共查询到20条相似文献,搜索用时 15 毫秒
1.
Reinhard Farwig Christian Komo 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):303-321
Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain
W ì \mathbbR3{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u
0, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u
0 and f or on the solution u itself such as Serrin’s condition || u ||Ls(0,T; Lq(W)) < ¥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with
2 < s < ¥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math
J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g.
u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||Ls¢(t-d,t; Lq(W)){\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy
\frac12|| u(t) ||22{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}. 相似文献
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Qionglei Chen Changxing Miao Zhifei Zhang 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2009
In this paper, we improve some known uniqueness results of weak solutions for the 3D Navier–Stokes equations. The proof uses the Fourier localization technique and the losing derivative estimates. 相似文献
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The purpose of this work is to study the global-in-time existence of weak solutions of a viscous capillary model of plasma expressed as a so-called Navier–Stokes–Poisson–Korteweg model for large data in three-dimensional space. Using the compactness argument, we prove the existence of global weak solutions in the classical sense to such system with a cold pressure. 相似文献
8.
Hyunseok Kim 《Annali dell'Universita di Ferrara》2009,55(2):279-287
We study the stationary Navier–Stokes equations in a bounded domain Ω of R
3 with smooth connected boundary. The notion of very weak solutions has been introduced by Marušić-Paloka (Appl. Math. Optim.
41:365–375, 2000), Galdi et al. (Math. Ann. 331:41–74, 2005) and Kim (Arch. Ration. Mech. Anal. 193:117–152, 2009) to obtain
solvability results for the Navier–Stokes equations with very irregular data. In this article, we prove a complete solvability
result which unifies those in Marušić-Paloka (Appl. Math. Optim. 41:365–375, 2000), Galdi et al. (Math. Ann. 331:41–74, 2005)
and Kim (Arch. Ration. Mech. Anal. 193:117–152, 2009) by adapting the arguments in Choe and Kim (Preprint) and Kim and Kozono
(Preprint). 相似文献
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《Journal de Mathématiques Pures et Appliquées》2005,84(11):1496-1514
In this paper we obtain a new regularity criterion for weak solutions to the 3-D Navier–Stokes equations. We show that if any one component of the velocity field belongs to with , , then the weak solution actually is regular and unique. 相似文献
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Zuoshunhua Shi 《Journal of Differential Equations》2018,264(3):1550-1580
In this paper, we mainly study the existence of self-similar solutions of stationary Navier–Stokes equations for dimension . For , if the external force is axisymmetric, scaling invariant, continuous away from the origin and small enough on the sphere , we shall prove that there exists a family of axisymmetric self-similar solutions which can be arbitrarily large in the class . Moreover, for axisymmetric external forces without swirl, corresponding to this family, the momentum flux of the flow along the symmetry axis can take any real number. However, there are no regular () axisymmetric self-similar solutions provided that the external force is a large multiple of some scaling invariant axisymmetric F which cannot be driven by a potential. In the case of dimension 4, there always exists at least one self-similar solution to the stationary Navier–Stokes equations with any scaling invariant external force in . 相似文献
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We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in [15] for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier–Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data. 相似文献
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H. B. de Oliveira 《NoDEA : Nonlinear Differential Equations and Applications》2013,20(3):797-824
In this work we consider the generalized Navier–Stokes equations with the presence of a damping term in the momentum equation. The problem studied here derives from the set of equations which govern isothermal flows of incompressible and homogeneous non-Newtonian fluids. For the generalized Navier–Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any ${q > \frac{2N}{N+2}}$ and any σ > 1, where q is the exponent of the diffusion term and σ is the exponent which characterizes the damping term. 相似文献
16.
Fei Jiang 《Nonlinear Analysis: Real World Applications》2011,12(3):1733-1735
In [A. Jüngel, Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal. 42 (2010) 1025–1045], Jüngel proved the global existence of the barotropic compressible quantum Navier–Stokes equations for when the viscosity constant is bigger than the scaled Planck constant. Recently, Dong [J. Dong, A note on barotropic compressible quantum Navier–Stokes equations, Nonlinear Anal. TMA 73 (2010) 854–856] extended Jüngel’s result to the case where the viscosity constant is equal to the scaled Planck constant by using a new estimate of the square root of the solutions. In this paper we show that Jüngel’s existence result still holds when the viscosity constant is bigger than the scaled Planck constant. Consequently, with our result, the existence for all physically interesting cases of the scaled Planck and viscosity constants is obtained. 相似文献
17.
On partial regularity of suitable weak solutions to the Navier–Stokes equations in unbounded domains
Tomoyuki Suzuki 《manuscripta mathematica》2008,125(4):471-493
Consider the nonstationary Navier–Stokes equations in Ω × (0, T), where Ω is a general unbounded domain with non-compact boundary in R
3. We prove the regularity of suitable weak solutions for large |x|. It should be noted that our result also holds near the boundary. Our result extends the previous ones by Caffarelli–Kohn–Nirenberg
in R
3 and Sohr-von Wahl in exterior domains to general domains. 相似文献
18.
A. Mikhailov 《Journal of Mathematical Sciences》2010,166(1):40-52
A class of sufficient conditions for the local boundary regularity of suitable weak solutions of nonstationary three-dimensional
Navier–Stokes equations is discussed. The corresponding results are stated in terms of functionals invariant with respect
to the scaling of Navier–Stokes equations. Bibliography: 26 titles. 相似文献
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