American Institute of Mathematical Sciences

October  2013, 6(5): 1277-1289. doi: 10.3934/dcdss.2013.6.1277

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

 1 Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914

Received  November 2011 Revised  January 2012 Published  March 2013

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.
Citation: Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277
References:
 [1] K. Abe and Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions,, Hokkaido University Preprint Series in Mathematics, 980 (2011). Google Scholar [2] D. Chae, Liouville type of theorems for the Euler and the Navier-Stokes equations,, Adv. Math., 228 (2011), 2855. doi: 10.1016/j.aim.2011.07.020. Google Scholar [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. Google Scholar [4] D. Chae, Note on the incompressible Euler and related equations on $\mathbfR^N$,, Chin. Ann. Math. Ser. B, 30 (2009), 513. doi: 10.1007/s11401-009-0107-4. Google Scholar [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 (). doi: 10.1093/imrn/rnn016. Google Scholar [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. doi: 10.1080/03605300902793956. Google Scholar [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. doi: 10.1512/iumj.1993.42.42034. Google Scholar [8] E. De Giorgi, "Frontiere Orientate di Misura Minima,", Seminario di Matematica della Scuola Normale Superiore di Pisa, (1961), 1960. Google Scholar [9] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Commun. Partial Differential Equations, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar [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 (2010). doi: 10.1007/978-0-8176-4651-6. Google Scholar [11] Y. Giga, A bound for global solutions of semilinear heat equations,, Comm. Math. Phys., 103 (1986), 415. Google Scholar [12] Y. Giga and R. V. Kohn, Characterizing blow-up using similarity variables,, Indiana Univ. Math. J., 36 (1987), 1. doi: 10.1512/iumj.1987.36.36001. Google Scholar [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. doi: 10.1007/s00220-011-1197-x. Google Scholar [14] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Monograph in Mathematics, 80 (1984). Google Scholar [15] R. Hamilton, The formation of singularities in the Ricci flow,, in, (1995), 7. Google Scholar [16] P.-Y. Hsu and Y. Maekawa, On nonexistence for stationary solutions to the Navier-Stokes equations with a linear strain,, preprint, (2011). Google Scholar [17] K. Kang, Unbounded normal derivative for the Stokes system near boundary,, Math. Annal., 331 (2005), 87. doi: 10.1007/s00208-004-0575-5. Google Scholar [18] G. Koch, N. Nadirashvilli, G. Seregin and V. Svěrák, Liouville theorems for the Navier-Stokes equations and applications,, Acta Math., 203 (2009), 83. doi: 10.1007/s11511-009-0039-6. Google Scholar [19] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1968). Google Scholar [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). Google Scholar [21] T. Ohyama, Interior regularity of weak solutions to the time-dependent Navier-Stokes equation,, Proc. Japan Acad., 36 (1960), 273. Google Scholar [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. doi: 10.1512/iumj.2007.56.2911. Google Scholar [23] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Prentice-Hall, (1967). Google Scholar [24] P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2007). Google Scholar [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. doi: 10.1080/03605300802683687. Google Scholar [26] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187. Google Scholar [27] M. Struwe, Geometric evolution problems,, in, 2 (1996), 257. Google Scholar

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References:
 [1] K. Abe and Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions,, Hokkaido University Preprint Series in Mathematics, 980 (2011). Google Scholar [2] D. Chae, Liouville type of theorems for the Euler and the Navier-Stokes equations,, Adv. Math., 228 (2011), 2855. doi: 10.1016/j.aim.2011.07.020. Google Scholar [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. Google Scholar [4] D. Chae, Note on the incompressible Euler and related equations on $\mathbfR^N$,, Chin. Ann. Math. Ser. B, 30 (2009), 513. doi: 10.1007/s11401-009-0107-4. Google Scholar [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 (). doi: 10.1093/imrn/rnn016. Google Scholar [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. doi: 10.1080/03605300902793956. Google Scholar [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. doi: 10.1512/iumj.1993.42.42034. Google Scholar [8] E. De Giorgi, "Frontiere Orientate di Misura Minima,", Seminario di Matematica della Scuola Normale Superiore di Pisa, (1961), 1960. Google Scholar [9] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Commun. Partial Differential Equations, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar [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 (2010). doi: 10.1007/978-0-8176-4651-6. Google Scholar [11] Y. Giga, A bound for global solutions of semilinear heat equations,, Comm. Math. Phys., 103 (1986), 415. Google Scholar [12] Y. Giga and R. V. Kohn, Characterizing blow-up using similarity variables,, Indiana Univ. Math. J., 36 (1987), 1. doi: 10.1512/iumj.1987.36.36001. Google Scholar [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. doi: 10.1007/s00220-011-1197-x. Google Scholar [14] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Monograph in Mathematics, 80 (1984). Google Scholar [15] R. Hamilton, The formation of singularities in the Ricci flow,, in, (1995), 7. Google Scholar [16] P.-Y. Hsu and Y. Maekawa, On nonexistence for stationary solutions to the Navier-Stokes equations with a linear strain,, preprint, (2011). Google Scholar [17] K. Kang, Unbounded normal derivative for the Stokes system near boundary,, Math. Annal., 331 (2005), 87. doi: 10.1007/s00208-004-0575-5. Google Scholar [18] G. Koch, N. Nadirashvilli, G. Seregin and V. Svěrák, Liouville theorems for the Navier-Stokes equations and applications,, Acta Math., 203 (2009), 83. doi: 10.1007/s11511-009-0039-6. Google Scholar [19] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1968). Google Scholar [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). Google Scholar [21] T. Ohyama, Interior regularity of weak solutions to the time-dependent Navier-Stokes equation,, Proc. Japan Acad., 36 (1960), 273. Google Scholar [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. doi: 10.1512/iumj.2007.56.2911. Google Scholar [23] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Prentice-Hall, (1967). Google Scholar [24] P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2007). Google Scholar [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. doi: 10.1080/03605300802683687. Google Scholar [26] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187. Google Scholar [27] M. Struwe, Geometric evolution problems,, in, 2 (1996), 257. Google Scholar
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