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A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations

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  • We construct a Poiseuille type flow which is a bounded entire solution of the nonstationary Navier-Stokes and the Stokes equations in a half space with non-slip boundary condition. Our result in particular implies that there is a nontrivial solution for the Liouville problem under the non-slip boundary condition. A review for cases of the whole space and a slip boundary condition is included.
    Mathematics Subject Classification: Primary: 35Q30; Secondary: 35B53, 76D05.


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  • [1]

    K. Abe and Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions, Hokkaido University Preprint Series in Mathematics, 980 (2011).


    D. Chae, Liouville type of theorems for the Euler and the Navier-Stokes equations, Adv. Math., 228 (2011), 2855-2868.doi: 10.1016/j.aim.2011.07.020.


    D. Chae, On the Liouville type of theorems with weights for the Navier-Stokes equations and the Euler equations, Differential Integral Equations, 25 (2012), 403-416.


    D. Chae, Note on the incompressible Euler and related equations on $\mathbfR^N$, Chin. Ann. Math. Ser. B, 30 (2009), 513-526.doi: 10.1007/s11401-009-0107-4.


    C.-C. Chen, R. M. Strain, H.-T. Yau and T.-P. TsaiLower bound on the blow-up rate of the axisymmetric Navier-Stokes equations, Int. Math. Res. Not. IMRN, 2008, Art. ID rnn016, 31 pp. doi: 10.1093/imrn/rnn016.


    C.-C. Chen, R. M. Strain, T.-P. Tsai and H.-T. Yau, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations. II, Comm. Partial Differential Equations, 34 (2009), 203-232.doi: 10.1080/03605300902793956.


    P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42 (1993), 775-789.doi: 10.1512/iumj.1993.42.42034.


    E. De Giorgi, "Frontiere Orientate di Misura Minima," Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61, Editrice Tecnico Scientifica, Pisa, 1961.


    B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differential Equations, 6 (1981), 883-901.doi: 10.1080/03605308108820196.


    M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations - Asymptotic Behavior of Solutions and Self-Similar Solutions," Progress in Nonlinear Differential Equations and Their Applications, 79, Birkhäuser Boston, Inc., Boston, MA, 2010.doi: 10.1007/978-0-8176-4651-6.


    Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys., 103 (1986), 415-421.


    Y. Giga and R. V. Kohn, Characterizing blow-up using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40.doi: 10.1512/iumj.1987.36.36001.


    Y. Giga and H. Miura, On vorticity directions near singularities for the Navier-Stokes flows with infinite energy, Comm. Math. Phys., 303 (2011), 289-300.doi: 10.1007/s00220-011-1197-x.


    E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monograph in Mathematics, 80, Birkhäuser Verlag, Basel, 1984.


    R. Hamilton, The formation of singularities in the Ricci flow, in "Surveys in differential geometry, Vol II" (Cambridge, MA, 1993), Int. Press, Cambridge, MA, (1995), 7-136.


    P.-Y. Hsu and Y. Maekawa, On nonexistence for stationary solutions to the Navier-Stokes equations with a linear strain, preprint, (2011).


    K. Kang, Unbounded normal derivative for the Stokes system near boundary, Math. Annal., 331 (2005), 87-109.doi: 10.1007/s00208-004-0575-5.


    G. Koch, N. Nadirashvilli, G. Seregin and V. Svěrák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105.doi: 10.1007/s11511-009-0039-6.


    O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Math. Soc., Providence, RI, 1968.


    Y. Maekawa, Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit, preprint, (2011).


    T. Ohyama, Interior regularity of weak solutions to the time-dependent Navier-Stokes equation, Proc. Japan Acad., 36 (1960), 273-277.


    P. Polácik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. II. Parabolic equations, Indiana Univ. Math. J., 56 (2007), 879-908.doi: 10.1512/iumj.2007.56.2911.


    M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967.


    P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States," Birkhäuser Advanced Texts: Basler Lehrbücher, [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007.


    G. Seregin and V. Šverák, On type I singularities of the local axi-symmetric solutions of the Navier-Stokes equations, Comm. Partial Differential Equations, 34 (2009), 171-201.doi: 10.1080/03605300802683687.


    J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.


    M. Struwe, Geometric evolution problems, in "Nonlinear Partial Differential Equations in Differential Geometry" (Park City, UT, 1992), IAS/Park City Math. Ser., 2, Amer. Math. Soc., Providence, RI, (1996), 257-339.

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