arXiv subjects

Shinar Kouranbaeva

Publications and source records attributed to Shinar Kouranbaeva.

3 recordsLinked to original sources

arXiv1999

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 $Y$ denote the configuration fiber bundle, we show that both the multisymplectic structure on $J^3Y$ 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.

math.DG↗

arXiv1999

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 $α$; one interpretation is that they are obtained by ensemble averaging the Euler equations in Lagrangian representation over rapid fluctuations whose amplitudes are of order $α$. 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 $H^1$ metric instead of the $L^2$ 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}.

chao-dyn↗

arXiv1998

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 $κ=0$ is the geodesic spray of the weak Riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the $H^1$ inner product over the entire group. This paper uses the right-trivialisation technique to rigorously verify that the Euler-Poincaré 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}).

math-ph↗