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

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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

Abstract
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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.

Keywords: Navier-Stokes equations, Liouville problem, non-slip boundary condition, Poiseuille type flow..

Mathematics Subject Classification: Primary: 35Q30; Secondary: 35B53, 76D0.

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).
[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.
[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.
[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.
[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.
[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.
[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.
[8]	E. De Giorgi, "Frontiere Orientate di Misura Minima,", Seminario di Matematica della Scuola Normale Superiore di Pisa, (1961), 1960.
[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.
[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.
[11]	Y. Giga, A bound for global solutions of semilinear heat equations,, Comm. Math. Phys., 103 (1986), 415.
[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.
[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.
[14]	E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Monograph in Mathematics, 80 (1984).
[15]	R. Hamilton, The formation of singularities in the Ricci flow,, in, (1995), 7.
[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. 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. 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, (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.
[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.
[23]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Prentice-Hall, (1967).
[24]	P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (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. 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.
[27]	M. Struwe, Geometric evolution problems,, in, 2 (1996), 257.

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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).
[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.
[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.
[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.
[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.
[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.
[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.
[8]	E. De Giorgi, "Frontiere Orientate di Misura Minima,", Seminario di Matematica della Scuola Normale Superiore di Pisa, (1961), 1960.
[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.
[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.
[11]	Y. Giga, A bound for global solutions of semilinear heat equations,, Comm. Math. Phys., 103 (1986), 415.
[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.
[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.
[14]	E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Monograph in Mathematics, 80 (1984).
[15]	R. Hamilton, The formation of singularities in the Ricci flow,, in, (1995), 7.
[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. 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. 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, (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.
[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.
[23]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Prentice-Hall, (1967).
[24]	P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (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. 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.
[27]	M. Struwe, Geometric evolution problems,, in, 2 (1996), 257.

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