共查询到20条相似文献,搜索用时 31 毫秒
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.
3.
This paper uses a variational approach to establish existence of solutions (σ
t
, v
t
) for the 1-d Euler–Poisson system by minimizing an action. We assume that the initial and terminal points σ
0, σ
T
are prescribed in , the set of Borel probability measures on the real line, of finite second-order moments. We show existence of a unique minimizer
of the action when the time interval [0,T] satisfies T < π. These solutions conserve the Hamiltonian and they yield a path t → σ
t
in . When σ
t
= δ
y(t) is a Dirac mass, the Euler–Poisson system reduces to . The kinetic version of the Euler–Poisson, that is the Vlasov–Poisson system was studied in Ambrosio and Gangbo (Comm Pure
Appl Math, to appear) as a Hamiltonian system.
WG gratefully acknowledges the support provided by NSF grants DMS-02-00267,
DMS-03-54729 and DMS-06-00791.
TN gratefully acknowledges the postdoctoral support provided by NSF grants DMS-03-
54729 and the School of Mathematics.
AT gratefully acknowledges the support provided by the School of Mathematics. 相似文献
4.
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
5.
K. Pileckas 《Journal of Mathematical Fluid Mechanics》2006,8(4):542-563
The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite
cylinder is proved. We assume that the initial data and the external forces do not depend on x3 and find the solution (u, p) having the following form
where x′ = (x1, x2). Such solution generalize the nonstationary Poiseuille solutions. 相似文献
6.
Hans-Otto Walther 《Journal of Dynamics and Differential Equations》2009,21(1):195-232
Systems of the form
generalize differential equations with delays r(t) < 0 which are given implicitly by the history x
t
of the state. We show that the associated initial value problem generates a semiflow with differentiable solution operators
on a Banach manifold. The theory covers reaction delays, signal transmission delays, threshold delays, and delays depending
on the present state x(t) only. As an application we consider a model for the regulation of the density of white blood cells and study monotonicity
properties of the delayed argument function . There are solutions (r, x) with τ′(t) > 0 and others with τ′(t) < 0. These other solutions correspond to feedback which reverses temporal order; they are short-lived and less abundant.
Transient behaviour with a sign change of τ′ is impossible.
相似文献
7.
In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles.
We find a fractional Lagrangian L(x(t), where
a
c
D
t
α
x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange
At last, exact solutions for some Euler–Lagrange equations are presented. In particular, we consider the following equations
where g(t) and f(t) are suitable functions.
D. Baleanu is on leave of absence from Institute of Space Sciences, P.O. BOX MG-23, 76900 Magurele-Bucharest, Romania. e-mail:
baleanu@venus.nipne.ro. 相似文献
(1) |
(2) |
(3) |
8.
In this paper we study linear reaction–hyperbolic systems of the form , (i = 1, 2, ..., n) for x > 0, t > 0 coupled to a diffusion equation for p
0 = p
0(x, y, θ, t) with “near-equilibrium” initial and boundary data. This problem arises in a model of transport of neurofilaments in axons.
The matrix (k
ij
) is assumed to have a unique null vector with positive components summed to 1 and the v
j
are arbitrary velocities such that . We prove that as the solution converges to a traveling wave with velocity v and a spreading front, and that the convergence rate in the uniform norm is , for any small positive α. 相似文献
9.
In association with multi-inhomogeneity problems, a special class of eigenstrains is discovered to give rise to disturbance
stresses of interesting nature. Some previously unnoticed properties of Eshelby’s tensors prove useful in this accomplishment.
Consider the set of nested similar ellipsoidal domains {Ω1, Ω2,⋯,Ω
N+1}, which are embedded in an infinite isotropic medium. Suppose that
in which and ξ
t
a
p
, p=1,2,3 are the principal half axes of Ω
t
. Suppose, the distribution of eigenstrain, ε
ij
*(x) over the regions Γ
t
=Ω
t+1−Ω
t
, t=1,2,⋯,N can be expressed as
where x
k
x
l
⋯x
m
is of order n, and f
ijkl ⋯m
(t) represents 3N(n+2)(n+1) different piecewise continuous functions whose arguments are ∑
p=1
3
x
p
2 /a
p
2. The nature of the disturbance stresses due to various classes of the piecewise nonuniform distribution of eigenstrains,
obtained via superpositions of Eq. (‡) is predicted and an infinite number of impotent eigenstrains are introduced. The present theory not only provides a general
framework for handling a broad range of nonuniform distribution of eigenstrains exactly, but also has great implications in
employing the equivalent inclusion method (EIM) to study the behavior of composites with functionally graded reinforcements.
The paper is dedicated to professor Toshio Mura. 相似文献
(‡) |
10.
Crack Initiation in Brittle Materials 总被引:1,自引:0,他引:1
Antonin Chambolle Alessandro Giacomini Marcello Ponsiglione 《Archive for Rational Mechanics and Analysis》2008,188(2):309-349
In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during
a load process through a time-dependent boundary datum of the type t → t
g(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation
is brutal, that is, a big crack appears after a positive time t
i
> 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large
class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the
problem is a local minimality result for the functional where , k > 0 and f is a suitable Carathéodory function. We prove that if the uncracked configuration u of Ω relative to a boundary displacement ψ has at most uniformly weak singularities, then configurations (uΓ, Γ) with small enough are such that . 相似文献
11.
Yoshihisa Morita Hirokazu Ninomiya 《Journal of Dynamics and Differential Equations》2006,18(4):841-861
We deal with a reaction–diffusion equation u
t
= u
xx
+ f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ1(x + c
1
t) (c
1 < 0) and ψ2(x + c
2
t) (c
2 > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all
. We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ1(x + c
1
t) and ψ2(x + c
2
t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c
1, we show the existence of an entire solution which behaves as ψ1( − x + c
1
t) in
and φ(x + ct) in
for t≈ − ∞. 相似文献
12.
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 .
相似文献
13.
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
14.
David N. Cheban 《Journal of Dynamics and Differential Equations》2008,20(3):669-697
In the present article we consider a special class of equations
when the function (E is a strictly convex Banach space) is V-monotone with respect to (w.r.t.) , i.e. there exists a continuous non-negative function , which equals to zero only on the diagonal, so that the numerical function α(t):= V(x
1(t), x
2(t)) is non-increasing w.r.t. , where x
1(t) and x
2(t) are two arbitrary solutions of (1) defined on . The main result of this article states that every V-monotone Levitan almost periodic (almost automorphic, Bohr almost periodic) Eq. (1) with bounded solutions admits at least
one Levitan almost periodic (almost automorphic, Bohr almost periodic) solution. In particulary, we obtain some new criterions
of existence of almost recurrent (Levitan almost periodic, almost automophic, recurrent in the sense of Birkgoff) solutions
of forced vectorial Liénard equations.
相似文献
15.
Hyunseok Kim 《Archive for Rational Mechanics and Analysis》2009,193(1):117-152
We consider the stationary Navier–Stokes equations in a bounded domain Ω in R
n
with smooth connected boundary, where n = 2, 3 or 4. In case that n = 3 or 4, existence of very weak solutions in L
n
(Ω) is proved for the data belonging to some Sobolev spaces of negative order. Moreover we obtain complete L
q
-regularity results on very weak solutions in L
n
(Ω). If n = 2, then similar results are also proved for very weak solutions in with any q
0 > 2. We impose neither smallness conditions on the external force nor boundary data for our existence and regularity results. 相似文献
16.
Xinyu He 《Journal of Mathematical Fluid Mechanics》2007,9(3):398-410
Let
be the exterior of the closed unit ball. Consider the self-similar Euler system
Setting α = β = 1/2 gives the limiting case of Leray’s self-similar Navier–Stokes equations. Assuming smoothness and smallness of the boundary
data on ∂Ω, we prove that this system has a unique solution
, vanishing at infinity, precisely
The self-similarity transformation is v(x, t) = u(y)/(t* − t)α, y = x/(t* − t)β, where v(x, t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x, t) blows up at (x*, t*), x* = 0, t* < + ∞. This isolated singularity has bounded energy with unbounded L
2 − norm of curl v. 相似文献
17.
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
18.
Lorenzo Brandolese 《Archive for Rational Mechanics and Analysis》2009,192(3):375-401
We study the solutions of the nonstationary incompressible Navier–Stokes equations in , of self-similar form , obtained from small and homogeneous initial data a(x). We construct an explicit asymptotic formula relating the self-similar profile U(x) of the velocity field to its corresponding initial datum a(x). 相似文献
19.
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
20.
Shui -Nee Chow Kening Lu John Mallet-Paret 《Archive for Rational Mechanics and Analysis》1995,129(3):245-304
For linear scalar parabolic equations such as
on a finite interval 0x, with various boundary conditions, we obtain canonical Floquet solutions u
n
(t, x). These solutions are characterized by the property that z(u
n
(t, x))=n for all t, where z(·) denotes the zero crossing (lap) number of Matano. The coefficients a(t, x) and b(t, x) are not assumed to be periodic in t, but if they are, the solutions u
n
(t, x) reduce to the standard Floquet solutions. Our results may naturally be expressed in the language of linear skew product flows. In this context, we obtain for each N1 an exponential dichotomy between the bundles span {u
n
(·,·)}
n
=1/N
and
. 相似文献