共查询到20条相似文献,搜索用时 31 毫秒
1.
We study rates of convergence of solutions in L
2 and H
1/2 for a family of elliptic systems {Le}{\{\mathcal{L}_\varepsilon\}} with rapidly oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence,
we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {Le}{\{\mathcal{L}_\varepsilon\}} . Most of our results, which rely on the recently established uniform estimates for the L
2 Dirichlet and Neumann problems in Kenig and Shen (Math Ann 350:867–917, 2011; Commun Pure Appl Math 64:1–44, 2011) are new even for smooth domains. 相似文献
2.
Martin Fuchs 《Journal of Mathematical Fluid Mechanics》2012,14(1):43-54
We investigate the steady flow of a shear thickening generalized Newtonian fluid under homogeneous boundary conditions on
a domain in
\mathbbR2{\mathbb{R}^{2}}. We assume that the stress tensor is generated by a potential of the form H = h (|e(u)|){H = h (|\varepsilon (u)|)}, e(u){\varepsilon (u)} denoting the symmetric part of the velocity gradient. We prove the existence of strong solutions for a large class of functions
h having the property that h′ (t)/t increases (shear thickening case). 相似文献
3.
Roberto Alicandro Marco Cicalese Antoine Gloria 《Archive for Rational Mechanics and Analysis》2011,200(3):881-943
This article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices. Under
polynomial growth assumptions, we prove that the energy functionals Fe{F_\varepsilon} stored in the deformation of an e{{\varepsilon}}-scaling of a stochastic lattice Γ-converge to a continuous energy functional when e{{\varepsilon}} goes to zero. In particular, the limiting energy functional is of integral type, and deterministic if the lattice is ergodic.
We also generalize, to systems and nonlinear settings, well-known results on stochastic homogenization of discrete elliptic
equations. As an application of the main result, we prove the convergence of a discrete model for rubber towards the nonlinear
theory of continuum mechanics. We finally address some mechanical properties of the limiting models, such as frame-invariance,
isotropy and natural states. 相似文献
4.
Giada Basile Stefano Olla Herbert Spohn 《Archive for Rational Mechanics and Analysis》2010,195(1):171-203
We consider lattice dynamics with a small stochastic perturbation of order ${\varepsilon}We consider lattice dynamics with a small stochastic perturbation of order e{\varepsilon} and prove that for a space–time scale of order e-1{\varepsilon^{-1}} the local spectral density (Wigner function) evolves according to a linear transport equation describing inelastic collisions.
For an energy and momentum conserving chain, the transport equation predicts a slow decay, as 1/?t{1/\sqrt t} , for the energy current correlation in equilibrium. This is in agreement with previous studies using a different method. 相似文献
5.
We consider periodic travelling gravity waves at the surface of an infinitely deep perfect fluid. The pattern is non-symmetric
with respect to the propagation direction of the waves and we consider a general non-resonant situation. Defining a couple
of amplitudes e1,e2{\varepsilon_{1},\varepsilon_{2}} along the basis of wave vectors which satisfy the dispersion relation, we first give the formal asymptotic expansion of bifurcating
solutions in powers of e1,e2.{\varepsilon_{1},\varepsilon_{2}.} Then, introducing an additional equation for the unknown diffeomorphism of the torus associated with an irrational rotation
number, which allows us to transform the differential at the successive points of the Newton iteration method into a differential
equation with two constant main coefficients, we are able to use a descent method leading to an invertible differential. Then,
by using an adapted Nash–Moser theorem, we prove the existence of solutions with the above asymptotic expansion for values
of the couple (e12,e22){(\varepsilon_{1}^{2},\varepsilon_{2}^{2})} in a subset of the first quadrant of the plane with asymptotic full measure at the origin. 相似文献
6.
Olivier Lafitte Mark Williams Kevin Zumbrun 《Archive for Rational Mechanics and Analysis》2012,204(1):141-187
The rigorous study of spectral stability for strong detonations was begun by Erpenbeck (Phys. Fluids 5:604–614 1962). Working with the Zeldovitch–von Neumann–D?ring (ZND) model (more precisely, Erpenbeck worked with an extension of ZND to
general chemistry and thermodynamics), which assumes a finite reaction rate but ignores effects such as viscosity corresponding
to second order derivatives, he used a normal mode analysis to define a stability function V(t,e){V(\tau,\epsilon)} whose zeros in ${\mathfrak{R}\tau > 0}${\mathfrak{R}\tau > 0} correspond to multidimensional perturbations of a steady detonation profile that grow exponentially in time. Later in a remarkable
paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966; Stability of detonations for disturbances of small transverse wavelength, 1965) he provided strong evidence, by a combination of formal and rigorous arguments, that for certain classes of steady ZND profiles,
unstable zeros of V exist for perturbations of sufficiently large transverse wavenumber e{\epsilon} , even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense defined (nearly 20 years
later) by Majda. In spite of a great deal of later numerical work devoted to computing the zeros of V(t,e){V(\tau,\epsilon)} , the paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966) remains one of the few works we know of [another is Erpenbeck (Phys. Fluids 7:684–696, 1964), which considers perturbations for which the ratio of longitudinal over transverse components approaches ∞] that presents
a detailed and convincing theoretical argument for detecting them. The analysis in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) points the way toward, but does not constitute, a mathematical proof that such unstable zeros exist. In this paper we identify
the mathematical issues left unresolved in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) and provide proofs, together with certain simplifications and extensions, of the main conclusions about stability and instability
of detonations contained in that paper. The main mathematical problem, and our principal focus here, is to determine the precise
asymptotic behavior as e?¥{\epsilon\to\infty} of solutions to a linear system of ODEs in x, depending on e{\epsilon} and a complex frequency τ as parameters, with turning points x
* on the half-line [0,∞). 相似文献
7.
Augusto Visintin 《Archive for Rational Mechanics and Analysis》2010,198(2):569-611
A class of nonlinear first-order processes is formulated as a minimization principle. In the presence of oscillating data, a two-scale model is then derived, via Nguetseng’s notion of two-scale convergence. The dependence on the fine-scale variable is eliminated by averaging with respect to the fine-scale (scale-integration or upscaling); conversely, any two-scale solution is retrieved from a coarse-scale one (scale-disintegration or downscaling). These results are first developed in a general functional framework, and are then applied to the homogenization of a relaxation dynamics in magnetic composites: $ \mathcal{A}(x/\varepsilon) {\partial B_\varepsilon\over \partial t} + \alpha(B_\varepsilon, x/\varepsilon) \ni H_\varepsilon $ $ \nabla \cdot B_\varepsilon=0, \quad \nabla \times H_\varepsilon =J(x) \quad \forall\;\varepsilon > 0. $ Here J is a prescribed current density. ${\mathcal{A}(y)}A class of nonlinear first-order processes is formulated as a minimization principle. In the presence of oscillating data,
a two-scale model is then derived, via Nguetseng’s notion of two-scale convergence. The dependence on the fine-scale variable
is eliminated by averaging with respect to the fine-scale (scale-integration or upscaling); conversely, any two-scale solution is retrieved from a coarse-scale one (scale-disintegration or downscaling). These results are first developed in a general functional framework, and are then applied to the homogenization of a relaxation
dynamics in magnetic composites:
A(x/e) [(?Be)/(?t)] + a(Be, x/e) ' He \mathcal{A}(x/\varepsilon) {\partial B_\varepsilon\over \partial t} + \alpha(B_\varepsilon, x/\varepsilon) \ni H_\varepsilon 相似文献
8.
We study a two-dimensional nonconvex and nonlocal energy in micromagnetics defined over S
2-valued vector fields. This energy depends on two small parameters, β and e{\varepsilon} , penalizing the divergence of the vector field and its vertical component, respectively. Our objective is to analyze the
asymptotic regime b << e << 1{\beta \ll \varepsilon \ll 1} through the method of Γ-convergence. Finite energy configurations tend to become divergence-free and in-plane in the magnetic
sample except in some small regions of typical width e{\varepsilon} (called Bloch walls) where the magnetization connects two directions on S
2. We are interested in quantifying the limit energy of the transition layers in terms of the jump size between these directions.
For one-dimensional transition layers, we show by Γ-convergence analysis that the exact line density of the energy is quadratic
in the jump size. We expect the same behaviour for the two-dimensional model. In order to prove that, we investigate the concept
of entropies. In the prototype case of a periodic strip, we establish a quadratic lower bound for the energy with a non-optimal
constant. Then we introduce and study a special class of Lipschitz entropies and obtain lower bounds coinciding with the one-dimensional
Γ-limit in some particular cases. Finally, we show that entropies are not appropriate in general for proving the expected
sharp lower bound. 相似文献
9.
This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water
of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy
E{{\mathcal E}} subject to the constraint I=?2m{{\mathcal I}=\sqrt{2}\mu}, where I{{\mathcal I}} is the wave momentum and 0 < m << 1{0 < \mu \ll 1} . Since E{{\mathcal E}} and I{{\mathcal I}} are both conserved quantities a standard argument asserts the stability of the set D
μ
of minimisers: solutions starting near D
μ
remain close to D
μ
in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water
waves of the present kind are modelled as solutions of the nonlinear Schr?dinger equation with cubic focussing nonlinearity.
We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model
equation as mˉ 0{\mu \downarrow 0} . 相似文献
10.
Vieri Benci Marco Ghimenti Anna Maria Micheletti 《Archive for Rational Mechanics and Analysis》2012,205(2):467-492
We study the behavior of the soliton solutions of the equation i\frac?y?t = - \frac12m Dy+ \frac12We¢(y) + V(x)y,i\frac{\partial\psi}{{\partial}t} = - \frac{1}{2m} \Delta\psi + \frac{1}{2}W_{\varepsilon}^{\prime}(\psi) + V(x){\psi}, 相似文献
11.
In this study, fully developed heat and fluid flow in a parallel plate channel partially filled with porous layer is analyzed
both analytically and numerically. The porous layer is located at the center of the channel and uniform heat flux is applied
at the walls. The heat and fluid flow equations for clear fluid and porous regions are separately solved. Continues shear
stress and heat flux conditions at the interface are used to determine the interface velocity and temperature. The velocity
and temperature profiles in the channel for different values of Darcy number, thermal conductivity ratio, and porous layer
thickness are plotted and discussed. The values of Nusselt number and friction factor of a fully clear fluid channel (Nu
cl = 4.12 and fRe
cl = 24) are used to define heat transfer increment ratio (eth = Nup/Nucl)({\varepsilon _{\rm th} =Nu_{\rm p}/Nu_{\rm cl})} and pressure drop increment ratio (ep = fRep/fRecl )({\varepsilon_{\rm p} =fRe_{\rm p}/fRe_{\rm cl} )} and observe the effects of an inserted porous layer on the increase of heat transfer and pressure drop. The heat transfer
and pressure drop increment ratios are used to define an overall performance (e = eth/ep)({\varepsilon = \varepsilon_{\rm th}/\varepsilon_{\rm p})} to evaluate overall benefits of an inserted porous layer in a parallel plate channel. The obtained results showed that for
a partially porous filled channel, the value of e{\varepsilon} is highly influenced from Darcy number, but it is not affected from thermal conductivity ratio (k
r) when k
r > 2. For a fully porous material filled channel, the value of e{\varepsilon} is considerably affected from thermal conductivity ratio as the porous medium is in contact with the channel walls. 相似文献
12.
We study the problem of an elastic shell-like inclusion with high rigidity in a three-dimensional domain by means of the asymptotic
expansion method. The analysis is carried out in a general framework of curvilinear coordinates. After defining a small real
adimensional parameter ε, we characterize the limit problems when the rigidity of the inclusion has order of magnitude
\frac1e\frac{1}{\varepsilon } and
\frac1e3\frac{1}{\varepsilon^{3}} with respect to the rigidities of the surrounding bodies. Moreover, we prove the strong convergence of the solution of the
initial three-dimensional problem towards the solution of the simplified limit problem. 相似文献
13.
The often used “Boussinesq equations” for the determination of the coupled flow and temperature field in natural convection are systematically deduced by an asymptotic approach. With the nondimensional temperature difference that drives the flow, ?, as a perturbation parameter the leading order equations are identified as the appropriate equations, named “asymptotic Boussinesq equations”. These equations appear as the distinguished limit $\varepsilon\rightarrow0
14.
Enzo Vitillaro 《Archive for Rational Mechanics and Analysis》1999,149(2):155-182
We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea where 0 < T £ ¥0
15.
C. Grotta-Ragazzo Coraci Pereira Malta K. Pakdaman 《Journal of Dynamics and Differential Equations》2010,22(2):203-252
We consider the scalar delayed differential equation e[(x)\dot](t)=-x(t) +f(x(t-1)){\epsilon\dot x(t)=-x(t)\,+f(x(t-1))}, where ${\epsilon\,{>}\,0}${\epsilon\,{>}\,0} and f verifies either df/dx > 0 or df/dx < 0 and some other conditions. We present theorems indicating that a generic initial condition with sign changes generates
a solution with a transient time of order exp(c/e){{\rm exp}(c{/}\epsilon)}, for some c > 0. We call it a metastable solution. During this transient a finite time span of the solution looks like that of a periodic
function. It is remarkable that if df/dx > 0 then f must be odd or present some other very special symmetry in order to support metastable solutions, while this condition is
absent in the case df/dx < 0. Explicit e{\epsilon}-asymptotics for the motion of zeroes of a solution and for the transient time regime are presented. 相似文献
16.
Katrin Schumacher 《Journal of Mathematical Fluid Mechanics》2009,11(4):552-571
We investigate the solvability of the instationary Navier–Stokes equations with fully inhomogeneous data in a bounded domain
W ì \mathbbRn \Omega \subset {{\mathbb{R}}^{n}} . The class of solutions is contained in Lr(0, T; Hb, qw (W))L^{r}(0, T; H^{\beta, q}_{w} (\Omega)), where Hb, qw (W){H^{\beta, q}_{w}} (\Omega) is a Bessel-Potential space with a Muckenhoupt weight w. In this context we derive solvability for small data, where this smallness can be realized by the restriction to a short
time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions,
or with very weak solutions. 相似文献
17.
We show that if a Leray–Hopf solution u of the three-dimensional Navier–Stokes equation belongs to C((0,T]; B-1¥,¥){C((0,T]; B^{-1}_{\infty,\infty})} or its jumps in the B-1¥,¥{B^{-1}_{\infty,\infty}}-norm do not exceed a constant multiple of viscosity, then u is regular for (0, T]. Our method uses frequency local estimates on the nonlinear term, and yields an extension of the classical Ladyzhenskaya–Prodi–Serrin
criterion. 相似文献
18.
Heinrich Freistühler Peter Szmolyan 《Archive for Rational Mechanics and Analysis》2010,195(2):353-373
A planar viscous shock profile of a hyperbolic–parabolic system of conservation laws is a steady solution in a moving coordinate
frame. The asymptotic stability of viscous profiles and the related vanishing-viscosity limit are delicate questions already
in the well understood case of one space dimension and even more so in the case of several space dimensions. It is a natural
idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile. The Evans
function method provides a geometric dynamical-systems framework to study the eigenvalue problem. In this approach eigenvalues
correspond to zeros of an essentially analytic function E(rl,rw){\mathcal{E}(\rho\lambda,\rho\omega)} which detects nontrivial intersections of the so-called stable and unstable spaces, that is, spaces of solutions that decay
on one (“−∞”) or the other side (“ + ∞”) of the shock wave, respectively. In a series of pioneering papers, Kevin Zumbrun
and collaborators have established in various contexts that spectral stability, that is, the non-vanishing of E(rl,rw){\mathcal{E}(\rho\lambda,\rho\omega)} and the non-vanishing of the Lopatinski–Kreiss–Majda function Δ(λ,ω), imply nonlinear stability of viscous shock profiles in several space dimensions. In this paper we show that these conditions
hold true for small amplitude extreme shocks under natural assumptions. This is done by exploiting the slow-fast nature of
the small-amplitude limit, which was used in a previous paper by the authors to prove spectral stability of small-amplitude
shock waves in one space dimension. Geometric singular perturbation methods are applied to decompose the stable and unstable
spaces into subbundles with good control over their limiting behavior. Three qualitatively different regimes are distinguished
that relate the small strength e{\epsilon} of the shock wave to appropriate ranges of values of the spectral parameters (ρλ, ρ
ω). Various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of
transversality. 相似文献
19.
The Darcy Model with the Boussinesq approximation is used to study natural convection in a horizontal annular porous layer
filled with a binary fluid, under the influence of a centrifugal force field. Neumann boundary conditions for temperature
and concentration are applied on the inner and outer boundary of the enclosure. The governing parameters for the problem are
the Rayleigh number, Ra, the Lewis number, Le, the buoyancy ratio, j{\varphi } , the radius ratio of the cavity, R, the normalized porosity, e{\varepsilon } , and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in a thin annular layer (R → 1), analytical solutions for the stream function, temperature and concentration fields are obtained using a concentric
flow approximation and an integral form of the energy equation. The critical Rayleigh number for the onset of supercritical
convection is predicted explicitly by the present model. Also, results are obtained from the analytical model for finite amplitude
convection for which the flow and heat and mass transfer are presented in terms of the governing parameters of the problem.
Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement
is observed between the analytical model and the numerical simulations. 相似文献
20.
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
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