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1.
Concerning to the non-stationary Navier–Stokes flow with a nonzero constant velocity at infinity, just a few results have
been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary
solutions for the Navier–Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata
(J Math Fluid Mech 7:339–367, 2005), in L
p
spaces for p ≥ 3. In this article, we first extend their result to the case
\frac32 < p{\frac{3}{2} < p} by modifying the method in Bae and Jin (J Math Fluid Mech 10:423–433, 2008) that was used to obtain weighted estimates for the Navier–Stokes flow with the zero velocity at infinity. Then, by using
our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier–Stokes flow with
a nonzero velocity at infinity. 相似文献
2.
We consider the three-dimensional flow through an aperture in a plane either with a prescribed flux or pressure drop condition.
We discuss the existence and uniqueness of solutions for small data in weighted spaces and derive their complete asymptotic
behaviour at infinity. Moreover, we show that each solution with a bounded Dirichlet integral, which has a certain weak additional
decay, behaves like O(r
−2) as r=|x|→∞ and admits a wide jet region. These investigations are based on the solvability properties of the linear Stokes system
in a half space ℝ
+
3
. To investigate the Stokes problem in ℝ
+
3
, we apply the Mellin transform technique and reduce the Stokes problem to the determination of the spectrum of the corresponding
invariant Stokes-Beltrami operator on the hemisphere. 相似文献
3.
We perform a mathematical analysis of the steady flow of a viscous liquid, L{\mathcal{L}} , past a three-dimensional elastic body, B{\mathcal{B}} . We assume that L{\mathcal{L}} fills the whole space exterior to B{\mathcal{B}} , and that its motion is governed by the Navier–Stokes equations corresponding to non-zero velocity at infinity, v
∞. As for B{\mathcal{B}} , we suppose that it is a St. Venant–Kirchhoff material, held in equilibrium either by keeping an interior portion of it
attached to a rigid body or by means of appropriate control body force and surface traction. We treat the problem as a coupled
steady state fluid-structure problem with the surface of B{\mathcal{B}} as a free boundary. Our main goal is to show existence and uniqueness for the coupled system liquid-body, for sufficiently
small |v
∞|. This goal is reached by a fixed point approach based upon a suitable reformulation of the Navier–Stokes equation in the
reference configuration, along with appropriate a priori estimates of solutions to the corresponding Oseen linearization and
to the elasticity equations. 相似文献
4.
We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields
are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This
toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided
the initial data belong to a “large” set in the Sobolev space H
1. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the
class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions
of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show
how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear)
inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits
a short and simple proof of the existence of strong solutions for all time. 相似文献
5.
Hyeong-Ohk Bae 《Journal of Mathematical Fluid Mechanics》2008,10(4):503-530
We estimate the time decay rates in L
1, in the Hardy space and in L
∞ of the gradient of solutions for the Stokes equations on the half spaces. For the estimates in the Hardy space we adopt the
ideas in [7], and also use the heat kernel and the solution formula for the Stokes equations.
We also estimate the temporal-spatial asymptotic estimates in L
q
, 1 < q < ∞, for the Stokes solutions.
This work was supported by grant No. (R05-2002-000-00002-0(2002)) from the Basic Research Program of the Korea Science & Engineering
Foundation. 相似文献
6.
We investigate the asymptotic behavior of solutions of linear differential equations with almost constant coefficients and
pulse action at fixed times as t tends to infinity. We establish conditions for the times of pulse action under which there exist values of pulse action for
which the solution of the considered Cauchy problem with initial conditions that coincide with the initial conditions for
a certain (arbitrary but fixed) solution of the original equation without pulse action is bounded, unbounded, or tending to
infinity.
__________
Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 444–455, October–December, 2005. 相似文献
7.
We consider the 3-D evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, see (1.2), and study
the problem of the strong convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations
under the zero-flux boundary condition. We prove here, in the flat boundary case, convergence in Sobolev spaces W
k, p
(Ω), for arbitrarily large k and p (for previous results see Xiao and Xin in Comm Pure Appl Math 60:1027–1055, 2007 and Beir?o da Veiga and Crispo in J Math
Fluid Mech, 2009, doi:). However this problem is still open for non-flat, arbitrarily smooth, boundaries. The main obstacle consists in some boundary
integrals, which vanish on flat portions of the boundary. However, if we drop the convective terms (Stokes problem), the inviscid,
strong limit result holds, as shown below. The cause of this different behavior is quite subtle. As a by-product, we set up
a very elementary approach to the regularity theory, in L
p
-spaces, for solutions to the Navier–Stokes equations under slip type boundary conditions. 相似文献
8.
Marshall Slemrod 《Archive for Rational Mechanics and Analysis》1999,149(1):1-22
. We study the asymptotic behavior as time goes to infinity of solutions to the initial‐boundary‐value problem on the half
space for a one‐dimensional model system for the isentropic flow of a compressible viscous gas, the so‐called p‐system with viscosity. As boundary conditions, we prescribe the constant state at infinity and require that the velocity
be zero at the boundary . When the velocity at infinity is negative and satisfies a condition on the magnitude, we prove that if the initial data
are suitably close to those for the corresponding outgoing viscous shock profile, which is suitably far from the boundary,
then a unique solution exists globally in time and tends toward the properly shifted viscous shock profile as the time goes
to infinity. The proof is given by an elementary energy method.
(Accepted March 2, 1998) 相似文献
9.
In this paper, we establish analyticity of the Navier–Stokes equations with small data in critical Besov spaces . The main method is Gevrey estimates, the choice of which is motivated by the work of Foias and Temam (Contemp Math 208:151–180, 1997). We show that mild solutions are Gevrey regular, that is, the energy bound holds in , globally in time for p < ∞. We extend these results for the intricate limiting case p = ∞ in a suitably designed E ∞ space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spaces. 相似文献
10.
This paper is concerned with the stationary problem of the Stokes equation in an infinite layer and provides a condition on
the external force sufficient for the existence of the solution. Since the Poiseuille flow is a solution to the homogeneous
equation, the solution is not unique when p = ∞. It is also proved that, under some suitable conditions, solutions to the homogeneous equation are limited only to the
Poiseuille flow. 相似文献
11.
James P. Kelliher 《Journal of Dynamics and Differential Equations》2009,21(4):631-661
We develop the concept of an infinite-energy statistical solution to the Navier–Stokes and Euler equations in the whole plane.
We use a velocity formulation with enough generality to encompass initial velocities having bounded vorticity, which includes
the important special case of vortex patch initial data. Our approach is to use well-studied properties of statistical solutions
in a ball of radius R to construct, in the limit as R goes to infinity, an infinite-energy solution to the Navier–Stokes equations. We then construct an infinite-energy statistical
solution to the Euler equations by making a vanishing viscosity argument. 相似文献
12.
This paper is concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier–Stokes equations in a compressible fluid in the Eulerian coordinate in the half space. This is the
second one of our series of papers on this subject. In this paper, firstly we classify completely the time-asymptotic states,
according to some parameters, that is the spatial-asymptotic states and boundary conditions, for this initial boundary value
problem, and some pictures for the classification of time-asymptotic states are drawn in the state space. In order to prove
the stability of the rarefaction wave, we use the solution to Burgers’ equation to construct a suitably smooth approximation
of the rarefaction wave and establish some time-decay estimates in L
p
-norm for the smoothed rarefaction wave. We then employ the L
2-energy method to prove that the rarefaction wave is non-linearly stable under a small perturbation, as time goes to infinity.
P. Zhu was supported by JSPS postdoctoral fellowship under P99217. 相似文献
13.
This paper concerns the existence of a steadily translating finger solution in a Hele-Shaw cell for small but non-zero surface
tension (ɛ2). Though there are numerous numerical and formal asymptotic results for this problem, we know of no mathematically rigorous
results that address the selection problem. We rigorously conclude that for relative finger width λ in the range , with small, analytic symmetric finger solutions exist in the asymptotic limit of surface tension if and only if the Stokes constant for a relatively simple nonlinear differential equation is zero. This Stokes constant
S depends on the parameter and earlier calculations by a number of authors have shown it to be zero for a discrete set of values of a.
The methodology consists of proving the existence and uniqueness of analytic solutions for a weak half-strip problem for any
λ in a compact subset of (0, 1). The weak problem is shown to be equivalent to the original finger problem in the function
space considered, provided we invoke a symmetry condition. Next, we consider the behavior of the solution in a neighborhood
of an appropriate complex turning point for the restricted case , for some . This turning point accounts for exponentially small terms in ɛ, as ɛ→0+ that generally violate the symmetry condition. We prove that the symmetry condition is satisfied for small ɛ when the parameter
a is constrained appropriately.
(Accepted July 4, 2002 Published online January 15, 2003)
Communicated by F. OTTO 相似文献
14.
15.
The existence and linear stability problem for the Stokes periodic wavetrain on fluids of finite depth is formulated in terms
of the spatial and temporal Hamiltonian structure of the water-wave problem. A proof, within the Hamiltonian framework, of
instability of the Stokes periodic wavetrain is presented. A Hamiltonian center-manifold analysis reduces the linear stability
problem to an ordinary differential eigenvalue problem on ℝ4. A projection of the reduced stability problem onto the tangent space of the 2-manifold of periodic Stokes waves is used
to prove the existence of a dispersion relation Λ(λ,σ, I
1, I
2)=0 where λ ε ℂ is the stability exponent for the Stokes wave with amplitude I
1 and mass flux I
2 and σ is the “sideband’ or spatial exponent. A rigorous analysis of the dispersion relation proves the result, first discovered
in the 1960's, that the Stokes gravity wavetrain of sufficiently small amplitude is unstable for F ε (0,F0) where F
0 ≈ 0.8 and F is the Froude number. 相似文献
16.
Konstantin Pileckas 《Journal of Mathematical Fluid Mechanics》2008,10(2):272-309
The time-dependent Navier–Stokes system is studied in a two-dimensional domain with strip-like outlets to infinity in weighted
Sobolev function spaces. It is proved that under natural compatibility conditions there exists a unique solution with prescribed
fluxes over cross-sections of outlets to infinity which tends in each outlet to the corresponding time-dependent Poiseuille
flow. The obtained results are proved for arbitrary large norms of the data (in particular, for arbitrary fluxes) and globally
in time.
The authors are supported by EC FP6 MC–ToK programme SPADE2, MTKD–CT–2004–014508. 相似文献
17.
Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T
0 > 0, ν
0 > 0 and a unique continuous family of strong solutions u
ν
(0 ≤ ν < ν
0) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T
0). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary
conditions imposed on curl
u
ν
and curl
2
u
ν
.
相似文献
18.
We prove Lp-Lq estimates of the Oseen semigroup in n-dimensional exterior domains
which refine and improve those obtained by Kobayashi and Shibata [15]. As an application, we give a globally in time stability theory for the stationary Navier–Stokes flow whose velocity at infinity is a non-zero constant vector. We thus extend the result of Shibata [21]. In particular, we find an optimal rate of convergence of solutions of the non-stationary problem to those of the corresponding stationary problem. 相似文献
19.
Eric Lombardi 《Archive for Rational Mechanics and Analysis》1997,137(3):227-304
In this paper a class of reversible analytic vector fields is investigated near an equilibrium. For these vector fields,
the part of the spectrum of the differential at the equilibrium which lies near the imaginary axis comes from the perturbation
of a double eigenvalue 0 and two simple eigenvalues , .
In the first part of this paper, we study the 4-dimensional problem. The existence of a family of solutions homoclinic to
periodic orbits of size less than μN for any fixed N, where μ is the bifurcation parameter, is known for vector fields. Using the analyticity of the vector field, we prove here the existence of solutions homoclinic to a periodic
orbit the size of which is exponentially small ( of order . This result receives its significance from the still unsolved question of whether there exist solutions that are homoclinic
to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum.
In the second part of this paper we prove that the exponential estimates still hold in infinite dimensions. This result cannot
be simply obtained from the study of the 4-dimensional analysis by a center-manifold reduction since this result is based
on analyticity of the vector field.
One example of such a vector field in infinite dimensions occurs when describing the irrotational flow of an inviscid fluid
layer under the influence of gravity and small surface tension (Bond number ) for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalized solitary wave. Our work shows
that there exist generalized solitary waves with exponentially small oscillations at infinity. More precisely, we prove that
for each F close enough to 1, there exist two reversible solutions homoclinic to a periodic orbit, the size of which is less than , l being any number between 0 and π and satisfying .
(Accepted October 2, 1995) 相似文献
20.
Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space R n+(n≥2) is considered around a given constant equilibrium. A solution formula for the linearized problem is derived, and Lp estimates for solutions of the linearized problem are obtained for 2≤p≤∞. It is shown that, as in the case of the Cauchy problem, the leading part of the solution of the linearized problem is decomposed into two parts, one that behaves like diffusion waves and the other one purely diffusively. There, however, are some aspects different from the Cauchy problem, especially in considering spatial derivatives. It is also shown that the solution of the linearized problem approaches for large times the solution of the nonstationary Stokes problem in some Lp spaces; and, as a result, a solution formula for the nonstationary Stokes problem is obtained. Large-time behavior of solutions of the nonlinear problem is then investigated in Lp spaces for 2≤p≤∞ by applying the results on the linearized analysis and the weighted energy method. The results indicate that there may be some nonlinear interaction phenomena not appearing in the Cauchy problem. 相似文献