共查询到20条相似文献,搜索用时 250 毫秒
1.
Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces 总被引:1,自引:0,他引:1
Katrin Schumacher 《Annali dell'Universita di Ferrara》2008,54(1):123-144
We investigate very weak solutions to the stationary Stokes and Stokes resolvent problem in function spaces with Muckenhoupt
weights. The notion used here is similar but even more general than the one used in Amann (Nonhomogeneous Navier–Stokes equations
with integrable low-regularity data. Int. Math. Ser., pp. 1–26. Kluwer Academic/Plenum Publishing, New York, 2002) or Galdi
et al. (Math. Ann. 331, 41–74, 2005). Consequently the class of solutions is enlarged. To describe boundary conditions we restrict ourselves to
more regular data. We introduce a Banach space that admits a restriction operator and that contains the solutions according
to such data.
相似文献
2.
The purpose of this work is to solve the exterior Stokes problem in the half-space . We study the existence and the uniqueness of generalized solutions in weighted L
p
theory with 1 < p < ∞. Moreover, we consider the case of strong solutions and very weak solutions. This paper extends the studies done in Alliot,
Amrouche (Math. Methods Appl. 23:575–600, 2000) for an exterior Stokes problem in the whole space and in Amrouche, Bonzom
(Exterior Problems in the Half-space, submitted) for the Laplace equation in the same geometry as here.
相似文献
3.
Solution of the spectral problem for the curl and Stokes operators with periodic boundary conditions
R. S. Saks 《Journal of Mathematical Sciences》2006,136(2):3794-3811
In this paper, we establish relations between eigenvalues and eigenfunctions of the curl operator and Stokes operator (with
periodic boundary conditions). These relations show that the curl operator is the square root of the Stokes operator with
ν = 1. The multiplicity of the zero eigenvalue of the curl operator is infinite. The space L
2(Q, 2π) is decomposed into a direct sum of eigenspaces of the operator curl. For any complex number λ, the equation rot
u + λu = f and the Stokes equation −ν(Δv + λ 2v) + ∇p = f, div v = 0, are solved. Bibliography: 15 titles.
Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 246–276. 相似文献
4.
In this paper, we prove that any weak solution to the non-stationary Stokes system in 3D with right hand side -div f satisfying (1.4) below, belongs to C( ]0, T[; C
α (Ω)). The proof is based on Campanato-type inequalities and the existence of a local pressure introduced in Wolf [13].
Proc. Conference “Variational analysis and PDE’s”. Intern. Centre “E. Majorana”, Erice, July 5–14, 2006. 相似文献
5.
In this paper, we investigate -estimates for the solution of the Stokes equation in a half space H where . It is shown that the solution of the Stokes equation is governed by an analytic semigroup on or . From the operatortheoretical point of view it is a surprising fact that the corresponding result for does not hold true. In fact, there exists an -function f satisfying such that the solution of the corresponding resolvent equation with right hand side f does not belong to . Taking into account however a recent result of Kozono on the nonlinear Navier-Stokes equation, the -result is not surprising and even natural. We also show that the Stokes operator admits a R-bounded -calculus on for 1 < p < and obtain as a consequence maximal -regularity for the solution of the Stokes equation.
Received August 24, 2000; accepted September 30, 2000. 相似文献
6.
A fully discrete multi-level spectral Galerkin method in space–time for the two-dimensional nonstationary Navier–Stokes problem
is considered. The method is a multi-scale method in which the fully nonlinear Navier–Stokes problem is only solved on the
lowest-dimensional space
with the largest time step Δt
1; subsequent approximations are generated on a succession of higher-dimensional spaces
with small time step Δt
j by solving a linearized Navier–Stokes problem about the solution on the previous level. Some error estimates are also presented
for the J-level spectral Galerkin method. The scaling relations of the dimensional numbers and time mesh widths that lead to optimal
accuracy of the approximate solution in H
1-norm and L
2-norm are investigated, i.e., m
j∼m
j−1
3/2
, Δt
j∼Δt
j−1
3/2
, j=2,. . .,J. We demonstrate theoretically that a fully discrete J-level spectral Galerkin method is significantly more efficient than the standard one-level spectral Galerkin method.
Mathematics subject classifications (2000) 35L70, 65N30, 76D06
Subsidized by the Special Funds for Major State Basic Research Projects G1999032801-07, NSF of China 10371095 and the City
University of Hong Kong Research Project 7001093, NSF of China 50323001. 相似文献
7.
In this paper we consider a resolvent problem of the Stokes operator with some boundary condition in the half space, which is obtained as a model problem arising in evolution free boundary problems for viscous, incompressible fluid flow. We show standard resolvent estimates in the Lq framework (1 < q < ∞), applying some kernel estimates to concrete solution formulas. The Volevich trick in [21] plays a fundamental role in estimating solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
8.
In the first part of the paper, we give a satisfactory definition of the Stokes operator in Lipschitz domains in
\mathbbRn {\mathbb{R}^n} when boundary conditions of Neumann type are considered. We then proceed to establish optimal global Sobolev regularity results
for vector fields in the domains of fractional powers of this Neumann–Stokes operator. Finally, we study the existence, regularity,
and uniqueness of mild solutions of the Navier–Stokes system with Neumann boundary conditions. Bibliography: 43 titles. Illustrations:
2 figures. 相似文献
9.
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
10.
Helmut Abels 《Mathematische Nachrichten》2006,279(4):351-367
In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω0 = ℝn –1 × (–1, 1), n ≥ 2, in Lq ‐Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω+0 = ℝn –1 × {1} and non‐slip boundary condition on the “lower boundary” ∂Ω–0 = ℝn –1 × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an Lq ‐theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L 2‐spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
11.
T. Junges Miotto 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):337-353
Based on a new Liouville theorem, we study a superlinear Ambrosetti–Prodi problem for the p-Laplacian operator, 1 < p < N. For this, we use the sub and supersolution method, blow up technique and the Leray–Schauder degree theory. 相似文献
12.
Cyril Odasso 《Probability Theory and Related Fields》2008,140(1-2):41-82
We establish a general criterion which ensures exponential mixing of parabolic stochastic partial differential equations (SPDE)
driven by a non additive noise which is white in time and smooth in space. We apply this criterion on two representative examples:
2D Navier–Stokes (NS) equations and Complex Ginzburg–Landau (CGL) equation with a locally Lipschitz noise. Due to the possible
degeneracy of the noise, Doob theorem cannot be applied. Hence, a coupling method is used in the spirit of Kuksin and Shirikyan
(J. Math. Pures Appl. 1:567–602, 2002) and Mattingly (Commun. Math. Phys. 230:421–462, 2002).
Previous results require assumptions on the covariance of the noise which might seem restrictive and artificial. For instance,
for NS and CGL, the covariance operator is supposed to be diagonal in the eigenbasis of the Laplacian and not depending on
the high modes of the solutions. The method developed in the present paper gets rid of such assumptions and only requires
that the range of the covariance operator contains the low modes. 相似文献
13.
A new method is given to establish Lq,r-estimates for solutions of the nonstationary Stokes problem. The method is based on estimates for heat potentials in these
spaces, and it is not connected with investigation of the resolvent for the Stokes operator. It is expected that the method
is applicable to a wide class of parabolic initial boundary-value problems. Bibliography:11 titles.
Translated fromZapiski Nauchnykh Seminarow POMI, Vol. 222, 1994, pp. 124–151. 相似文献
14.
T. S. Kopaliani 《Ukrainian Mathematical Journal》2008,60(12):2006-2014
We point out that if the Hardy–Littlewood maximal operator is bounded on the space L
p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L
p
(ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L
p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L
p(t) (ℝ
n
), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ
n
, if and only if p(t) = const.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008. 相似文献
15.
We consider the full system of compressible Navier–Stokes equations for heat conducting fluid. We show that the temperature
is uniformly positive for t ≥ t
0 (for any t
0 > 0) for any solutions with finite initial entropy. The assumptions on the viscosity and conductivity coefficients are minimal
(for instance, the solutions constructed by Feireisl in (Oxford Lecture Series in Mathematics and its Applications, vol 26,
2004) verify all the requirements).
相似文献
16.
In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity ω0 and the circulation γ of the initial flow around the obstacle. We prove that, if γ is sufficiently small, the limit flow satisfies the full-plane Navier–Stokes system, with initial vorticity ω0 + γδ, where δ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in Iftimie et al. (Comm. Part. Differ. Eqn. 28, 349–379 (2003)), where the effect of the small obstacle appears in the coefficients of the PDE and not only in the initial data. The main ingredients of the proof are L
p
− L
q
estimates for the Stokes operator in an exterior domain, a priori estimates inspired on Kato’s fixed point method, energy estimates, renormalization and interpolation. 相似文献
17.
Takayuki Abe 《Mathematical Methods in the Applied Sciences》2004,27(9):1007-1048
This paper is concerned with the standard Lp estimate of solutions to the resolvent problem for the Stokes operator on an infinite layer with ‘Neumann–Dirichlet‐type’ boundary condition. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
18.
Xiang-Dong Li 《Probability Theory and Related Fields》2008,141(1-2):247-281
Under the condition that the Bakry–Emery Ricci curvature is bounded from below, we prove a probabilistic representation formula
of the Riesz transforms associated with a symmetric diffusion operator on a complete Riemannian manifold. Using the Burkholder
sharp L
p
-inequality for martingale transforms, we obtain an explicit and dimension-free upper bound of the L
p
-norm of the Riesz transforms on such complete Riemannian manifolds for all 1 < p < ∞. In the Euclidean and the Gaussian cases, our upper bound is asymptotically sharp when p→ 1 and when p→ ∞.
Research partially supported by a Delegation in CNRS at the University of Paris-Sud during the 2005–2006 academic year. 相似文献
19.
Yinnian He Yan Zhang Yueqiang Shang Hui Xu 《Numerical Methods for Partial Differential Equations》2012,28(5):1620-1642
A combination method of the Newton iteration and two‐level finite element algorithm is applied for solving numerically the steady Navier‐Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the m Newton iterations for solving the Navier‐Stokes problem on a coarse grid and computing the Stokes problem on a fine grid. Then, the uniform stability and convergence with respect to ν of the two‐level Newton iterative solution are analyzed for the large m and small H and h << H. Finally, some numerical tests are made to demonstrate the effectiveness of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012 相似文献
20.
María del Mar González 《Calculus of Variations and Partial Differential Equations》2009,36(2):173-210
We prove a Γ-convergence result for an energy functional related to some fractional powers of the Laplacian operator, (−Δ)
s
for 1/2 < s < 1, with two singular perturbations, that leads to a two-phase problem. The case (−Δ)1/2 was considered by Alberti–Bouchitté–Seppecher in relation to a model in capillarity with line tension effect. However, the
proof in our setting requires some new ingredients such as the Caffarelli–Silvestre extension for the fractional Laplacian
and new trace inequalities for weighted Sobolev spaces. 相似文献