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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.
It is well-known that a KAM torus can be considered as a graph of smooth viscosity solution. Salamon and Zehnder (Comment
Math Helv 64:84–132, 1989) have proved that there exist invariant tori having prescribed Diophantine frequencies for nearly
integrable and positively definite Lagrangian systems with associated Hamiltonian H, whose Diophantine index is τ. If the invariant torus is represented as in the cotangent bundle , then we can show that for any viscosity solution u (x, P), which satisfies the H-J Eq. (1.1),
when is small enough.
For the more exact form, please see Theorem 2 for details. 相似文献
3.
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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4.
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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5.
Changyou Wang 《Archive for Rational Mechanics and Analysis》2008,188(2):351-369
For any compact n-dimensional Riemannian manifold (M, g) without boundary, a compact Riemannian manifold without boundary, and 0 < T ≦ +∞, we prove that for n ≧ 4, if u : M × (0, T] → N is a weak solution to the heat flow of harmonic maps such that , then u ∈C
∞(M × (0, T], N). As a consequence, we show that for n ≧3, if 0 < T < +∞ is the maximal time interval for the unique smooth solution u ∈C
∞(M × [0, T), N) of (1.1), then blows up as t ↑ T. 相似文献
6.
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
7.
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
8.
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
9.
In this work, positive solutions to a doubly nonlinear parabolic equation with a nonlinear boundary condition are considered.
We study the problem where 0 < m, r, α < ∞ are parameters. It is known that for some values of the parameters there are solutions that blow up in finite time.
We determine in terms of α ,m, r the blow-up sets for these solutions. We prove that single point blow-up occurs if
max{m, r} < α,
global blow-up appears for the range of parameters
0 < m < α < r
and regional blow-up takes place if
0 < m < α = r and .
In this case the blow-up set consists of the interval . 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.
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
12.
Role of the Pressure for Validity of the Energy Equality for Solutions of the Navier–Stokes Equation
Igor Kukavica 《Journal of Dynamics and Differential Equations》2006,18(2):461-482
We prove that a weak solution
of the Navier–Stokes system satisfies the energy equality if the associated pressure is locally square integrable. A similar statement is shown to hold for the Euler system. 相似文献
13.
Peter Takáč 《Journal of Dynamics and Differential Equations》2006,18(3):693-765
We are concerned with the existence of a weak solution
to the degenerate quasi-linear Dirichlet boundary value problem
It is assumed that 1 < p < ∞, p ≠ 2, Ω is a bounded domain in
is a given function, and λ stands for the (real) spectral parameter near the first (smallest) eigenvalue λ1 of the positive p-Laplacian − Δ
p
, where
. Eigenvalue λ1 being simple, let φ1 denote the eigenfunction associated with it. We show the existence of a solution for problem (P) when f “nearly” satisfies the orthogonality condition ∫Ω
f φ1 dx = 0 and λ ≤ λ1 + δ (with δ > 0 small enough). Moreover, we obtain at least three distinct solutions if either p < 2 and λ1 − δ ≤ λ < λ1, or else p > 2 and λ1 < λ ≤ λ1 + δ. The proofs use a minimax principle for the corresponding energy functional performed in the orthogonal decomposition
induced by the inner product in L
2(Ω). First, the global minimum is taken over
, and then either a local minimum or a local maximum over lin {φ1}. If the latter is a local minimum, the local minimizer in
thus obtained provides a solution to problem (P). On the other hand, if it is a local maximum, one gets only a pair of sub- and supersolutions to problem (P), which is then used to obtain a solution by a topological degree argument. 相似文献
14.
Some Results on the <Emphasis Type="BoldItalic">m</Emphasis>-Laplace Equations with Two Growth Terms
We prove the existence of positive radial solutions of the following equation:
and give sufficient conditions on the positive functions K1(r) and K2 (r) for the existence and nonexistence of ground states (G.S.) and Singular ground states (S.G.S.), when
or
. We also give sufficient conditions for the existence of radial S.G.S. and G.S. of equation
when
and
, respectively. We are also able to classify all the S.G.S. of this equation. The proofs use a new Emden–Fowler transform which allow us to use techniques taken from dynamical system theory, in particular the ones developed in Johnson et al. (Nonlinear Anal, T.M.A. 20, 1279–1302 (1993)) for the problems obtained by substituting the ordinary Laplacian Δ for the m-Laplacian Δm in the preceding equations.MSC: 37B55, 35H30, 35J70 相似文献
15.
Georgy M. Kobelkov 《Journal of Mathematical Fluid Mechanics》2007,9(4):588-610
For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the
large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component
u
3 under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary
time interval [0, T], any viscosity coefficients and any initial conditions
a weak solution exists and is unique and and the norms are continuous in t.
The work was carried out under partial support of Russian Foundation for Basic Research (project 05-01-00864). 相似文献
16.
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
17.
Under certain assumptions on f and g we prove that positive, global and bounded solutions u of the non-autonomous heat equation
in
(N ≥ 3) converge to a steady state.
Dedicated to Prof. Pavol Brunovsky on the occasion of his 70th birthday. 相似文献
18.
This paper deals with connected branches of nonstationary periodic trajectories of Hamilton equations
emanating from the degenerate stationary point
for H being the generalized Hénon-Heiles (HH) Hamiltonian:
or the generalized Yang-Mills (YM) Hamiltonian:
The existence of such branches has been proved. Minimal periods of searched trajectories near x0 have been described. 相似文献
19.
Adriana Valentina Busuioc Dragoş Iftimie 《Journal of Dynamics and Differential Equations》2006,18(2):357-379
We consider in this paper the equations of motion of third grade fluids on a bounded domain of
or
with Navier boundary conditions. Under the assumption that the initial data belong to the Sobolev space H
2, we prove the existence of a global weak solution. In dimension two, the uniqueness of such solutions is proven. Additional regularity of bidimensional initial data is shown to imply the same additional regularity for the solution. No smallness condition on the data is assumed. 相似文献
20.
S. V. Zelik 《Journal of Dynamics and Differential Equations》2007,19(1):1-74
We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇
x
)u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor
in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in
. In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on
. As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of
. 相似文献