905 research outputs found
Existence of Multiple Vortices in Supersymmetric Gauge Field Theory
Two sharp existence and uniqueness theorems are presented for solutions of
multiple vortices arising in a six-dimensional brane-world supersymmetric gauge
field theory under the general gauge symmetry group and
with Higgs scalar fields in the fundamental representation of .
Specifically, when the space of extra dimension is compact so that vortices are
hosted in a 2-torus of volume |\Om|, the existence of a unique multiple
vortex solution representing respectively prescribed vortices
arising in the species of the Higgs fields is established under the
explicitly stated necessary and sufficient condition \[ n_i<\frac{g^2v^2}{8\pi
N}|\Om|+\frac{1}{N}(1-\frac{1}{N}[\frac{g}{e}]^2)n,\quad i=1,...,N,] where
and are the U(1) electromagnetic and SU(N) chromatic coupling constants,
measures the energy scale of broken symmetry, and is
the total vortex number; when the space of extra dimension is the full plane,
the existence and uniqueness of an arbitrarily prescribed -vortex solution
of finite energy is always ensured. These vortices are governed by a system of
nonlinear elliptic equations, which may be reformulated to allow a variational
structure. Proofs of existence are then developed using the methods of calculus
of variations.Comment: 23 page
Note on Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes equations
X.Y. is partially supported by the Discovery Grant No. RES0020476 from NSERC.In this article we prove new regularity criteria of the Prodi-Serrin-Ladyzhenskaya type for the Cauchy problem of the three-dimensional incompressible Navier-Stokes equations. Our results improve the classical Lr(0,T;Ls) regularity criteria for both velocity and pressure by factors of certain nagative powers of the scaling invariant norms ||u||L3 and ||u||H1/2.PostprintPeer reviewe
Universal estimate of the gradient for parabolic equations
We suggest a modification of the estimate for weighted Sobolev norms of
solutions of parabolic equations such that the matrix of the higher order
coefficients is included into the weight for the gradient. More precisely, we
found the upper limit estimate that can be achieved by variations of the zero
order coefficient. As an example of applications, an asymptotic estimate was
obtained for the gradient at initial time. The constant in the estimates is the
same for all possible choices of the dimension, domain, time horizon, and the
coefficients of the parabolic equation. As an another example of application,
existence and regularity results are obtained for parabolic equations with time
delay for the gradient.Comment: 15 page
Thermoacoustic tomography with an arbitrary elliptic operator
Thermoacoustic tomography is a term for the inverse problem of determining of
one of initial conditions of a hyperbolic equation from boundary measurements.
In the past publications both stability estimates and convergent numerical
methods for this problem were obtained only under some restrictive conditions
imposed on the principal part of the elliptic operator. In this paper
logarithmic stability estimates are obatined for an arbitrary variable
principal part of that operator. Convergence of the Quasi-Reversibility Method
to the exact solution is also established for this case. Both complete and
incomplete data collection cases are considered.Comment: 16 page
Asymptotic expansion of the solution of the steady Stokes equation with variable viscosity in a two-dimensional tube structure
The Stokes equation with the varying viscosity is considered in a thin tube
structure, i.e. in a connected union of thin rectangles with heights of order
and with bases of order 1 with smoothened boundary. An
asymptotic expansion of the solution is constructed: it contains some
Poiseuille type flows in the channels (rectangles) with some boundary layers
correctors in the neighborhoods of the bifurcations of the channels. The
estimates for the difference of the exact solution and its asymptotic
approximation are proved.Comment: 22 pages, 20 figure
The Construction of a Partially Regular Solution to the Landau-Lifshitz-Gilbert Equation in
We establish a framework to construct a global solution in the space of
finite energy to a general form of the Landau-Lifshitz-Gilbert equation in
. Our characterization yields a partially regular solution,
smooth away from a 2-dimensional locally finite Hausdorff measure set. This
construction relies on approximation by discretization, using the special
geometry to express an equivalent system whose highest order terms are linear
and the translation of the machinery of linear estimates on the fundamental
solution from the continuous setting into the discrete setting. This method is
quite general and accommodates more general geometries involving targets that
are compact smooth hypersurfaces.Comment: 43 pages, 2 figure
Navier-Stokes equations on the flat cylinder with vorticity production on the boundary
We study the two-dimensional Navier-Stokes system on a flat cylinder with the
usual Dirichlet boundary conditions for the velocity field u. We formulate the
problem as an infinite system of ODE's for the natural Fourier components of
the vorticity, and the boundary conditions are taken into account by adding a
vorticity production at the boundary. We prove equivalence to the original
Navier-Stokes system and show that the decay of the Fourier modes is
exponential for any positive time in the periodic direction, but it is only
power-like in the other direction.Comment: 25 page
Stochastic Interactions of Two Brownian Hard Spheres in the Presence of Depletants
A quantitative analysis is presented for the stochastic interactions of a
pair of Brownian hard spheres in non-adsorbing polymer solutions. The hard
spheres are hypothetically trapped by optical tweezers and allowed for random
motion near the trapped positions. The investigation focuses on the long-time
correlated Brownian motion. The mobility tensor altered by the polymer
depletion effect is computed by the boundary integral method, and the
corresponding random displacement is determined by the fluctuation-dissipation
theorem. From our computations it follows that the presence of depletion layers
around the hard spheres has a significant effect on the hydrodynamic
interactions and particle dynamics as compared to pure solvent and pure polymer
solution (no depletion) cases. The probability distribution functions of random
walks of the two interacting hard spheres that are trapped clearly shifts due
to the polymer depletion effect. The results show that the reduction of the
viscosity in the depletion layers around the spheres and the entropic force due
to the overlapping of depletion zones have a significant influence on the
correlated Brownian interactions.Comment: 30 pages, 9 figures, 1 appendix, 40 formulas inside the text, 5
formulas in appendi
Homogenization of the planar waveguide with frequently alternating boundary conditions
We consider Laplacian in a planar strip with Dirichlet boundary condition on
the upper boundary and with frequent alternation boundary condition on the
lower boundary. The alternation is introduced by the periodic partition of the
boundary into small segments on which Dirichlet and Neumann conditions are
imposed in turns. We show that under the certain condition the homogenized
operator is the Dirichlet Laplacian and prove the uniform resolvent
convergence. The spectrum of the perturbed operator consists of its essential
part only and has a band structure. We construct the leading terms of the
asymptotic expansions for the first band functions. We also construct the
complete asymptotic expansion for the bottom of the spectrum
Estimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number
The tradition in Navier-Stokes analysis of finding estimates in terms of the
Grashof number \bG, whose character depends on the ratio of the forcing to
the viscosity , means that it is difficult to make comparisons with other
results expressed in terms of Reynolds number \Rey, whose character depends
on the fluid response to the forcing. The first task of this paper is to apply
the approach of Doering and Foias \cite{DF} to the two-dimensional
Navier-Stokes equations on a periodic domain by estimating
quantities of physical relevance, particularly long-time averages
\left, in terms of the Reynolds number \Rey = U\ell/\nu, where
U^{2}= L^{-2}\left and is the forcing scale. In
particular, the Constantin-Foias-Temam upper bound \cite{CFT} on the attractor
dimension converts to a_{\ell}^{2}\Rey(1 + \ln\Rey)^{1/3}, while the estimate
for the inverse Kraichnan length is (a_{\ell}^{2}\Rey)^{1/2}, where
is the aspect ratio of the forcing. Other inverse length scales,
based on time averages, and associated with higher derivatives, are estimated
in a similar manner. The second task is to address the issue of intermittency :
it is shown how the time axis is broken up into very short intervals on which
various quantities have lower bounds, larger than long time-averages, which are
themselves interspersed by longer, more quiescent, intervals of time.Comment: 21 pages, 1 figure, accepted for publication from J. Math. Phys. for
the special issue on mathematical fluid mechanic
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