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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 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
3.
Chérif Amrouche M. Ángeles Rodríguez-Bellido 《Archive for Rational Mechanics and Analysis》2011,199(2):597-651
The concept of very weak solution introduced by Giga (Math Z 178:287–329, 1981) for the Stokes equations has hardly been studied in recent years for either the Navier–Stokes
equations or the Navier–Stokes type equations. We treat the stationary Stokes, Oseen and Navier–Stokes systems in the case
of a bounded open set, connected of class C1,1{\mathcal{C}^{1,1}} of
\mathbbR3{\mathbb{R}^3}. Taking up once again the duality method introduced by Lions and Magenes (Problèmes aus limites non-homogènes et applications, vols. 1 & 2, Dunod, Paris, 1968) and Giga (Math Z 178:287–329, 1981) for open sets of class C¥{\mathcal{C}^{\infty}} [see also chapter 4 of Necas (Les méthodes directes en théorie des équations elliptiques. (French) Masson et Cie, éd., Paris; Academia, éditeurs, Prague,
1967), which considers the Hilbertian case p = 2 for general elliptic operators], we give a simpler proof of the existence of a very weak solution for stationary Oseen
and Navier–Stokes equations when data are not regular enough, based on density arguments and a functional framework adequate
for defining more rigourously the traces of non-regular vector fields. In the stationary Navier–Stokes case, the results will
be valid for external forces not necessarily small, which lets us extend the uniqueness class of solutions for these equations.
Considering more regular data, regularity results in fractional Sobolev spaces will also be discussed for the three systems.
All these results can be extended to other dimensions. 相似文献
4.
This article characterizes the singularities of very weak solutions of 3D stationary Navier–Stokes equations in a punctured
ball which are sufficiently small in weak L
3. 相似文献
5.
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
6.
Luan Thach Hoang 《Journal of Mathematical Fluid Mechanics》2010,12(3):435-472
This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier–Stokes equations in a three-dimensional
domain Ωɛ, whose thickness is of order O(ɛ) as ɛ → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the
bottom and top boundaries of Ωɛ, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ɛ3/4) as ɛ → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H1(Ωɛ), respectively, L2(Ωɛ), then the strong solution of the Navier–Stokes equations exists for all time. Our proofs rely on the study of the dependence
of the Stokes operator on ɛ, and the non-linear estimate in which the contributions of the boundary integrals are non-trivial. 相似文献
7.
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
8.
9.
Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional
Navier–Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the
North and South poles of the sphere. We prove analytically for the linearized Navier–Stokes equations that the stationary
flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of
the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically
stable for all Reynolds numbers.
相似文献
10.
G. Seregin 《Journal of Mathematical Fluid Mechanics》2007,9(1):34-43
A sufficient condition of regularity for solutions to the Navier–Stokes equations is proved. It generalizes the so-called
L
3,∞-case. 相似文献
11.
Magnus Fontes 《Journal of Mathematical Fluid Mechanics》2010,12(3):412-434
In this work we study the fully nonhomogeneous initial boundary value problem for the two-dimensional time-dependent Navier–Stokes
equations in a general open space domain in R2 with low regularity assumptions on the initial and the boundary value data. We show that the perturbed Navier–Stokes operator
is a diffeomorphism from a suitable function space onto its own dual and as a corollary we get that the Navier–Stokes equations
are uniquely solvable in these spaces and that the solution depends smoothly on all involved data. Our source data space and
solution space are in complete natural duality and in this sense, without any smallness assumptions on the data, we solve
the equations for data with optimally low regularity in both space and time. 相似文献
12.
In this article we present a Ladyženskaja–Prodi–Serrin Criteria for regularity of solutions for the Navier–Stokes equation
in three dimensions which incorporates weak L
p
norms in the space variables and log improvement in the time variable. 相似文献
13.
Explicit formulae for the fundamental solution of the linearized time dependent Navier–Stokes equations in three spatial dimensions
are obtained. The linear equations considered in this paper include those used to model rigid bodies that are translating
and rotating at a constant velocity. Estimates extending those obtained by Solonnikov in [23] for the fundamental solution
of the time dependent Stokes equations, corresponding to zero translational and angular velocity, are established. Existence
and uniqueness of solutions of these linearized problems is obtained for a class of functions that includes the classical
Lebesgue spaces Lp(R3), 1 < p < ∞. Finally, the asymptotic behavior and semigroup properties of the fundamental solution are established. 相似文献
14.
Tsuyoshi Yoneda 《Archive for Rational Mechanics and Analysis》2011,200(1):225-237
We consider existence of solutions, for large times, to the Navier–Stokes equations in a rotating frame with spatially almost
periodic large data provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models
of meteorology and geophysics dealing with large-scale phenomena. To show existence of solutions for large times, we use the
ℓ
1-norm of amplitudes. Existence for large times is proven by means of techniques of fast singular oscillating limits and bootstrapping
from a global-in-time unique solution to the limit equation. 相似文献
15.
It is known that the three-dimensional Navier–Stokes system for an incompressible fluid in the whole space has a one parameter
family of explicit stationary solutions that are axisymmetric and homogeneous of degree −1. We show that these solutions are
asymptotically stable under any L
2-perturbation. 相似文献
16.
We study the vanishing viscosity limit of the compressible Navier–Stokes equations to the Riemann solution of the Euler equations
that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a
family of smooth solutions to the compressible Navier–Stokes equations that converges to the Riemann solution away from the
initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first
mathematical justification of this limit for the Navier–Stokes equations to the Riemann solution that contains these two typical
nonlinear hyperbolic waves. 相似文献
17.
We consider spectral semi-Galerkin approximations for the strong solutions of the nonhomogeneous Navier–Stokes equations.
We derive an optimal uniform in time error bound in the H1 norm for approximations of the velocity. We also derive an error estimate for approximations of the density in some spaces
Lr.
P. Braz e Silva was supported for this work by FAPESP/Brazil, #02/13270-1 and is currently supported in part by CAPES/MECD-DGU
Brazil/Spain, #117/06. M. Rojas-Medar is partially supported by CAPES/MECD-DGU Brazil/Spain, #117/06 and project BFM2003-06446-CO-01,
Spain. 相似文献
18.
New sufficient conditions of local regularity for suitable weak solutions to the non-stationary three-dimensional Navier–Stokes
equations are proved. They contain the celebrated Caffarelli–Kohn–Nirenberg theorem as a particular case.
相似文献
19.
We consider the Navier–Stokes equations in the thin 3D domain , where is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly,
when ε ≪ 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges
(as ε → 0) to a unique stationary measure for the Navier–Stokes equation on . Thus, the 2D Navier–Stokes equations on surfaces describe asymptotic in time, and limiting in ε, statistical properties of 3D solutions in thin 3D domains. 相似文献