共查询到20条相似文献,搜索用时 31 毫秒
1.
We deal with consistent first order non-relativistic corrections (i.e. in the small parameter , where c is the speed of light) of the Dirac–Maxwell system. We discuss a selfconsistent modeling of the Pauli equation as the O(ɛ) approximation of the Dirac equation. We suggest a coupling to the “magnetostatic”O(ɛ) approximation of the Maxwell equations consisting of Poisson equations for the four components of the potential. We sketch
the semiclassical/nonrelativistic limits of this model.
(Received 22 May 2000) 相似文献
2.
This paper proposes and analyzes a finite element method for a nonlinear singular elliptic equation arising from the black
hole theory in the general relativity. The nonlinear equation, which was derived and analyzed by Huisken and Ilmanen in (J
Diff Geom 59:353–437), represents a level set formulation for the inverse mean curvature flow describing the evolution of
a hypersurface whose normal velocity equals the reciprocal of its mean curvature. We first propose a finite element method
for a regularized flow which involves a small parameter ɛ; a rigorous analysis is presented to study well-posedness and convergence of the scheme under certain mesh-constraints, and
optimal rates of convergence are verified. We then prove uniform convergence of the finite element solution to the unique
weak solution of the nonlinear singular elliptic equation as the mesh size h and the regularization parameter ɛ both tend to zero. Computational results are provided to show the efficiency of the proposed finite element method and to
numerically validate the “jumping out” phenomenon of the weak solution of the inverse mean curvature flow. Numerical studies
are presented to evidence the existence of a polynomial scaling law between the mesh size h and the regularization parameter ɛ for optimal convergence of the proposed scheme. Finally, a numerical convergence study for another approach recently proposed
by R. Moser (The inverse mean curvature flow and p-harmonic functions. preprint U Bath, 2005) for approximating the inverse mean curvature flow via p-harmonic functions is also included. 相似文献
3.
E. Marušic-Paloka 《Applied Mathematics and Optimization》2001,44(3):245-272
We consider an injection of incompressible viscous fluid in a curved pipe with a smooth central curve γ . The one-dimensional model is obtained via singular perturbation of the Navier—Stokes system as ɛ , the ratio between the cross-section area and the length of the pipe, tends to zero. An asymptotic expansion of the flow
in powers of ɛ is computed. The first term in the expansion depends only on the tangential injection along the central curve γ of the pipe and the velocity as well as the pressure drop are in the tangential direction. The second term contains the
effects of the curvature (flexion) of γ in the direction of the tangent while the effects of torsion appear in the direction of the normal and the binormal to γ . The boundary layers at the ends of the pipe are studied. The error estimate is proved.
Accepted 21 March 2001. Online publication 9 August 2001. 相似文献
4.
Eduard Maru?i?-Paloka Igor Pa?anin 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,24(5):81-99
In this paper we study the flow of incompressible Newtonian fluid through a helical pipe with prescribed pressures at its
ends. Pipe’s thickness and the helix step are considered as the small parameter ɛ. By rigorous asymptotic analysis, as ɛ→
0 , the effective behaviour of the flow is found. The error estimate for the approximation is proved. 相似文献
5.
H.G. Rotstein et al. proposed a nonconserved phase-field system characterized by the presence of memory terms both in the heat conduction and
in the order parameter dynamics. These hereditary effects are represented by time convolution integrals whose relaxation kernels
k and h are nonnegative, smooth and decreasing. Rescaling k and h properly, we obtain a system of coupled partial integrodifferential equations depending on two relaxation times ɛ and σ.
When ɛ and σ tend to 0, the formal limiting system is the well-known nonconserved phase-field model proposed by G. Caginalp.
Assuming the exponential decay of the relaxation kernels, the rescaled system, endowed with homogeneous Neumann boundary conditions,
generates a dissipative strongly continuous semigroup Sɛ, σ(t) on a suitable phase space, which accounts for the past histories of the temperature as well as of the order parameter. Our
main result consists in proving the existence of a family of exponential attractors
for Sɛ, σ(t), with ɛ, σ ∈ [0, 1], whose symmetric Hausdorff distance from
tends to 0 in an explicitly controlled way. 相似文献
6.
F. J. Solis 《Applied Mathematics and Optimization》2000,41(3):331-342
The local adaptive Galerkin bases for large-dimensional dynamical systems, whose long-time behavior is confined to a finite-dimensional
manifold, are optimal bases chosen by a local version of a singular decomposition analysis. These bases are picked out by
choosing directions of maximum bending of the manifold restricted to a ball of radius ɛ . We show their geometrical meaning by analyzing the eigenvalues of a certain self-adjoint operator. The eigenvalues scale
according to the information they carry, the ones that scale as ɛ
2
have a common factor that depends only on the dimension of the manifold, the ones that scale as ɛ
4
give the different curvatures of the manifold, the ones that scale as ɛ
6
give the third invariants, as the torsion for curves, and so on. In this way we obtain a decomposition of phase space into
orthogonal spaces E
m
, where E
m
is spanned by the eigenvectors whose corresponding eigenvalues scale as ɛ
m
. This decomposition is analogous to the Frenet frames for curves. We also discover a practical way to compute the dimension
and local structure of the invariant manifold.
Accepted 14 October 1998 相似文献
7.
We study the semi-classical limit of the dynamic of electrons in a stratified medium. The medium is assumed to be periodic
in one direction and slowly varying in the other directions. In this case, a small parameter ɛ is introduced and corresponds
both to the Planck constant and to scaled lattice thickness. The limit behavior is studied by means of Wigner measures. The
limit process is described by infinitely many transport (Vlasov) equations.
(Received 5 May 1999; in revised from 18 November 1999) 相似文献
8.
The Golovach problem, also known as the ɛ-search problem, is as follows. A team of pursuers pursues an evader on a topological
graph. The objective of the pursuers is to catch the evader, that is, approach the evader to a distance not exceeding a given
nonnegative number ɛ. It is assumed that the evader is invisible to the pursuers and is fully informed beforehand about the
search program of the pursuers. The problem is to find the ɛ-search number, i.e., the least number of pursuers sufficient
for capturing the evader. Graphs with monotone ɛ-search number are studied; the ɛ-search number of a graph G is said to be monotone if it is not exceeded by the ɛ-search numbers of all connected subgraphs H of G. It is known that the ɛ-search number of any tree is monotone for all nonnegative ɛ. The edgesearch number, which is equal
to the 0-search number, is monotone for all connected subgraphs of an arbitrary graph. A sufficient monotonicity condition
for the ɛ-search number of any graph is obtained. This result is improved in the case of complete subgraphs. The Golovach
function is constructed for graphs obtained by removing one edge from complete graphs with unit edges. 相似文献
9.
In this paper we study the asymptotic behavior of solutions u
ɛ of the elliptic variational inequality for the Laplace operator in domains periodically perforated by balls with radius of
size C
0ɛα, C
0 > 0, α = n/n−2, and distributed with period ɛ. On the boundary of balls, we have the following nonlinear restrictions u
ɛ ≥ 0, ∂ν
u
ɛ ≥ −ɛ−ασ(x, u
ɛ), u
ɛ(∂ν
u
ɛ + ɛ−ασ(x, u
ɛ)) = 0. The weak convergence of the solutions u
ɛ to the solution of an effective variational equality is proved. In this case, the effective equation contains a nonlinear
term which has to be determined as solution of a functional equation. Furthermore, a corrector result with respect to the
energy norm is given. 相似文献
10.
I. Kh. Khusnullin 《Computational Mathematics and Mathematical Physics》2010,50(4):646-664
A perturbed two-parameter boundary value problem is considered for a second-order differential operator on an interval with
Dirichlet conditions. The perturbation is described by the potential μ−1
V((x − x
0)ɛ−1), where 0 < ɛ ≪ 1 and μ is an arbitrary parameter such that there exists δ > 0 for which ɛ/μ = o(ɛδ). It is shown that the eigenvalues of this operator converge, as ɛ → 0, to the eigenvalues of the operator with no potential.
Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed. 相似文献
11.
T. V. Dudnikova 《Theoretical and Mathematical Physics》2011,169(3):1668-1682
We study the dynamics of lattice systems in ℤd, d ≥ 1. We assume that the initial data are random functions. We introduce the family of initial measures {μ0ɛ, ɛ > 0}. The measures μ0ɛ are assumed to be locally homogeneous or “slowly changing” under spatial shifts of the order o(ɛ−1
) and inhomogeneous under shifts of the order ɛ−1
. Moreover, correlations of the measures μ0ɛ decrease uniformly in ɛ at large distances. For all τ ∈ ℝ \ 0, r ∈ ℝd, and κ > 0, we consider distributions of a random solution at the instants t = τ/ɛκ at points close to [r/ɛ] ∈ ℤd. Our main goal is to study the asymptotic behavior of these distributions as ɛ → 0 and to derive the limit hydrodynamic equations of the Euler and Navier-Stokes type. 相似文献
12.
Vlad Bally 《Probability Theory and Related Fields》1998,111(3):453-467
Recently the connection between control and game problems and Backward Stochastic Differential Equations has been established.
This allows us to use an approximation scheme for such equations in order to construct an ɛ-optimal control.
Received: 13 November 1995 / Revised version: 11 February 1998 相似文献
13.
Shu-Cheng Chang 《Annals of Global Analysis and Geometry》2006,30(1):37-63
In this paper, we consider a fourth-order gradient flow of the quadratic Riemannian functional ɛ of traceless Ricci curvature on closed 3 -manifolds with a fixed conformal class. We show that the L
2-curvature pinching locally conformally flat 3-manifolds can be deformed to space forms through such gradient flow. More precisely, for the suitable small initial energy functional ɛ, the gradient flow exists for all times and converges smoothly to space forms as the time goes to infinity. As a consequence, we prove the stability for any background metric whose such gradient flow converges to an Einstein metric.Mathematics Subject Classifications (2000): Primary: 53C21; Secondary: 58JOS.Communicated by: Claude LeBrun (Stony Brook) 相似文献
14.
Paul C. Fife 《Annali di Matematica Pura ed Applicata》1971,90(1):99-147
Summary The paper treats elliptic operators of the form L(ɛ∂1, ..., ɛ∂n), where L is a polynomial in a variables of order 2m1, and ɛ is a small parameter. Solutionsu
ɛ of Lu=0 in a half space satisfyng conditions Bj(ɛ∂1, ɛ∂2, ..., ɛ∂n)u=ɛγjϕj(x)(j=1, ..., m1) on the boundary are constructed and estimated using H?lder norms, Poisson kernels, and an elaborate potential theory. Properties
of the interior limit u0=u
ɛ(κ) are studied. The paper is preparatory to a detailed investigation of Schauder estimates for such problems with variables
coefficients.
Supported in part by N. S. F. Grant GP-11660.
Entrata in Redazione il 9 gennaio 1971. 相似文献
15.
We introduce the concept of ɛ-consistent equilibrium where each player plays a ɛ-best response after every history reached
with positive probability. In particular, an ɛ-consistent equilibrium induces an ɛ-equilibrium in any subgame reached along
the play path. The existence of ɛ-consistent equilibrium is examined in various repeated games. The main result is the existence
in stochastic games with absorbing states.
Received January 1995/Revised version October 1996/Final version September 1997 相似文献
16.
In this article, we study the motion of an incompressible homogeneous Newtonian fluid in a rigid porous medium of infinite
extent. The fluid is bounded below by a fixed layer having an external source (with an injection rate b), and above by a free surface moving under the influence of gravity. The flow is governed by Darcy’s law. If b(c) = 0 for some c > 0 then the system admits (u, f) ≡ (c, c) as an equilibrium solution. We shall prove that the stability properties of this equilibrium are determined by the slope
of b in c : The equilibrium is unstable if b′(c) < 0, whereas b′(c) > 0 implies exponential stability.
Zhaoyong Feng: He is grateful to the DFG for financial support through the Graduiertenkolleg 615 “Interaction of Modeling,
Computation Methods and Software Concepts for Scientific-Technological Problems”. 相似文献
17.
Iryna Ryzhkova 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(2):246-261
We discuss the problem of non-linear oscillations of a clamped thermoelastic plate in a subsonic gas flow. The dynamics of
the plate is described by von Kármán system in the presence of thermal effects. No mechanical damping is assumed. To describe
the influence of the gas flow we apply the linearized theory of potential flows. Our main result states that each weak solution
of the problem considered tends to the set of the stationary points of the problem.
A similar problem was considered in [27], but with rotational inertia accounted for, i.e. with the additional term −αΔutt,α > 0, and the same result on stabilization was obtained. There was introduced the decomposition of the solution such that
the one term tends to zero and the other is compact in special (“local energy”) topology. This decomposition enables us to
prove the main result. But the case of rotational inertia neglected (α = 0) appears more difficult. Low a priori smoothness
of ut in the case α = 0 prevents us to construct such a decomposition. In order to prove additional smoothness of ut we use analyticity of the corresponding thermoelastic semigroup proved in [25].
The isothermal variant of this problem with additional mechanical damping term −εΔut
, ε > 0 was considered in [13] and stabilization to the set of stationary solutions to the problem was proved. The problem,
considered in the present work can also be regarded as an extension of the result of [18] to the case when gas occupies an
unbounded domain. 相似文献
18.
Thomas Bartsch Shuangjie Peng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(5):778-804
We study the radially symmetric Schr?dinger equation
with N ≥ 1, ɛ > 0 and p > 1. As ɛ→ 0, we prove the existence of positive radially symmetric solutions concentrating simultaneously on k spheres. The radii are localized near non-degenerate critical points of the function
Supported by the Alexander von Humboldt foundation in Germany and NSFC (No:10571069) in China. 相似文献
19.
We consider the collision dynamics produced by three beads with masses (m
1, m
2, m
3) sliding without friction on a ring, where the masses are scaled so that m
1 = 1/ɛ, m
2 = 1, m
3 = 1 − ɛ, for 0 ⩽ ɛ ⩾ 1. The singular limits ɛ = 0 and ɛ = 1 correspond to two equal mass beads colliding on the ring with a wall, and without a wall respectively. In both these
cases, all solutions are periodic and the eigenvalue distributions (around the unit circle) associated with the products of
collision matrices are discrete. We then numerically examine the regime which parametrically connects these two states, i.e.
0 < ɛ < 1, and show that the eigenvalue distribution is generically uniform around the unit circle, which implies that the dynamics
are no longer periodic. By a sequence of careful numerical experiments, we characterize how the uniform spectrum collapses
from continuous to discrete in the two singular limits ɛ → 0 and ɛ → 1 for an ensemble of initial velocities sampled uniformly on a fixed energy surface. For the limit ɛ → 0, the distribution forms Gaussian peaks around the discrete limiting values ± 1, ± i, with variances that scale in power law form as σ
2 ∼ αɛ
β. By contrast, the convergence in the limit ɛ → 1 to the discrete values ±1 is shown to follow a logarithmic power-law σ
2 ∼ log(ɛ
β). 相似文献