-
Previous Article
On the anisotropic Orlicz spaces applied in the problems of continuum mechanics
- DCDS-S Home
- This Issue
-
Next Article
$H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions
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 |
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), 2855-2868.
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), 403-416. |
[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. |
[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), 203-232.
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), 775-789.
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, 1960-61, Editrice Tecnico Scientifica, Pisa, 1961. |
[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. |
[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. |
[11] |
Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys., 103 (1986), 415-421. |
[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. |
[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. |
[14] |
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monograph in Mathematics, 80, Birkhäuser Verlag, Basel, 1984. |
[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. |
[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), 87-109.
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), 83-105.
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, Vol. 23, American Math. Soc., Providence, RI, 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), 273-277. |
[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. |
[23] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967. |
[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. |
[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. |
[26] |
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195. |
[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. |
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). |
[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. |
[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. |
[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. |
[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), 203-232.
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), 775-789.
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, 1960-61, Editrice Tecnico Scientifica, Pisa, 1961. |
[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. |
[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. |
[11] |
Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys., 103 (1986), 415-421. |
[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. |
[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. |
[14] |
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monograph in Mathematics, 80, Birkhäuser Verlag, Basel, 1984. |
[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. |
[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), 87-109.
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), 83-105.
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, Vol. 23, American Math. Soc., Providence, RI, 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), 273-277. |
[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. |
[23] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967. |
[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. |
[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. |
[26] |
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195. |
[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. |
[1] |
Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete and 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 and 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 and 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 and Related Models, 2018, 11 (3) : 469-490. doi: 10.3934/krm.2018021 |
[5] |
Quanrong Li, Shijin Ding. Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3561-3581. doi: 10.3934/cpaa.2021121 |
[6] |
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 and Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148 |
[7] |
Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769 |
[8] |
Dongho Chae, Shangkun Weng. Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5267-5285. doi: 10.3934/dcds.2016031 |
[9] |
Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153 |
[10] |
Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3489-3530. doi: 10.3934/dcds.2021005 |
[11] |
Jean-Pierre Raymond. Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1537-1564. doi: 10.3934/dcdsb.2010.14.1537 |
[12] |
Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783 |
[13] |
Boris Muha, Zvonimir Tutek. Note on evolutionary free piston problem for Stokes equations with slip boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1629-1639. doi: 10.3934/cpaa.2014.13.1629 |
[14] |
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 and Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207 |
[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 and 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 and Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219 |
[17] |
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 and Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234 |
[18] |
Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201 |
[19] |
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 and Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041 |
[20] |
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 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]