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1.
We study the global attractor of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form . If the functions g
0(x, t) and g
1 (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor is bounded in the space H, however, its norm may be unbounded as since the magnitude of the external force is growing. Assuming that the function g
1 (z, t) has a divergence representation of the form where the functions (see Section 3), we prove that the global attractors of the N.–S. equations are uniformly bounded with respect to for all . We also consider the “limiting” 2D N.–S. system with external force g
0(x, t). We have found an estimate for the deviation of a solution of the original N.–S. system from a solution u
0(x, t) of the “limiting” N.–S. system with the same initial data. If the function g
1 (z, t) admits the divergence representation, the functions g
0(x, t) and g
1 (z, t) are translation compact in the corresponding spaces, and , then we prove that the global attractors converges to the global attractor of the “limiting” system as in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of from of the form: in the case, when the global attractor is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).
相似文献
2.
Michael Winkler 《Journal of Dynamics and Differential Equations》2008,20(1):87-113
The paper deals with positive solutions of the initial-boundary value problem for with zero Dirichlet data in a smoothly bounded domain . Here is positive on (0,∞) with f(0) = 0, and λ1 is exactly the first Dirichlet eigenvalue of −Δ in Ω. In this setting, (*) may possess oscillating solutions in presence
of a sufficiently strong degeneracy. More precisely, writing , it is shown that if then there exist global classical solutions of (*) satisfying and . Under the additional structural assumption , s > 0, this result can be sharpened: If then (*) has a global solution with its ω-limit set being the ordered arc that consists of all nonnegative multiples of the
principal Laplacian eigenfunction. On the other hand, under the above additional assumption the opposite condition ensures that all solutions of (*) will stabilize to a single equilibrium.
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3.
Let be the set of m × m matrices A(λ) depending analytically on a parameter λ in a closed interval . Consider one-parameter families of quasi-periodic linear differential equations: , where is analytic and sufficiently small. We prove that there is an open and dense set in , such that for each the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost
all in Lebesgue measure sense provided that g is sufficiently small. The result gives an affirmative answer to a conjecture of Eliasson (In: Proceeding of Symposia in
Pure Mathematics).
Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday 相似文献
4.
S. M. Bruschi A. N. Carvalho J. W. Cholewa Tomasz Dlotko 《Journal of Dynamics and Differential Equations》2006,18(3):767-814
For
, we consider a family of damped wave equations
, where − Λ denotes the Laplacian with zero Dirichlet boundary condition in L
2(Ω). For a dissipative nonlinearity f satisfying a suitable growth restrictions these equations define on the phase space
semigroups
which have global attractors A
η,
. We show that the family
, behaves upper and lower semicontinuously as the parameter η tends to 0+. 相似文献
5.
Giuseppe Da Prato Arnaud Debussche 《Journal of Dynamics and Differential Equations》2008,20(2):301-335
We study the long time behavior of the solution X(t, s, x) of a 2D-Navier–Stokes equation subjected to a periodic time dependent forcing term. We prove in particular that as , approaches a periodic orbit independently of s and x for any continuous and bounded real function .
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6.
Robert Jensen Changyou Wang Yifeng Yu 《Archive for Rational Mechanics and Analysis》2008,190(2):347-370
For a bounded domain and , assume that is convex and coercive, and that has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation:
For H = H(p, x) depending on x, we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to Aronsson’s equation and
that of the Hamilton–Jacobi equation .
Supported by NSF DMS 0601162.
Supported by NSF DMS 0601403. 相似文献
7.
Joel Avrin 《Journal of Dynamics and Differential Equations》2008,20(2):479-518
We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in
which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let P
m
be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Q
m
=I − P
m
, then we add to the NSE operators μ A
φ in a general family such that A
φ≥Q
m
A
α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral
vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers
past a cutoff λ
m0
where m
0 ≤ m, so that for large enough m
0 the inertial-range wavenumbers see only standard NSE viscosity.
We first obtain estimates on the Hausdorff and fractal dimensions of the attractor (respectively and ). For a constant K
α on the order of unity we show if μ ≥ ν that and if μ ≤ ν that where ν is the standard viscosity coefficient, l
0 = λ1−1/2 represents characteristic macroscopic length, and is the Kolmogorov length scale, i.e. where is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and K
α are dimensionless and scale-invariant. The estimate grows in m due to the term λ
m
/λ1 but at a rate lower than m
3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition , the estimates become for μ ≥ ν and for μ ≤ ν. This result holds independently of α, with K
α and c
α independent of m. In an SVV example μ ≥ ν, and for μ ≤ ν aspects of SEV theory and observation suggest setting for 1/c within α orders of magnitude of unity, giving the estimate where c
α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz
predictions for the number of degrees of freedom in 3D turbulent flow with m so large that (e.g. in the distinguished-class case for m
0 large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE
solutions. We would expect lower choices of λ
m
(e.g. with a > 1) to still give good NSE approximation with lower powers on l
0/l
ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice
, motivated by the Chapman–Enskog expansion in the case m = 0, the condition becomes , giving agreement with Landau–Lifschitz for smaller values of λ
m
then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial
manifold for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such
an of dimension N > m for the general class of operators A
φ if α > 5/2.
The special class of A
φ such that P
m
A
φ = 0 and Q
m
A
φ ≥ Q
m
A
α has a unique spectral-gap property which we can use whenever α ≥ 3/2 to show that we have an inertial manifold of dimension m if m is large enough. As a corollary, for most of the cases of the operators A
φ in the distinguished-class case that we expect will be typically used in practice we also obtain an , now of dimension m
0 for m
0 large enough, though under conditions requiring generally larger m
0 than the m in the special class. In both cases, for large enough m (respectively m
0), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics,
and in particular trajectories on are controlled by essentially NSE dynamics.
相似文献
8.
Jorge Lewowicz 《Journal of Dynamics and Differential Equations》2006,18(4):1095-1102
We consider diffeomorphisms f of a smooth compact riemannian mainfold M and its suspension flow
. Assuming some regularity of the stable (unstable) sets at the points
we prove the persistence in the future of {f
n
(x), n ≥ 0} or
, i.e., that C
0 small perturbations g of f have a semi-trajectory that closely shadows {f
n
(x), n ≥ 0} and that the suspension of g has also a semi-trajectory that closely shadows
. In case x belongs to a minimal set of f we show that the assumptions concerning the regularity of stable and unstable sets could be reduced to a neighbourhood of x. 相似文献
9.
Carlos Rocha 《Journal of Dynamics and Differential Equations》2007,19(3):571-591
We consider the set of 2π-periodic solutions of the ordinary differential equation u′′ + g(u) = 0 for a nonlinearity , satisfying a dissipative condition of the form for , and under the generic assumption that the potential G, given by , is a Morse function. Under these assumptions, we characterize the period maps realizable by planar Hamiltonian systems of
the form . Considering the Morse type of G, the set of periodic orbits in the phase space is decomposed into disks and annular regions. Then, the realizable period maps are described in terms of sets of sequences
of positive integers corresponding to the lap numbers of the 2π-periodic solutions. This leads to a characterization of the
classes of Morse–Smale attractors that are realizable by dissipative semilinear parabolic equations of the form defined on the circle, .
相似文献
10.
Hildebrando M. Rodrigues J. Solà-Morales 《Journal of Dynamics and Differential Equations》2006,18(4):961-974
We present an example of a contraction diffeomorphism in infinite dimensions that is not
-linearizable, and we construct a regular ordinary differential equation in a Hilbert space whose time-one map is that diffeomorphism. With this we have an example of an asymptotically stable ODE that is not
-conjugate to its linear part. 相似文献
11.
We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation
where is a ring-shaped domain, a and μ are given positive constants, is the Heaviside maximal monotone graph: if s > 0, if s < 0. Such equations arise in climatology (the so-called Budyko energy balance model), as well as in other contexts such as
combustion. We show that under certain conditions on the initial data the level sets are n-dimensional hypersurfaces in the (x, t)-space and show that the dynamics of Γ
μ
is governed by a differential equation which generalizes the classical Darcy law in filtration theory. This differential
equation expresses the velocity of advancement of the level surface Γ
μ
through spatial derivatives of the solution u. Our approach is based on the introduction of a local set of Lagrangian coordinates: the equation is formally considered
as the mass balance law in the motion of a fluid and the passage to Lagrangian coordinates allows us to watch the trajectory
of each of the fluid particles. 相似文献
12.
Two-Phase Inertial Flow in Homogeneous Porous Media: A Theoretical Derivation of a Macroscopic Model
The purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian
fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ, the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic
mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities
involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting
macroscopic momentum equation relates the phase-averaged pressure gradient to the filtration or Darcy velocity in a coupled nonlinear form explicitly given by
or equivalently
In these equations, and are the inertial and coupling inertial correction tensors that are functions of flow-rates. The dominant and coupling permeability tensors and and the permeability and viscous drag tensors and are intrinsic and are those defined in the conventional manner as in (Whitaker, Chem Eng Sci 49:765–780, 1994) and (Lasseux
et al., Transport Porous Media 24(1):107–137, 1996). All these tensors can be determined from closure problems that are to
be solved using a spatially periodic model of a porous medium. The practical procedure to compute these tensors is provided. 相似文献
13.
I. A. Guerra 《Journal of Dynamics and Differential Equations》2007,19(1):243-263
Consider the problem
where Ω is a bounded convex domain in
, N > 2, with smooth boundary
. We study the asymptotic behaviour of the least energy solutions of this system as
. We show that the solution remain bounded for p large. In the limit, we find that the solution develops one or two peaks away from the boundary, and when a single peak occurs, we have a characterization of its location.This research was supported by FONDECYT 1061110 and 3040059. 相似文献
14.
We study the limit of the hyperbolic–parabolic approximation
The function is defined in such a way as to guarantee that the initial boundary value problem is well posed even if is not invertible. The data and are constant. When is invertible, the previous problem takes the simpler form
Again, the data and are constant. The conservative case is included in the previous formulations. Convergence of the , smallness of the total variation and other technical hypotheses are assumed, and a complete characterization of the limit
is provided. The most interesting points are the following: First, the boundary characteristic case is considered, that is,
one eigenvalue of can be 0. Second, as pointed out before, we take into account the possibility that is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta
relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if
this condition is not satisfied, then pathological behaviors may occur. 相似文献
15.
We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension
where , is a smooth matrix-valued map and the initial data is assumed to have small total variation. We present a front tracking algorithm that generates piecewise constant approximate
solutions converging in to the vanishing viscosity solution of (1), which, by the results in [6], is the unique limit of solutions to the (artificial)
viscous parabolic approximation
as . In the conservative case where A(u) is the Jacobian matrix of some flux function F(u) with values in , the limit of front tracking approximations provides a weak solution of the system of conservation laws u
t
+ F(u)
x
= 0, satisfying the Liu admissibility conditions.
These results are achieved under the only assumption of strict hyperbolicity of the matrices A(u), . In particular, our construction applies to general, strictly hyperbolic systems of conservation laws with characteristic
fields that do not satisfy the standard conditions of genuine nonlinearity or of linear degeneracy in the sense of Lax[17], or in the generalized sense of Liu[23].
Dedicated to Prof. Tai Ping Liu on the occasion of his 60
th
birthday 相似文献
16.
17.
18.
Thierry Cazenave Flávio Dickstein Fred B. Weissler 《Journal of Dynamics and Differential Equations》2007,19(3):789-818
In this paper, we construct solutions u(t,x) of the heat equation on such that has nontrivial limit points in as t → ∞ for certain values of μ > 0 and β > 1/2. We also show the existence of solutions of this type for nonlinear heat equations.
相似文献
19.
Let be an infinite cylinder of , n ≥ 3, with a bounded cross-section of C
1,1-class. We study resolvent estimates and maximal regularity of the Stokes operator in for 1 < q, r < ∞ and for arbitrary Muckenhoupt weights ω ∈ A
r
with respect to x′ ∈ Σ. The proofs use an operator-valued Fourier multiplier theorem and techniques of unconditional Schauder decompositions
based on the -boundedness of the family of solution operators for a system in Σ parametrized by the phase variable of the one-dimensional
partial Fourier transform.
Supported by the Gottlieb Daimler- und Karl Benz-Stiftung, grant no. S025/02-10/03. 相似文献
20.
We investigate the dynamics of the semiflow φ induced on H01(Ω) by the Cauchy problem of the semilinear parabolic equation
on Ω. Here
is a bounded smooth domain, and
has subcritical growth in u and satisfies
. In particular we are interested in the case when f is definite superlinear in u. The set
of attraction of 0 contains a decreasing family of invariant sets
distinguished by the rate of convergence
. We prove that the Wk’s are global submanifolds of
, and we find equilibria in the boundaries
. We also obtain results on nodal and comparison properties of these equilibria. In addition the paper contains a detailed exposition of the semigroup approach for semilinear equations, improving earlier results on stable manifolds and asymptotic behavior near an equilibrium.Supported by DFG Grant BA 1009/15-1. 相似文献