# 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), 2855-2868. 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
 [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, 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, 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, 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, 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, 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] Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3489-3530. doi: 10.3934/dcds.2021005 [10] 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 [11] 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 [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] 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, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207 [14] Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234 [15] 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 [16] 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 [17] 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 [18] 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 [19] 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 [20] 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

2019 Impact Factor: 1.233