American Institute of Mathematical Sciences

Advanced
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.  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

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.  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

[1]

Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045
[2]

Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353
[3]

Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719
[4]

Linjie Xiong. Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary. Kinetic & Related Models, 2018, 11 (3) : 469-490. doi: 10.3934/krm.2018021
[5]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
[6]

Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769
[7]

Dongho Chae, Shangkun Weng. Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5267-5285. doi: 10.3934/dcds.2016031
[8]

Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153
[9]

Jean-Pierre Raymond. Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1537-1564. doi: 10.3934/dcdsb.2010.14.1537
[10]

Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783
[11]

Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
[12]

Boris Muha, Zvonimir Tutek. Note on evolutionary free piston problem for Stokes equations with slip boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1629-1639. doi: 10.3934/cpaa.2014.13.1629
[13]

Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 741-766. doi: 10.3934/nhm.2012.7.741
[14]

Franck Boyer, Pierre Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219
[15]

Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201
[16]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[17]

Yoshihiro Shibata. On the local wellposedness of free boundary problem for the Navier-Stokes equations in an exterior domain. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1681-1721. doi: 10.3934/cpaa.2018081
[18]

Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135
[19]

Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17
[20]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences