651 research outputs found
A Note on the Regularity of Inviscid Shell Model of Turbulence
In this paper we continue the analytical study of the sabra shell model of
energy turbulent cascade initiated in \cite{CLT05}. We prove the global
existence of weak solutions of the inviscid sabra shell model, and show that
these solutions are unique for some short interval of time. In addition, we
prove that the solutions conserve the energy, provided that the components of
the solution satisfy , for
some positive absolute constant , which is the analogue of the Onsager's
conjecture for the Euler's equations. Moreover, we give a Beal-Kato-Majda type
criterion for the blow-up of solutions of the inviscid sabra shell model and
show the global regularity of the solutions in the ``two-dimensional''
parameters regime
Coulomb plasmas in outer envelopes of neutron stars
Outer envelopes of neutron stars consist mostly of fully ionized, strongly
coupled Coulomb plasmas characterized by typical densities about 10^4-10^{11}
g/cc and temperatures about 10^4-10^9 K. Many neutron stars possess magnetic
fields about 10^{11}-10^{14} G. Here we briefly review recent theoretical
advances which allow one to calculate thermodynamic functions and electron
transport coefficients for such plasmas with an accuracy required for
theoretical interpretation of observations.Comment: 4 pages, 2 figures, latex2e using cpp2e.cls (included). Proc. PNP-10
Workshop, Greifswald, Germany, 4-9 Sept. 2000. Accepted for publication in
Contrib. Plasma Phys. 41 (2001) no. 2-
Stochastic attractors for shell phenomenological models of turbulence
Recently, it has been proposed that the Navier-Stokes equations and a
relevant linear advection model have the same long-time statistical properties,
in particular, they have the same scaling exponents of their structure
functions. This assertion has been investigate rigorously in the context of
certain nonlinear deterministic phenomenological shell model, the Sabra shell
model, of turbulence and its corresponding linear advection counterpart model.
This relationship has been established through a "homotopy-like" coefficient
which bridges continuously between the two systems. That is, for
one obtains the full nonlinear model, and the corresponding linear
advection model is achieved for . In this paper, we investigate the
validity of this assertion for certain stochastic phenomenological shell models
of turbulence driven by an additive noise. We prove the continuous dependence
of the solutions with respect to the parameter . Moreover, we show the
existence of a finite-dimensional random attractor for each value of
and establish the upper semicontinuity property of this random attractors, with
respect to the parameter . This property is proved by a pathwise
argument. Our study aims toward the development of basic results and techniques
that may contribute to the understanding of the relation between the long-time
statistical properties of the nonlinear and linear models
Baxter T-Q Equation for Shape Invariant Potentials. The Finite-Gap Potentials Case
The Darboux transformation applied recurrently on a Schroedinger operator
generates what is called a {\em dressing chain}, or from a different point of
view, a set of supersymmetric shape invariant potentials. The finite-gap
potential theory is a special case of the chain. For the finite-gap case, the
equations of the chain can be expressed as a time evolution of a Hamiltonian
system. We apply Sklyanin's method of separation of variables to the chain. We
show that the classical equation of the separation of variables is the Baxter
T-Q relation after quantization.Comment: 25 pages, no figures Extended section 10, one reference added.
Version accepted for publication in Jurnal of Mathematical Physic
Vorticity alignment results for the three-dimensional Euler and Navier-Stokes equations
We address the problem in Navier-Stokes isotropic turbulence of why the
vorticity accumulates on thin sets such as quasi-one-dimensional tubes and
quasi-two-dimensional sheets. Taking our motivation from the work of Ashurst,
Kerstein, Kerr and Gibson, who observed that the vorticity vector
{\boldmath\omega} aligns with the intermediate eigenvector of the strain
matrix , we study this problem in the context of both the three-dimensional
Euler and Navier-Stokes equations using the variables \alpha =
\hat{{\boldmath\xi}}\cdot S\hat{{\boldmath\xi}} and {\boldmath\chi} =
\hat{{\boldmath\xi}}\times S\hat{{\boldmath\xi}} where
\hat{{\boldmath\xi}} = {\boldmath\omega}/\omega. This introduces the
dynamic angle , which lies between
{\boldmath\omega} and S{\boldmath\omega}. For the Euler equations a
closed set of differential equations for and {\boldmath\chi} is
derived in terms of the Hessian matrix of the pressure . For
the Navier-Stokes equations, the Burgers vortex and shear layer solutions turn
out to be the Lagrangian fixed point solutions of the equivalent
(\alpha,{\boldmath\chi}) equations with a corresponding angle .
Under certain assumptions for more general flows it is shown that there is an
attracting fixed point of the (\alpha,\bchi) equations which corresponds to
positive vortex stretching and for which the cosine of the corresponding angle
is close to unity. This indicates that near alignment is an attracting state of
the system and is consistent with the formation of Burgers-like structures.Comment: To appear in Nonlinearity Nov. 199
Pores in Bilayer Membranes of Amphiphilic Molecules: Coarse-Grained Molecular Dynamics Simulations Compared with Simple Mesoscopic Models
We investigate pores in fluid membranes by molecular dynamics simulations of
an amphiphile-solvent mixture, using a molecular coarse-grained model. The
amphiphilic membranes self-assemble into a lamellar stack of amphiphilic
bilayers separated by solvent layers. We focus on the particular case of
tension less membranes, in which pores spontaneously appear because of thermal
fluctuations. Their spatial distribution is similar to that of a random set of
repulsive hard discs. The size and shape distribution of individual pores can
be described satisfactorily by a simple mesoscopic model, which accounts only
for a pore independent core energy and a line tension penalty at the pore
edges. In particular, the pores are not circular: their shapes are fractal and
have the same characteristics as those of two dimensional ring polymers.
Finally, we study the size-fluctuation dynamics of the pores, and compare the
time evolution of their contour length to a random walk in a linear potential
Entire solutions of hydrodynamical equations with exponential dissipation
We consider a modification of the three-dimensional Navier--Stokes equations
and other hydrodynamical evolution equations with space-periodic initial
conditions in which the usual Laplacian of the dissipation operator is replaced
by an operator whose Fourier symbol grows exponentially as \ue ^{|k|/\kd} at
high wavenumbers . Using estimates in suitable classes of analytic
functions, we show that the solutions with initially finite energy become
immediately entire in the space variables and that the Fourier coefficients
decay faster than \ue ^{-C(k/\kd) \ln (|k|/\kd)} for any . The
same result holds for the one-dimensional Burgers equation with exponential
dissipation but can be improved: heuristic arguments and very precise
simulations, analyzed by the method of asymptotic extrapolation of van der
Hoeven, indicate that the leading-order asymptotics is precisely of the above
form with . The same behavior with a universal constant
is conjectured for the Navier--Stokes equations with exponential
dissipation in any space dimension. This universality prevents the strong
growth of intermittency in the far dissipation range which is obtained for
ordinary Navier--Stokes turbulence. Possible applications to improved spectral
simulations are briefly discussed.Comment: 29 pages, 3 figures, Comm. Math. Phys., in pres
Wetting and Minimal Surfaces
We study minimal surfaces which arise in wetting and capillarity phenomena.
Using conformal coordinates, we reduce the problem to a set of coupled boundary
equations for the contact line of the fluid surface, and then derive simple
diagrammatic rules to calculate the non-linear corrections to the Joanny-de
Gennes energy. We argue that perturbation theory is quasi-local, i.e. that all
geometric length scales of the fluid container decouple from the
short-wavelength deformations of the contact line. This is illustrated by a
calculation of the linearized interaction between contact lines on two opposite
parallel walls. We present a simple algorithm to compute the minimal surface
and its energy based on these ideas. We also point out the intriguing
singularities that arise in the Legendre transformation from the pure Dirichlet
to the mixed Dirichlet-Neumann problem.Comment: 22 page
Finite time singularities in a class of hydrodynamic models
Models of inviscid incompressible fluid are considered, with the kinetic
energy (i.e., the Lagrangian functional) taking the form in 3D Fourier representation, where
is a constant, . Unlike the case (the usual Eulerian
hydrodynamics), a finite value of results in a finite energy for a
singular, frozen-in vortex filament. This property allows us to study the
dynamics of such filaments without the necessity of a regularization procedure
for short length scales. The linear analysis of small symmetrical deviations
from a stationary solution is performed for a pair of anti-parallel vortex
filaments and an analog of the Crow instability is found at small wave-numbers.
A local approximate Hamiltonian is obtained for the nonlinear long-scale
dynamics of this system. Self-similar solutions of the corresponding equations
are found analytically. They describe the formation of a finite time
singularity, with all length scales decreasing like ,
where is the singularity time.Comment: LaTeX, 17 pages, 3 eps figures. This version is close to the journal
pape
Turbulence and passive scalar transport in a free-slip surface
We consider the two-dimensional (2D) flow in a flat free-slip surface that
bounds a three-dimensional (3D) volume in which the flow is turbulent. The
equations of motion for the two-dimensional flow in the surface are neither
compressible nor incompressible but strongly influenced by the 3D flow
underneath the surface. The velocity correlation functions in the 2D surface
and in the 3D volume scale with the same exponents. In the viscous subrange the
amplitudes are the same, but in the inertial subrange the 2D one is reduced to
2/3 of the 3D amplitude. The surface flow is more strongly intermittent than
the 3D volume flow. Geometric scaling theory is used to derive a relation
between the scaling of the velocity field and the density fluctuations of a
passive scalar advected on the surface.Comment: 11 pages, 10 Postscript figure
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