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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.
We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ω ɛ that is the union of a domain Ω0 and a large number 2N of thin rods with thickness of order ɛ =
(N
−1). Depending on their lengths, the thin rods are divided into two levels. In addition, the rods from each level are ɛ-periodically
alternated. Inhomogeneous Neumann boundary conditions are given on the vertical sides of the thin rods of the first level,
and homogeneous Dirichlet boundary conditions are given on the vertical sides of the rods of the second level. We investigate
the asymptotic behavior of a solution of this problem as ɛ → 0 and prove a convergence theorem and the convergence of the
energy integral.
__________
Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 241–257, April–June, 2005. 相似文献
3.
We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction that is the union of a domain 0 and a large number 2N of thin rods with variable thickness of order =
(N
–1). The thin rods are divided into two levels, depending on their length. In addition, the thin rods from each level are -periodically alternated. We investigate the asymptotic behavior of the solution as 0 under the Robin conditions on the boundaries of the thin rods. By using some special extension operators, a convergence theorem is proved.Published in Neliniini Kolyvannya, Vol. 7, No. 3, pp. 336–355, July–September, 2004. 相似文献
4.
We consider the Navier–Stokes equations in the thin 3D domain , where is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly,
when ε ≪ 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges
(as ε → 0) to a unique stationary measure for the Navier–Stokes equation on . Thus, the 2D Navier–Stokes equations on surfaces describe asymptotic in time, and limiting in ε, statistical properties of 3D solutions in thin 3D domains. 相似文献
5.
Luan Thach Hoang 《Journal of Mathematical Fluid Mechanics》2010,12(3):435-472
This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier–Stokes equations in a three-dimensional
domain Ωɛ, whose thickness is of order O(ɛ) as ɛ → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the
bottom and top boundaries of Ωɛ, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ɛ3/4) as ɛ → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H1(Ωɛ), respectively, L2(Ωɛ), then the strong solution of the Navier–Stokes equations exists for all time. Our proofs rely on the study of the dependence
of the Stokes operator on ɛ, and the non-linear estimate in which the contributions of the boundary integrals are non-trivial. 相似文献
6.
Markus Lilli 《Journal of Elasticity》2007,87(1):73-94
We consider a non-convex variational problem (P) and the corresponding singular perturbed problem (P
ε
). The qualitative behavior of stable critical points of (P
ε
) depending on ε and a lower order term is discussed and we prove compactness of a sequence of stable critical points as ε ↘ 0. Moreover we show whether this limit is the global minimizer of (P). Furthermore uniform convergence is considered as well as the convergence rate depending on ε.
相似文献
7.
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
8.
Duvan Henao 《Journal of Elasticity》2009,94(1):55-68
We prove that energy minimizers for nonlinear elasticity in which cavitation is allowed only at a finite number of prescribed
flaw points can be obtained, in the limit as ε→0, by introducing micro-voids of radius ε in the domain at the prescribed locations and minimizing the energy without allowing for cavitation. This extends the result
by Sivaloganathan, Spector, and Tilakraj (SIAM J. Appl. Math. 66:736–757, 2006) to the case of multiple cavities, and constitutes a first step towards the numerical simulation of cavitation (in the nonradially-symmetric
case).
相似文献
9.
We develop a scheme for the investigation of the asymptotic behavior of eigenvalues and eigenvectors of a family of self-adjoint compact operators {A: > 0} that act in different spaces
and lose their compactness in the limit case 0. We prove the Hausdorff convergence of the spectrum of the operator A to the spectrum of the limit operator A0, obtain asymptotic estimates for this convergence both to points of the discrete spectrum and to points of the essential spectrum of the operator A0, and prove asymptotic estimates for eigenvectors of A. This scheme is applied to the investigation of the asymptotic behavior of eigenvalues and eigenfunctions of the Neumann problem in a thick singularly degenerate junction that consists of two domains connected by an -periodic system of thin rods of fixed length. 相似文献
10.
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
11.
Anne-Laure Dalibard 《Archive for Rational Mechanics and Analysis》2009,192(1):117-164
We study the limit as ε → 0 of the entropy solutions of the equation . We prove that the sequence u
ε
two-scale converges toward a function u(t, x, y), and u is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation
law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence
result in . 相似文献
12.
13.
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
14.
Diego R. Moreira 《Archive for Rational Mechanics and Analysis》2009,191(1):97-141
In this paper, we study a free boundary problem obtained as a limit as ε → 0 to the following regularizing family of semilinear equations , where β
ε
approximates the Dirac delta in the origin and F is a Lipschitz function bounded away from 0 and infinity. The least supersolution approach is used to construct solutions
satisfying geometric properties of the level surfaces that are uniform in ε. This allows to prove that the free boundary of a limit has the “right” weak geometry, in the measure theoretical sense.
By the construction of some barriers with curvature, the classification of global profiles of the blow-up analysis is carried
out and the limit functions are proven to be viscosity and pointwise solution ( almost everywhere) to a free boundary problem. Finally, the free boundary is proven to be a C
1,α surface around almost everywhere point.
An erratum to this article can be found at 相似文献
15.
Juan Dávila Manuel del Pino Monica Musso Juncheng Wei 《Archive for Rational Mechanics and Analysis》2006,182(2):181-221
We consider the boundary value problem
where Ω is a smooth and bounded domain in ℝ2 and λ > 0. We prove that for any integer k ≧ 1 there exist at least two solutions u
λ
with the property that the boundary flux satisfies up to subsequences λ → 0,
where the ξ
j
are points of ∂Ω ordered clockwise in j. 相似文献
16.
Stan Alama Lia Bronsard J. Alberto Montero 《Archive for Rational Mechanics and Analysis》2008,187(3):481-522
We construct local minimizers of the Gross–Pitaevskii energy, introduced to model Bose–Einstein condensates (BEC) in the Thomas–Fermi
regime which are subject to a uniform rotation. Our sample domain is taken to be a solid torus of revolution in with starshaped cross-section. We show that for angular speeds ωε = O(|ln ε|) there exist local minimizers of the energy which exhibit vortices, for small enough values of the parameter ε. These vortices concentrate at one or several planar arcs (represented by integer multiplicity rectifiable currents) which
minimize a line energy, obtained as a Γ-limit of the Gross–Pitaevskii functional. The location of these limiting vortex lines
can be described under certain geometrical hypotheses on the cross-sections of the torus. 相似文献
17.
Jiří Neustupa 《Journal of Mathematical Fluid Mechanics》2009,11(1):22-45
We derive a sufficient condition for stability of a steady solution of the Navier–Stokes equation in a 3D exterior domain
Ω. The condition is formulated as a requirement on integrability on the time interval (0, +∞) of a semigroup generated by
the linearized problem for perturbations, applied to a finite family of certain functions. The norm of the semigroup is measured
in a bounded sub-domain of Ω. We do not use any condition on “smallness” of the basic steady solution.
相似文献
18.
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
19.
Dorin Bucur Eduard Feireisl Šárka Nečasová 《Journal of Mathematical Fluid Mechanics》2008,10(4):554-568
We consider a stationary Navier–Stokes flow in a bounded domain supplemented with the complete slip boundary conditions. Assuming
the boundary of the domain is formed by a family of unidirectional asperities, whose amplitude as well as frequency is proportional
to a small parameter ε, we shall show that in the asymptotic limit the motion of the fluid is governed by the same system
of the Navier–Stokes equations, however, the limit boundary conditions are different. Specifically, the resulting boundary
conditions prevent the fluid from slipping in the direction of asperities, while the motion in the orthogonal direction is
allowed without any constraint.
The work of Š. N. supported by Grant IAA100190505 of GA ASCR in the framework of the general research programme of the Academy
of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503. 相似文献