Advanced Search
Article Contents
Article Contents

Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle

Abstract Related Papers Cited by
  • We consider the global existence of the two-dimensional Navier-Stokes flow in the exterior of a moving or rotating obstacle. Bogovski$\check{i}$ operator on a subset of $\mathbb{R}^2$ is used in this paper. One important thing is to show that the solution of the equations does not blow up in finite time in the sense of some $L^2$ norm. We also obtain the global existence for the 2D Navier-Stokes equations with linearly growing initial velocity.
    Mathematics Subject Classification: Primary: 35A01, 35B65; Secondary: 53C35.


    \begin{equation} \\ \end{equation}
  • [1]

    H. Amann, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 16-98.doi: 10.1007/s000210050018.


    A. Banin, A. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Indiana Univ. Math. J., 48 (1999), 1133-1176.doi: 10.1016/S0893-9659(99)00208-6.


    A. Banin, A. Mahalov and B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J., 50 (2001), 1-35.doi: 10.1512/iumj.2001.50.2155.


    H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.), 1 (1995), 197-263.doi: 10.1007/BF01671566.


    M. E. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR, 248 (1979), 1037-1040.


    W. Borchers, Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten, Habilitationschrift Universität Paderborn, 1992.


    M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics, 3, North-Holland, Amsterdam, 2004, 161-244.


    D. C. Chang, The dual of Hardy spaces on a domain in $\mathbbR^n$, Forum Math., 6 (1994), 65-81.doi: 10.1515/form.1994.6.65.


    D. C. Chang, G. Dafni and C. Sadosky, A div-curl lemma in BMO on a domain, Progr. Math., 238 (2005), 55-65.doi: 10.1007/0-8176-4416-4_5.


    D. C. Chang, G. Dafni and E. M. Stein, Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in $\mathbbR^n$, Trans. Amer. Math. Soc., 351 (1999), 1605-1661.doi: 10.1090/S0002-9947-99-02111-X.


    D. C. Chang, S. G. Krantz and E. M. Stein, $\mathcal H^p$ theory on a smooth domain in $\mathbbR^n$ and elliptic boundary value problems, J. Funct. Anal., 114 (1993), 286-347.doi: 10.1006/jfan.1993.1069.


    R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, {Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.


    Z. Chen and T. Miyakawa, Decay properties of weak solutions to a perturbed Navier-Stokes system in $\mathbbR^n$, Adv. Math. Sci. Appl., 7 (1997), 741-770.


    P. Cumsille and M. Tucsnak, Well-posedness for the Navier-Stokes flow in exterior of a rotating obstacle, Math. Methods in the Applied Sciences, 29 (2006), 595-623.doi: 10.1002/mma.702.


    D. Y. Fang, M. Hieber and T. Zhang, Density-dependent incompressible viscous fluid flow subject to linearly growing initial data, Applicable Analysis, 91 (2012), 1477-1493.doi: 10.1080/00036811.2011.608160.


    D. Y. Fang, B. Han and T. Zhang, Global existencefor the two dimensional incompressible viscous fluids with linearly growing velocity, Mathematical Methods in the Applied Sciences, 36 (2013), 921-935.doi: 10.1002/mma.2649.


    G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Tracts Natur. Philos., Springer-Verlag, New York, 1994.doi: 10.1007/978-1-4612-5364-8.


    M. Geissert, H. Heck and M. Hieber, $L^p$-theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, J. Reine Angew. Math., 596 (2006), 45-62.doi: 10.1515/CRELLE.2006.051.


    Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rat. Mech. Anal., 89 (1985), 267-281.doi: 10.1007/BF00276875.


    Y. Giga, S. Matsui and O. Sawada, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity, J. Math. Fluid Mech., 3 (2001), 302-315.doi: 10.1007/PL00000973.


    T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of rotating obstacle, Arch. Roational Mech. Anal., 150 (1999), 307-348.doi: 10.1007/s002050050190.


    T. Hishida, The Stokes operator with rotation effect in exterior domains, Analysis, 19 (1999), 51-67.doi: 10.1524/anly.1999.19.1.51.


    M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbbR^N$ with linearly growing initial data, Arch. Roational Mech. Anal., 175 (2005), 269-285.doi: 10.1007/s00205-004-0347-0.


    O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.


    J. Leray, Sur le mouvement d'un liquide visqueux remplissant l'espace, Acta mathematica, 63 (1934), 193-248.doi: 10.1007/BF02547354.


    A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math., 39 (1986), 187-220.doi: 10.1002/cpa.3160390711.


    H. Okamoto, Exact solutions of the Navier-Stokes equations via Leray's scheme, Japan J. Indust. Appl. Math., 14 (1997), 169-197.doi: 10.1007/BF03167263.


    H. Sohr, The Navier-Stokes Equations, Birkhäuser Advanced Texts, Basel 2001.


    V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Sov. Math., 8 (1977), 467-529.


    L. Tartar, An Introduction to Sobolev Space and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, Springer-Verlag Berlin Heidelberg, 2007.

  • 加载中

Article Metrics

HTML views() PDF downloads(259) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint