid Equations:
Submitted to the Journal of Mathematical Physic
A Nonlinear Analysis of the Averaged Euler Equations
This paper develops the geometry and analysis of the averaged Euler equations
for ideal incompressible flow in domains in Euclidean space and on Riemannian
manifolds, possibly with boundary. The averaged Euler equations involve a
parameter ; one interpretation is that they are obtained by ensemble
averaging the Euler equations in Lagrangian representation over rapid
fluctuations whose amplitudes are of order . The particle flows
associated with these equations are shown to be geodesics on a suitable group
of volume preserving diffeomorphisms, just as with the Euler equations
themselves (according to Arnold's theorem), but with respect to a right
invariant metric instead of the metric. The equations are also
equivalent to those for a certain second grade fluid. Additional properties of
the Euler equations, such as smoothness of the geodesic spray (the Ebin-Marsden
theorem) are also shown to hold. Using this nonlinear analysis framework, the
limit of zero viscosity for the corresponding viscous equations is shown to be
a regular limit, {\it even in the presence of boundaries}.Comment: 25 pages, no figures, Dedicated to Vladimir Arnold on the occasion of
his 60th birthday, Arnold Festschrift Volume 2 (in press
Second-order multisymplectic field theory: A variational approach to second-order multisymplectic field theory
This paper presents a geometric-variational approach to continuous and
discrete {\it second-order} field theories following the methodology of
\cite{MPS}. Staying entirely in the Lagrangian framework and letting denote
the configuration fiber bundle, we show that both the multisymplectic structure
on as well as the Noether theorem arise from the first variation of the
action function. We generalize the multisymplectic form formula derived for
first order field theories in \cite{MPS}, to the case of second-order field
theories, and we apply our theory to the Camassa-Holm (CH) equation in both the
continuous and discrete settings. Our discretization produces a
multisymplectic-momentum integrator, a generalization of the Moser-Veselov
rigid body algorithm to the setting of nonlinear PDEs with second order
Lagrangians
On a two-component -Camassa--Holm system
A novel -Camassa--Holm system is studied as a geodesic flow on a
semidirect product obtained from the diffeomorphism group of the circle. We
present the corresponding details of the geometric formalism for metric Euler
equations on infinite-dimensional Lie groups and compare our results to what
has already been obtained for the usual two-component Camassa--Holm equation.
Our approach results in well-posedness theorems and explicit computations of
the sectional curvature.Comment: 12 page
The Degasperis-Procesi equation as a non-metric Euler equation
In this paper we present a geometric interpretation of the periodic
Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric
linear connection on the diffeomorphism group of the circle. We also show that
for any evolution in the family of -equations there is neither gain nor loss
of the spatial regularity of solutions. This in turn allows us to view the
Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fr\'echet
space of all smooth functions on the circle.Comment: 17 page
Local and Global Well-posedness of the fractional order EPDiff equation on
Of concern is the study of fractional order Sobolev--type metrics on the
group of -diffeomorphism of and on its Sobolev
completions . It is shown that the
-Sobolev metric induces a strong and smooth Riemannian metric on the
Banach manifolds for . As
a consequence a global well-posedness result of the corresponding geodesic
equations, both on the Banach manifold and on
the smooth regular Fr\'echet-Lie group of all -diffeomorphisms is
obtained. In addition a local existence result for the geodesic equation for
metrics of order is derived.Comment: 37 page
Global generalized solutions for Maxwell-alpha and Euler-alpha equations
We study initial-boundary value problems for the Lagrangian averaged alpha
models for the equations of motion for the corotational Maxwell and inviscid
fluids in 2D and 3D. We show existence of (global in time) dissipative
solutions to these problems. We also discuss the idea of dissipative solution
in an abstract Hilbert space framework.Comment: 27 pages, to appear in Nonlinearit
Breakdown for the Camassa-Holm Equation Using Decay Criteria and Persistence in Weighted Spaces
We exhibit a sufficient condition in terms of decay at infinity of the
initial data for the finite time blowup of strong solutions to the
Camassa--Holm equation: a wave breaking will occur as soon as the initial data
decay faster at infinity than the solitons. In the case of data decaying slower
than solitons we provide persistence results for the solution in weighted
-spaces, for a large class of moderate weights. Explicit asymptotic
profiles illustrate the optimality of these results
High-precision molecular dynamics simulation of UO2-PuO2: superionic transition in uranium dioxide
Our series of articles is devoted to high-precision molecular dynamics
simulation of mixed actinide-oxide (MOX) fuel in the rigid ions approximation
using high-performance graphics processors (GPU). In this article we assess the
10 most relevant interatomic sets of pair potential (SPP) by reproduction of
the Bredig superionic phase transition (anion sublattice premelting) in uranium
dioxide. The measurements carried out in a wide temperature range from 300K up
to melting point with 1K accuracy allowed reliable detection of this phase
transition with each SPP. The {\lambda}-peaks obtained are smoother and wider
than it was assumed previously. In addition, for the first time a pressure
dependence of the {\lambda}-peak characteristics was measured, in a range from
-5 GPa to 5 GPa its amplitudes had parabolic plot and temperatures had linear
(that is similar to the Clausius-Clapeyron equation for melting temperature).Comment: 7 pages, 6 figures, 1 tabl
Recommended from our members
Self-Consistent Multiscale Theory of Internal Wave, Mean-Flow Interactions
This is the final report of a three-year, Laboratory Directed Research and Development (LDRD) project at Los Alamos National Laboratory (LANL). The research reported here produced new effective ways to solve multiscale problems in nonlinear fluid dynamics, such as turbulent flow and global ocean circulation. This was accomplished by first developing new methods for averaging over random or rapidly varying phases in nonlinear systems at multiple scales. We then used these methods to derive new equations for analyzing the mean behavior of fluctuation processes coupled self consistently to nonlinear fluid dynamics. This project extends a technology base relevant to a variety of multiscale problems in fluid dynamics of interest to the Laboratory and applies this technology to those problems. The project's theoretical and mathematical developments also help advance our understanding of the scientific principles underlying the control of complex behavior in fluid dynamical systems with strong spatial and temporal internal variability
- …