Article Contents
Article Contents

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

• 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.

 Citation:

•  [1] K. Abe and Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions, Hokkaido University Preprint Series in Mathematics, 980 (2011). [2] 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. [3] 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. [4] 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. [5] C.-C. Chen, R. M. Strain, H.-T. Yau and T.-P. Tsai, Lower 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. [6] 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. [7] 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. [8] E. De Giorgi, "Frontiere Orientate di Misura Minima," Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61, Editrice Tecnico Scientifica, Pisa, 1961. [9] 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. [10] 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. [11] Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys., 103 (1986), 415-421. [12] 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. [13] 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. [14] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monograph in Mathematics, 80, Birkhäuser Verlag, Basel, 1984. [15] 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. [16] P.-Y. Hsu and Y. Maekawa, On nonexistence for stationary solutions to the Navier-Stokes equations with a linear strain, preprint, (2011). [17] K. Kang, Unbounded normal derivative for the Stokes system near boundary, Math. Annal., 331 (2005), 87-109.doi: 10.1007/s00208-004-0575-5. [18] 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. [19] 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. [20] Y. Maekawa, Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit, preprint, (2011). [21] T. Ohyama, Interior regularity of weak solutions to the time-dependent Navier-Stokes equation, Proc. Japan Acad., 36 (1960), 273-277. [22] 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. [23] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967. [24] 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. [25] 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. [26] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195. [27] 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.