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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

Yoshikazu Giga 1,

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.
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). Google Scholar

[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.  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), 403-416.  Google Scholar

[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.  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), 203-232. 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), 775-789. 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, 1960-61, Editrice Tecnico Scientifica, Pisa, 1961.  Google Scholar

[9]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differential Equations, 6 (1981), 883-901. 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, Birkhäuser Boston, Inc., Boston, MA, 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), 415-421.  Google Scholar

[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.  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), 289-300. doi: 10.1007/s00220-011-1197-x.  Google Scholar

[14]

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

[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.  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), 87-109. 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), 83-105. 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, Vol. 23, American Math. Soc., Providence, RI, 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), 273-277.  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), 879-908. doi: 10.1512/iumj.2007.56.2911.  Google Scholar

[23]

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

[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.  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), 171-201. 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), 187-195.  Google Scholar

[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.  Google Scholar

show all references

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), 2855-2868. 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), 403-416.  Google Scholar

[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.  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), 203-232. 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), 775-789. 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, 1960-61, Editrice Tecnico Scientifica, Pisa, 1961.  Google Scholar

[9]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differential Equations, 6 (1981), 883-901. 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, Birkhäuser Boston, Inc., Boston, MA, 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), 415-421.  Google Scholar

[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.  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), 289-300. doi: 10.1007/s00220-011-1197-x.  Google Scholar

[14]

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

[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.  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), 87-109. 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), 83-105. 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, Vol. 23, American Math. Soc., Providence, RI, 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), 273-277.  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), 879-908. doi: 10.1512/iumj.2007.56.2911.  Google Scholar

[23]

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

[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.  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), 171-201. 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), 187-195.  Google Scholar

[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.  Google Scholar

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