共查询到20条相似文献,搜索用时 15 毫秒
1.
We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w, by means of a feedback boundary control. We first determine a feedback law for the linearized system around w. Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H3/2+ε,3/4+ε/2(Q), with 0<ε. In [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t=0, is achieved by choosing a time varying control operator in a neighbourhood of t=0. 相似文献
2.
In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions. 相似文献
3.
J. M. Bernard 《Nonlinear Analysis: Real World Applications》2003,4(5):805-839
This paper is concerned with the time-dependent Stokes and Navier–Stokes problems with nonstandard boundary conditions: the pressure is given on some part of the boundary. The stationary case was first studied by Bégue, Conca, Murat and Pironneau and, next, their study were completed by Bernard, mainly about regularity. In this paper, the Stokes problem is studied by a method analogous to that of Temam for the standard problem, combined with regularity results of Bernard for the nonstandard stationary case. We obtain existence, uniqueness and regularity H2. In addition, in two dimensions, a regularity W2,r, r2, is proved. Next, for the nonstandard Navier–Stokes problem, we present some existence, uniqueness and regularity H2 results. The proof of existence is based on a fixed point method. 相似文献
4.
Summary. Optimal control problems governed by the two-dimensional instationary Navier–Stokes equations and their spatial discretizations with finite elements are investigated. A concept of semi–discrete solutions to the control problem is introduced which is utilized to prove existence and uniqueness of discrete controls in neighborhoods of regular continuous solutions. Furthermore, an optimal error estimate in terms of the spatial discretization parameter is given.Correspondence to: M. Hinze 相似文献
5.
Wei Liu 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(18):7543-7561
In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a Banach space with locally monotone operators, which is a generalization of the classical result for monotone operators. In particular, we show that local monotonicity implies pseudo-monotonicity. The main results are applied to PDE of various types such as porous medium equations, reaction–diffusion equations, the generalized Burgers equation, the Navier–Stokes equation, the 3D Leray-α model and the p-Laplace equation with non-monotone perturbations. 相似文献
6.
We prove the existence of a weak solution to Navier–Stokes equations describing the isentropic flow of a gas in a convex and bounded region, ΩR2, with nonhomogeneous Dirichlet boundary conditions on ∂Ω. These results are also extended to flow domain surrounding an obstacle. 相似文献
7.
In previous article [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075], we introduced a system of equations (phase-lock equations) to model the superconductivity phenomena. We investigated its connection to Ginzburg–Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the steady-state problem associated with the phase-lock equations. We prove that the steady-state problem has multiple solutions and show that the solution set enjoys some structural properties as proved by Foias and Teman for the Navier–Stokes equations in [C. Foias, R. Teman, Structure of the set of stationary solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. XXX (1977) 149–164]. 相似文献
8.
Yasushi Taniuchi 《Mathematische Zeitschrift》2009,261(3):597-615
We present a uniqueness theorem for time-periodic solutions to the Navier–Stokes equations in unbounded domains. Thus far,
results on the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domain, roughly speaking,
have only found that a small time-periodic L
n
-solution is unique within the class of solutions which have sufficiently small L
∞(L
n
)-norm. In this paper, we show that a small time-periodic L
n
-solution is unique within the class of all time-periodic L
n
-solutions, which contains large solutions. We also consider the uniqueness of solutions in weak-L
n
space. The proof of the present uniqueness theorem is based on the method of dual equations.
相似文献
9.
The barotropic compressible Navier–Stokes equations in an unbounded domain are studied. We prove the unique existence of the solution (u, p) of the system (1.1) in the Sobolev spaceHk + 3 × Hk + 2provided that the derivatives of the data of the problem are sufficiently small, wherek ≥ 0 is any integer. The proof follows from an analysis of the linearized problem, the solvability of the continuity equation, and the Schauder fixed point theory. Similar smoothness results are obtained for a linearized form of (1.1). 相似文献
10.
The present work establishes a Navier–Stokes limit for the Boltzmann equation considered over the infinite spatial domain R
3. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations whose limit points (in the w-L
1 topology) are governed by Leray solutions of the limiting Navier–Stokes equations. This completes the arguments in Bardos-Golse-Levermore [Commun. Pure Appl. Math. 46(5), 667–753 (1993)] for the steady case, and in Lions-Masmoudi [Arch. Ration. Mech. Anal. 158(3), 173–193 (2001)] for the time-dependent case.Mathematics Subject Classification (2000) 35Q35, 35Q30, 82C40 相似文献
11.
We derive an exact formula for solutions to the Stokes equations in the half space with an external forcing term. This formula is used to establish local and global existence and uniqueness in a suitable Besov space for solutions to the Navier Stokes equations. In particular, wellposedness is proved for initial data in L3(R3 +). 相似文献
12.
We show that the Lp spatial–temporal decay rates of solutions of incompressible flow in an 2D exterior domain. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in He, Xin [C. He, Z. Xin, Weighted estimates for nonstationary Navier–Stokes equations in exterior domain, Methods Appl. Anal. 7 (3) (2000) 443–458], and our previous results [H.-O. Bae, B.J. Jin, Asymptotic behavior of Stokes solutions in 2D exterior domains, J. Math. Fluid Mech., in press; H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains, submitted for publication]. For the spatial decay rate estimate, we first extend temporal decay rate result of the Navier–Stokes solutions for general Lp space when the initial velocity is in , 1<rq<∞ (1<r<q=∞). 相似文献
13.
Takahiro Okabe 《Journal of Evolution Equations》2011,11(2):265-286
We consider the initial boundary value problem to the Navier–Stokes equations in a bounded domain with the inhomogeneous time-dependent
data b(t) ? H1/2(?W){\beta(t) \in H^{1/2}(\partial\Omega)} under the general flux condition. We establish a reproductive property for weak solutions of the Navier–Stokes equations.
Here, the reproductive property is regarded as the generalization of the time periodicity. As an application, we can prove
the existence of periodic weak solutions. 相似文献
14.
We prove, on one hand, that for a convenient body force with values
in the distribution space (H
-1(D))
d
, where D is the geometric
domain of the fluid, there exist a velocity u and a pressure p
solution of the stochastic Navier–Stokes equation in dimension
2, 3 or 4.
On the other hand, we prove that, for a body force with values in the
dual space V of the divergence free subspace V of (H
1
0(D))
d
,
in general it is not possible to solve the stochastic Navier–Stokes
equations.
More precisely, although such body forces have been considered, there
is no topological space in which Navier–Stokes equations could be
meaningful for them. 相似文献
15.
Sergueï A. Nazarov Adlia Sequeira Juha H. Videman 《Journal de Mathématiques Pures et Appliquées》2001,80(10):445
In this paper our objective is to provide physically reasonable solutions for the stationary Navier–Stokes equations in a two-dimensional domain with two outlets to infinity, a semi-strip Π− and a half-plane K. The same problem in an aperture domain, i.e. in a domain with two half-plane outlets to infinity, has been studied but only under symmetry restrictions on the data. Here, we assume that the main asymptotic term of the solution takes an antisymmetric form in K and apply the technique of weighted spaces with detached asymptotics, i.e. we use spaces where the functions have prescribed asymptotic forms in the outlets.After first showing that the corresponding Stokes problem admits a unique solution if and only if certain compatibility conditions are satisfied, we write the Navier–Stokes equations as a perturbation of the Stokes problem and the crucial compatibility condition as an algebraic equation by which the flux becomes determined. Assuming that the coefficient of the main (antisymmetric) asymptotic term of the solution in K does not vanish and that the data are sufficiently small, we use a contraction principle to solve the Navier–Stokes system coupled with the algebraic equation.Finally, we discuss the ill-posedness of the Navier–Stokes problem with prescribed flux. 相似文献
16.
Abstract. In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier—Stokes equation in bounded
and unbounded domains. These solutions are stochastic analogs of the classical Lions—Prodi solutions to the deterministic
Navier—Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability
space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions
to the Navier—Stokes martingale problem where the probability space is also obtained as a part of the solution. 相似文献
17.
P. Maremonti 《Journal of Mathematical Sciences》2009,159(4):486-523
The Cauchy problem and the initial boundary value problem in the half-space of the Stokes and Navier–Stokes equations are
studied. The existence and uniqueness of classical solutions (u, π) (considered at least C
2 × C
1 smooth with respect to the space variable and C
1 × C
0 smooth with respect to the time variable) without requiring convergence at infinity are proved. A priori the fields u and π are nondecreasing at infinity. In the case of the Stokes problem, the existence, for any t > 0, and the uniqueness of solutions with kinetic field and pressure field are established for some β ∈ (0, 1) and γ ∈ (0, 1 − β). In the case of Navier–Stokes equations, the existence (local in time) and the uniqueness of classical solutions to the
Navier–Stokes equations are shown under the assumption that the initial data are only continuous and bounded, by proving that,
for any t ∈ (0, T), the kinetic field u(x, t) is bounded and, for any γ ∈ (0, 1), the pressure field π(x, t) is O(1 + |x|
γ
). Bibliography: 20 titles.
To V. A. Solonnikov on his 75th birthday
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 176–240. 相似文献
18.
We prove a backward uniqueness result for the heat operator with variable lower order terms, which implies the full regularity of L
3,-solutions of the three-dimensional Navier–Stokes equations. Bibliography: 12 titles. 相似文献
19.
We show the existence of time periodic solutions of the Navier–Stokes equations in bounded domains of
\mathbb R3{\mathbb R^3} with inhomogeneous boundary conditions in the strong and weak sense. In particular, for weak solutions, we deal with more
generalized conditions on the boundary data for Leray’s problem. 相似文献
20.
Stochastic 2-D Navier—Stokes Equation 总被引:1,自引:0,他引:1
Abstract. In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier—Stokes equation in bounded
and unbounded domains. These solutions are stochastic analogs of the classical Lions—Prodi solutions to the deterministic
Navier—Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability
space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions
to the Navier—Stokes martingale problem where the probability space is also obtained as a part of the solution. 相似文献