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1.
We consider a mixed boundary-value problem for a Poisson equation in a plane two-level junction Ωε that is the union of a domain Ω0 and a large number 3N of thin rods with thickness of order
. The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The homogeneous Dirichlet conditions and inhomogeneous Neumann conditions are given on the sides
of the thin rods from the first level and the second level, respectively. Using the method of matched asymptotic expansions
and special junction-layer solutions, we construct an asymptotic approximation for the solution and prove the corresponding
estimates in the Sobolev space H
1(Ωε) as ε → 0 (N → +∞).
Published in Neliniini Kolyvannya, Vol. 9, No. 3, pp. 336–355, July–September, 2006. 相似文献
2.
Adriano Montanaro 《Archive for Rational Mechanics and Analysis》1998,143(4):375-400
We consider equilibria arising in a model for phase transitions which correspond to stable critical points of the constrained
variational problem Here W is a double‐well potential and is a strictly convex domain. For ε small, this is closely related to the problem of partitioning Ω into two subdomains of fixed volume, where the subdomain boundaries correspond to the transitional boundary between phases.
Motivated by this geometry problem, we show that in a strictly convex domain, stable critical points of the original variational
problem have a connected, thin transition layer separating the two phases. This relates to work in [GM] where special geometries
such as cylindrical domains were treated, and is analogous to the results in [CHo] which show that
in a convex domain, stable critical points of the corresponding unconstrained problem are constant. The proof of connectivity employs tools from geometric measure theory including the co‐area formula
and the isoperimetric inequality on manifolds. The thinness of the transition layer follows from a separate calculation establishing
spatial decay of solutions to the pure phases.
(Accepted July 15, 1996) 相似文献
3.
Marc Briane 《Archive for Rational Mechanics and Analysis》2006,182(2):255-267
The paper deals with the asymptotic behaviour as ε → 0 of a two-dimensional conduction problem whose matrix-valued conductivity a
ε
is ε-periodic and not uniformly bounded with respect to ε. We prove that only under the assumptions of equi-coerciveness and L
1-boundedness of the sequence a
ε
, the limit problem is a conduction problem of same nature. This new result points out a fundamental difference between the
two-dimensional conductivity and the three-dimensional one. Indeed, under the same assumptions of periodicity, equi-coerciveness
and L
1-boundedness, it is known that the high-conductivity regions can induce nonlocal effects in three (or greater) dimensions. 相似文献
4.
We consider a mixed boundary-value problem for the Poisson equation in a plane thick junction Ωε that is the union of a domain Ω0 and a large number of ε-periodically located thin rods. The nonuniform Signorini conditions are given on the vertical sides
of the thin rods. The asymptotic analysis of this problem is made as ε → 0, i.e., in the case where the number of thin rods
infinitely increases and their thickness tends to zero. With the help of the integral identity method, we prove the convergence
theorem and show that the nonuniform Signorini conditions are transformed (as ε → 0) into the limiting variational inequalities
in the domain that is filled up with thin rods when passing to the limit. The existence and uniqueness of a solution to this
nonstandard limit problem are established. The convergence of the energy integrals is proved as well.
Published in Neliniini Kolyvannya, Vol. 12, No. 1, pp. 44–58, January–March, 2009. 相似文献
5.
Yoshihiro Ueda Tohru Nakamura Shuichi Kawashima 《Archive for Rational Mechanics and Analysis》2010,198(3):735-762
This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space.
We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection.
In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t
−α/4 as t → ∞, provided that the initial perturbation is in the weighted space
L2a=L2(\mathbb R+; (1+x)a dx){L^2_\alpha=L^2({\mathbb R}_+;\,(1+x)^\alpha dx)} . This convergence rate t
−α/4 is weaker than the one for the non-degenerate case and requires the restriction α < α*(q), where α*(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative
in L2a{L^2_\alpha} for α > α*(q) with another critical value α*(q). Our stability analysis is based on the space–time weighted energy method in which the spatial weight is chosen as a function
of the degenerate stationary wave. 相似文献
6.
M. J. Giles 《Applied Scientific Research》1996,57(3-4):223-234
The exact expression for the probability distribution function (pdf),P(Δur), of a velocity difference Δur, over a distancer, in incompressible fluid turbulence, obtained from the Navier-Stokes equations, is used as a basis for deriving approximate
profiles forP(Δur). These approximate forms are deduced from an approximate factorisation of the underlying functional probability distribution
of the flow field, in which the individual factors capture different physical effects.P(Δur) is represented as the integral, with respect to the spatially averaged dissipation rateε
r, of the product of the conditionalpdf of Δur givenε
r, and thepdf ofε
r. The approximation yields the latter as a log-Poissonpdf, while the conditionalpdf is found to be a Gaussian for a transverse increment, and the product of a Gaussian and a cubic polynomial for a longitudinal
increment. This approximation is equivalent to the refined similarity hypothesis coupled with the log-Poisson distribution,
and it possesses the characteristic features ofP(Δur), including symmetric profiles for transverse increments, asymmetric profiles for longitudinal increments, and the development
of pronounced non-Gaussian features at small separations. The associated scaling exponents for longitudinal and transverse
structure functions are shown to be identical, in this approximation, and to assume the log-Poisson form. 相似文献
7.
IntroductionLetN ∈Rm0 ×n(m0 ≥n)beaverticalblockmatrixoftype(m1,… ,mn) ,q∈Rm0 beaconstantvectorpartitionedconformablywithN ,thatareN =N1Nn, q =q1qn, m0 = ni=1mi,whereNi ∈Rmi×n,qi ∈Rmi,i ∈I =1 ,… ,n .WeconsidertheverticallinearomplementarityproblemVLCP(N ,q)associatedwith (N ,q)offindingavectorx∈Rnsuchthatx≥ 0 , si(x) :=Nix+qi ≥ 0 , xi∏mij=1sij(x) =0 (i∈I) ,wherexi,sij(x)denotetheithcomponentofxandthejthcomponentofsi(x) ,respectively ;∏ mj=1ajdenotesa1…am.Iti… 相似文献
8.
A fully developed laminar Poiseuille flow subject to constant heat flux across the wall is analysed with respect to its stability
behavior by applying a weakly nonlinear stability theory. It is based on an expansion of the disturbance control equations
with respect to a perturbation parameter ε. This parameter is the small initial amplitude of the fundamental wave. This fundamental
wave which is the solution of the linear (Orr-Sommerfeld) first order equation triggers all higher order effects with respect
to ε. Heat transfer is accounted for asymptotically through an expansion with respect to a small heat transfer parameter ε
T
. Both perturbation parameters, ε and ε
T
, are linked by the assumption ε
T
=O(ε2) by which a certain distinguished limit is assumed. The results for a fluid with temperature dependent viscosity show that
heat transfer effects in the nonlinear range continue to act in the same way as in the initial linear range.
Received on 11 August 1997 相似文献
9.
The existence and linear stability problem for the Stokes periodic wavetrain on fluids of finite depth is formulated in terms
of the spatial and temporal Hamiltonian structure of the water-wave problem. A proof, within the Hamiltonian framework, of
instability of the Stokes periodic wavetrain is presented. A Hamiltonian center-manifold analysis reduces the linear stability
problem to an ordinary differential eigenvalue problem on ℝ4. A projection of the reduced stability problem onto the tangent space of the 2-manifold of periodic Stokes waves is used
to prove the existence of a dispersion relation Λ(λ,σ, I
1, I
2)=0 where λ ε ℂ is the stability exponent for the Stokes wave with amplitude I
1 and mass flux I
2 and σ is the “sideband’ or spatial exponent. A rigorous analysis of the dispersion relation proves the result, first discovered
in the 1960's, that the Stokes gravity wavetrain of sufficiently small amplitude is unstable for F ε (0,F0) where F
0 ≈ 0.8 and F is the Froude number. 相似文献
10.
TheT
ε
*
integral was calculated on the surface of single edge notched, three-point bend (SE(B)) specimens using experimentally obtained
displacements. Comparison was made withT
ε
*
calculated with the measured surface displacements andT
ε
*
calculated at several points through the thickness of a finite element (FE) model of the SE(B) specimen. Good comparison
was found between the surfaceT
ε
*
calculated from displacements extracted from the FE model and the surfaceT
ε
*
calculated from experimentally obtained displacements. The computedT
ε
*
integral was also observed to decrease as the crack front was traversed from the surface to the mid-plane of the specimen.
Mid-planeT
ε
*
values tend to be approximately 10% of the surface values. 相似文献
11.
H. Irago 《Journal of Elasticity》1999,57(1):55-83
Let u(ε) be a rescaled 3-dimensional displacement field solution of the linear elastic model for a free prismatic rod Ωε having cross section with diameter of order ε, and let u
(0) –Bernoulli–Navier displacement – and u
(2) be the two first terms derived from the asymptotic method. We analyze the residue r(ε) = u(ε) − (u
(0) + ε2
u
(2)) and if the cross section is star-shaped, we prove such residue presents a Saint-Venant"s phenomenon near the ends of the
rod.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
12.
Hansjörg Kielhöfer 《Archive for Rational Mechanics and Analysis》2000,155(4):261-276
We study the Cahn-Hilliard energy E
ɛ(u) over the unit square under the constraint of a constant mass m with (ɛ > 0) and without ɛ= 0) interfacial energy. Minimizers of E
0(u) have no preferred pattern and we select patterns via sequences of conditionally critical points of E
ɛ(u) converging to minimizers as ɛ tends to zero. Those critical points are not minimizers if the singular limit has no minimal
interface. We obtain them by a global bifurcation analysis of the Euler-Lagrange equations for E
ɛ(u) where the mass m is the bifurcation parameter. We make use of the symmetry of the unit square, and the elliptic maximum principle, in turn,
implies that the location of maxima and minima is fixed for all solutions on global branches. This property is used to guarantee
the existence
of a singular limit and to verify the Weierstrass-Erdmann corner condition which proves its minimizing property.
Accepted January 21, 2000?Published online November 24, 2000 相似文献
13.
Giovanni Alberti Guy Bouchitté Pierre Seppecher 《Archive for Rational Mechanics and Analysis》1998,144(1):1-46
We make the connection between the geometric model for capillarity with line tension and the Cahn‐Hilliard model of two‐phase
fluids. To this aim we consider the energies where u is a scalar density function and W and V are double‐well potentials. We show that the behaviour of F
ε
in the limit ε→0 and λ→∞ depends on the limit of ε log λ. If this limit is finite and strictly positive, then the singular limit of the energies F
ε
leads to a coupled problem of bulk and surface phase transitions, and under certain assumptions agrees with the relaxation
of the capillary energy with line tension. These results were announced in [ABS1] and [ABS2].
(Accepted November 5, 1997) 相似文献
14.
H. Watanabe Tomohiro Sato Motoyuki Hirose Kunihiro Osaki Ming-Long Yao 《Rheologica Acta》1998,37(6):519-527
Dielectric relaxation behavior was examined for 4-4′-n-pentyl-cyanobiphenyl (5CB) and 4-4′-n-heptyl-cyanobiphenyl (7CB) under flow. In quiescent states at all temperatures examined, both 5CB and 7CB exhibited dispersions
in their complex dielectric constant ε*(ω) at characteristic frequencies ω
c
above 106 rad s–1. This dispersion reflected orientational fluctuation of individual 5CB and 7CB molecules having large dipoles parallel to
their principal axis (in the direction of C≡N bond). In the isotropic state at high temperatures, these molecules exhibited no detectable changes of ε*(ω) under flow at
shear rates . In contrast, in the nematic state at lower temperatures the terminal relaxation intensity of ε*(ω) as well as the static
dielectric constant ε′(0) decreased under flow at . This rheo-dielectric change was discussed in relation to the flow effects on the nematic texture (director distribution)
and anisotropy in motion of individual molecules with respect to the director.
Received: 14 April 1998 Accepted: 29 July 1998 相似文献
15.
For elliptic equations ε2Δu − V(x) u + f(u) = 0, x ∈ R
N
, N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component
of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal.
An erratum to this article can be found at 相似文献
16.
This paper deals with the asymptotic analysis of the three-dimensional problem for a linearly elastic cantilever having an
open cross-section which is the union of rectangles with sides of order ε and ε
2, as ε goes to zero. Under suitable assumptions on the given loads and for homogeneous and isotropic material, we show that the
three-dimensional problem Γ-converges to the classical one-dimensional Vlassov model for thin-walled beams.
相似文献
17.
c ). To begin with, we assume that the flux-function f(u) is piecewise genuinely nonlinear, in the sense that it exhibits finitely many (at most p, say) points of lack of genuine nonlinearity along each wave curve. Importantly, our analysis applies to arbitrary large p, in the sense that the constant c restricting the total variation is independent of p. Second, by an approximation argument, we prove that the existence theory above extends to general flux-functions f(u) that can be approached by a sequence of piecewise genuinely nonlinear flux-functions f
ε(u).
The main contribution in this paper is the derivation of uniform estimates for the wave curves and wave interactions (which are entirely independent of the properties of the flux-function) together
with a new wave interaction potential which is decreasing in time and is a fully local functional depending upon the angle
made by any two propagating discontinuities. Our existence theory applies, for instance, to the p-system of gas dynamics for general pressure-laws p=p(v) satisfying solely the hyperbolicity condition p′(v)<0 but no convexity assumption.
(Accepted December 30, 2002)
Published online April 23, 2003
Communicated by C. M. Dafermos 相似文献
18.
Xiao-Biao Lin 《Journal of Dynamics and Differential Equations》2006,18(1):1-52
The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u = u
ε(X/T). In Lin and Schecter (2003, SIAM J. Math. Anal. 35, 884–921) it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/ε and slow eigenvalues of order one. The fast eigenvalues represent motion in an initial time layer, where near the shock waves solutions quickly converge to traveling-wave-like motion. The slow eigenvalues represent motion after the initial time layer, where motion between the shock waves is dominant. In this paper we use tools from dynamical systems and singular perturbation theory to study the slow eigenvalues. We show how to construct asymptotic expansions of eigenvalue-eigenfunction pairs to any order in ε. We also prove the existence of true eigenvalue-eigenfunction pairs near the asymptotic expansions. 相似文献
19.
Insights into the Relationships Among Capillary Pressure,Saturation, Interfacial Area and Relative Permeability Using Pore-Network Modeling 总被引:1,自引:0,他引:1
To gain insight in relationships among capillary pressure, interfacial area, saturation, and relative permeability in two-phase
flow in porous media, we have developed two types of pore-network models. The first one, called tube model, has only one element
type, namely pore throats. The second one is a sphere-and-tube model with both pore bodies and pore throats. We have shown
that the two models produce distinctly different curves for capillary pressure and relative permeability. In particular, we
find that the tube model cannot reproduce hysteresis. We have investigated some basic issues such as effect of network size,
network dimension, and different trapping assumptions in the two networks. We have also obtained curves of fluid–fluid interfacial
area versus saturation. We show that the trend of relationship between interfacial area and saturation is largely influenced
by trapping assumptions. Through simulating primary and scanning drainage and imbibition cycles, we have generated two surfaces
fitted to capillary pressure, saturation, and interfacial area (P
c
–S
w
–a
nw
) points as well as to relative permeability, saturation, and interfacial area (k
r
–S
w
–a
nw
) points. The two fitted three-dimensional surfaces show very good correlation with the data points. We have fitted two different
surfaces to P
c
–S
w
–a
nw
points for drainage and imbibition separately. The two surfaces do not completely coincide. But, their mean absolute difference
decreases with increasing overlap in the statistical distributions of pore bodies and pore throats. We have shown that interfacial
area can be considered as an essential variable for diminishing or eliminating the hysteresis observed in capillary pressure–saturation
(P
c
–S
w
) and the relative permeability–saturation (k
r
–S
w
) curves. 相似文献
20.
Chou Huan-wen 《应用数学和力学(英文版)》1983,4(6):855-863
In this paper,the method of composite expansions which was proposed by W. Z. Chien (1948)[5]is extended to investigate two-parameter boundary layer problems.For the problems of symmetric deformations of the spherical shells under the action of uniformly distribution load q, its nonlinear equilibrium equations can be written as follows: where ε and δ are undetermined parameters.If δ=1 and ε is a small parameter, the above-mentioned problem is called first boundary layer problem; if ε is a small parameter, and δ is a small parameter, too, the above-mentioned problem is called second boundary layer problem.For the above-mentioned problems, however, we assume that the constants ε, δ and p satisfy the following equation: εp=1-ε In defining this condition by using the extended method of composite expansions, we find the asymptotic solution of the above-mentioned problems with the clamped boundary conditions. 相似文献