431 research outputs found
Singularity theory study of overdetermination in models for L-H transitions
Two dynamical models that have been proposed to describe transitions between
low and high confinement states (L-H transitions) in confined plasmas are
analysed using singularity theory and stability theory. It is shown that the
stationary-state bifurcation sets have qualitative properties identical to
standard normal forms for the pitchfork and transcritical bifurcations. The
analysis yields the codimension of the highest-order singularities, from which
we find that the unperturbed systems are overdetermined bifurcation problems
and derive appropriate universal unfoldings. Questions of mutual equivalence
and the character of the state transitions are addressed.Comment: Latex (Revtex) source + 13 small postscript figures. Revised versio
Analytical Study of Certain Magnetohydrodynamic-alpha Models
In this paper we present an analytical study of a subgrid scale turbulence
model of the three-dimensional magnetohydrodynamic (MHD) equations, inspired by
the Navier-Stokes-alpha (also known as the viscous Camassa-Holm equations or
the Lagrangian-averaged Navier-Stokes-alpha model). Specifically, we show the
global well-posedness and regularity of solutions of a certain MHD-alpha model
(which is a particular case of the Lagrangian averaged
magnetohydrodynamic-alpha model without enhancing the dissipation for the
magnetic field). We also introduce other subgrid scale turbulence models,
inspired by the Leray-alpha and the modified Leray-alpha models of turbulence.
Finally, we discuss the relation of the MHD-alpha model to the MHD equations by
proving a convergence theorem, that is, as the length scale alpha tends to
zero, a subsequence of solutions of the MHD-alpha equations converges to a
certain solution (a Leray-Hopf solution) of the three-dimensional MHD
equations.Comment: 26 pages, no figures, will appear in Journal of Math Physics;
corrected typos, updated reference
Uncertainty estimates and L_2 bounds for the Kuramoto-Sivashinsky equation
We consider the Kuramoto-Sivashinsky (KS) equation in one spatial dimension
with periodic boundary conditions. We apply a Lyapunov function argument
similar to the one first introduced by Nicolaenko, Scheurer, and Temam, and
later improved by Collet, Eckmann, Epstein and Stubbe, and Goodman, to prove
that ||u||_2 < C L^1.5. This result is slightly weaker than that recently
announced by Giacomelli and Otto, but applies in the presence of an additional
linear destabilizing term. We further show that for a large class of Lyapunov
functions \phi the exponent 1.5 is the best possible from this line of
argument. Further, this result together with a result of Molinet gives an
improved estimate for L_2 boundedness of the Kuramoto-Sivashinsky equation in
thin rectangular domains in two spatial dimensions.Comment: 17 pages, 1 figure; typos corrected, references added; figure
modifie
Statistical properties of stochastic 2D Navier-Stokes equations from linear models
A new approach to the old-standing problem of the anomaly of the scaling
exponents of nonlinear models of turbulence has been proposed and tested
through numerical simulations. This is achieved by constructing, for any given
nonlinear model, a linear model of passive advection of an auxiliary field
whose anomalous scaling exponents are the same as the scaling exponents of the
nonlinear problem. In this paper, we investigate this conjecture for the 2D
Navier-Stokes equations driven by an additive noise. In order to check this
conjecture, we analyze the coupled system Navier-Stokes/linear advection system
in the unknowns . We introduce a parameter which gives a
system ; this system is studied for any
proving its well posedness and the uniqueness of its invariant measure
.
The key point is that for any the fields and
have the same scaling exponents, by assuming universality of the
scaling exponents to the force. In order to prove the same for the original
fields and , we investigate the limit as , proving that
weakly converges to , where is the only invariant
measure for the joint system for when .Comment: 23 pages; improved versio
Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise
The Primitive Equations are a basic model in the study of large scale Oceanic
and Atmospheric dynamics. These systems form the analytical core of the most
advanced General Circulation Models. For this reason and due to their
challenging nonlinear and anisotropic structure the Primitive Equations have
recently received considerable attention from the mathematical community.
In view of the complex multi-scale nature of the earth's climate system, many
uncertainties appear that should be accounted for in the basic dynamical models
of atmospheric and oceanic processes. In the climate community stochastic
methods have come into extensive use in this connection. For this reason there
has appeared a need to further develop the foundations of nonlinear stochastic
partial differential equations in connection with the Primitive Equations and
more generally.
In this work we study a stochastic version of the Primitive Equations. We
establish the global existence of strong, pathwise solutions for these
equations in dimension 3 for the case of a nonlinear multiplicative noise. The
proof makes use of anisotropic estimates, estimates on the
pressure and stopping time arguments.Comment: To appear in Nonlinearit
Quantum Zakharov Model in a Bounded Domain
We consider an initial boundary value problem for a quantum version of the
Zakharov system arising in plasma physics. We prove the global well-posedness
of this problem in some Sobolev type classes and study properties of solutions.
This result confirms the conclusion recently made in physical literature
concerning the absence of collapse in the quantum Langmuir waves. In the
dissipative case the existence of a finite dimensional global attractor is
established and regularity properties of this attractor are studied. For this
we use the recently developed method of quasi-stability estimates. In the case
when external loads are functions we show that every trajectory from
the attractor is both in time and spatial variables. This can be
interpret as the absence of sharp coherent structures in the limiting dynamics.Comment: 27 page
Optimal Constraint Projection for Hyperbolic Evolution Systems
Techniques are developed for projecting the solutions of symmetric hyperbolic
evolution systems onto the constraint submanifold (the constraint-satisfying
subset of the dynamical field space). These optimal projections map a field
configuration to the ``nearest'' configuration in the constraint submanifold,
where distances between configurations are measured with the natural metric on
the space of dynamical fields. The construction and use of these projections is
illustrated for a new representation of the scalar field equation that exhibits
both bulk and boundary generated constraint violations. Numerical simulations
on a black-hole background show that bulk constraint violations cannot be
controlled by constraint-preserving boundary conditions alone, but are
effectively controlled by constraint projection. Simulations also show that
constraint violations entering through boundaries cannot be controlled by
constraint projection alone, but are controlled by constraint-preserving
boundary conditions. Numerical solutions to the pathological scalar field
system are shown to converge to solutions of a standard representation of the
scalar field equation when constraint projection and constraint-preserving
boundary conditions are used together.Comment: final version with minor changes; 16 pages, 14 figure
Phase Slips and the Eckhaus Instability
We consider the Ginzburg-Landau equation, , with complex amplitude . We first analyze the phenomenon of
phase slips as a consequence of the {\it local} shape of . We next prove a
{\it global} theorem about evolution from an Eckhaus unstable state, all the
way to the limiting stable finite state, for periodic perturbations of Eckhaus
unstable periodic initial data. Equipped with these results, we proceed to
prove the corresponding phenomena for the fourth order Swift-Hohenberg
equation, of which the Ginzburg-Landau equation is the amplitude approximation.
This sheds light on how one should deal with local and global aspects of phase
slips for this and many other similar systems.Comment: 22 pages, Postscript, A
Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent
In this paper the long time behaviour of the solutions of 3-D strongly damped
wave equation is studied. It is shown that the semigroup generated by this
equation possesses a global attractor in H_{0}^{1}(\Omega)\times L_{2}(\Omega)
and then it is proved that this global attractor is a bounded subset of
H^{2}(\Omega)\times H^{2}(\Omega) and also a global attractor in
H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)
Isomerization dynamics of a buckled nanobeam
We analyze the dynamics of a model of a nanobeam under compression. The model
is a two mode truncation of the Euler-Bernoulli beam equation subject to
compressive stress. We consider parameter regimes where the first mode is
unstable and the second mode can be either stable or unstable, and the
remaining modes (neglected) are always stable. Material parameters used
correspond to silicon. The two mode model Hamiltonian is the sum of a
(diagonal) kinetic energy term and a potential energy term. The form of the
potential energy function suggests an analogy with isomerisation reactions in
chemistry. We therefore study the dynamics of the buckled beam using the
conceptual framework established for the theory of isomerisation reactions.
When the second mode is stable the potential energy surface has an index one
saddle and when the second mode is unstable the potential energy surface has an
index two saddle and two index one saddles. Symmetry of the system allows us to
construct a phase space dividing surface between the two "isomers" (buckled
states). The energy range is sufficiently wide that we can treat the effects of
the index one and index two saddles in a unified fashion. We have computed
reactive fluxes, mean gap times and reactant phase space volumes for three
stress values at several different energies. In all cases the phase space
volume swept out by isomerizing trajectories is considerably less than the
reactant density of states, proving that the dynamics is highly nonergodic. The
associated gap time distributions consist of one or more `pulses' of
trajectories. Computation of the reactive flux correlation function shows no
sign of a plateau region; rather, the flux exhibits oscillatory decay,
indicating that, for the 2-mode model in the physical regime considered, a rate
constant for isomerization does not exist.Comment: 42 pages, 6 figure
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