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Upper bounds for the attractor dimension of damped Navier-Stokes equations in $\mathbb R^2$
1. | Keldysh Institute of Applied Mathematics, Moscow 125047, Russian Federation |
2. | University of Surrey, Department of Mathematics, Guildford, GU2 7XH, United Kingdom |
3. | Department of Mathematics, University of Surrey, Guildford, GU2 7XH |
References:
[1] |
P. Anthony and S. Zelik, Infinite-energy solutions for the Navier-Stokes equations in a strip revisited, Commun. Pure Appl. Anal., 13 (2014), 1361-1393.
doi: 10.3934/cpaa.2014.13.1361. |
[2] |
A. Babin, The attractor of a Navier-Stokes system in unbounded channel-like domain, Jour. Dyn. Diff. Eqns., 4 (1992), 555-584.
doi: 10.1007/BF01048260. |
[3] |
A. Babin and M. Vishik, Attractors of evolution partial differential equations and estimates of their dimension, Uspekhi Mat. Nauk, 38 (1983), 133-187; English transl. Russian Math. Surveys., 38 (1983). |
[4] |
A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992. |
[5] |
J. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[6] |
V. Barcilon, P. Constantin and E. S. Titi, Existence of solutions to the Stommel-Charney model of the gulf stream, SIAM J. Math. Anal., 19 (1988), 1355-1364.
doi: 10.1137/0519099. |
[7] |
M. Bartuccelli, J. Deane and S. Zelik, Asymptotic expansions and extremals for the critical Sobolev and Gagliardo-Nirenberg inequalities on a torus, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 445-482.
doi: 10.1017/S0308210511000473. |
[8] |
V. Chepyzhov and A. Ilyin, On the fractal dimension of invariant sets: Applications to Navier-Stokes equations, Discrete Contin. Dyn. Syst., 10 (2004), 117-135. |
[9] |
V. Chepyzhov and A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Anal. Theory, Methods & Applications, 44 (2001), 811-819.
doi: 10.1016/S0362-546X(99)00309-0. |
[10] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Providence, RI: Amer. Math. Soc., 2002. (Amer. Math. Soc. Colloq. Publ. V.49.) |
[11] |
V. Chepyzhov, M. Vishik and S. Zelik, Strong trajectory attractors for dissipative Euler equations, J. Math. Pures Appl., 96 (2011), 395-407.
doi: 10.1016/j.matpur.2011.04.007. |
[12] |
V. Chepyzhov and S. Zelik, Infinite energy solutions for dissipative Euler equations in $R^2$, J. Math. Fluid Mech., 17 (2015), 513-532, arXiv:1501.00684.
doi: 10.1007/s00021-015-0213-x. |
[13] |
P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for the 2D Navier-Stokes equations, Comm. Pure Appl. Math., 38 (1985), 1-27.
doi: 10.1002/cpa.3160380102. |
[14] |
P. Constantin and C. Foias, Navier-Stokes Equations, Univ. of Chicago Press, Chicago, 1988. |
[15] |
P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Physica D, 30 (1988), 284-296.
doi: 10.1016/0167-2789(88)90022-X. |
[16] |
P. Constantin and F. Ramos, Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbb R^2$, Comm. Math. Phys., 275 (2007), 529-551.
doi: 10.1007/s00220-007-0310-7. |
[17] |
C. Doering and J. Gibbon, Note on the Constantin-Foias-Temam attractor dimension estimate for two-dimensional turbulence, Phys. D, 48 (1991), 471-480.
doi: 10.1016/0167-2789(91)90098-T. |
[18] |
J. Dolbeault, A. Laptev and M. Loss, Lieb-Thirring inequalities with improved constants, J. European Math. Soc., 10 (2008), 1121-1126.
doi: 10.4171/JEMS/142. |
[19] |
C. Foias, O. Manely, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge Univ. Press, Cambridge, 2001.
doi: 10.1017/CBO9780511546754. |
[20] |
Th. Gallay, Infinite energy solutions of the two-dimensional Navier-Stokes equations,, , ().
|
[21] |
Th. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math. Phys., 255 (2005), 97-129.
doi: 10.1007/s00220-004-1254-9. |
[22] |
A. A. Ilyin, Euler equations with dissipation, Mat. Sbornik, 182 (1991), 1729-1739; English transl. in Math. USSR, Sbornik, 74 (1993), 475-485. |
[23] |
A. Ilyin, On the spectrum of the Stokes operator, Funktsional. Anal. i Prilozhen, 43 (2009), 14-25; English transl. in Funct. Anal. Appl., 43 (2009), 254-263.
doi: 10.1007/s10688-009-0034-x. |
[24] |
A. Ilyin, A. Miranville and E. Titi, Small viscosity sharp estimate for the global attractor of the 2-D damped-driven Navier-Stokes equations, Commun. Math. Sciences, 2 (2004), 403-426.
doi: 10.4310/CMS.2004.v2.n3.a4. |
[25] |
A. A. Ilyin, Lieb-Thirring inequalities on some manifolds, J. Spectr. Theory, 2 (2012), 57-78.
doi: 10.4171/JST/21. |
[26] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[27] |
E. Lieb, On characteristic exponents in turbulence, Comm. Math. Phys., 92 (1984), 473-480.
doi: 10.1007/BF01215277. |
[28] |
V. X. Liu, A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier-Stokes equations, Comm. Math. Phys., 158 (1993), 327-339.
doi: 10.1007/BF02108078. |
[29] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of differential equations: Evolutionary equations. Vol. IV, Handb. Differ. Equ., Elsevier North-Holland, Amsterdam, (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[30] |
I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393.
doi: 10.1088/0951-7715/11/5/012. |
[31] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. |
[32] |
J. Pennant, A finite dimensional global attractor for infinite energy solutions of the damped Navier-Stokes equations in $\mathbb R^2$,, submitted., ().
|
[33] |
J. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011. |
[34] |
R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), 71-85.
doi: 10.1016/S0362-546X(97)00453-7. |
[35] |
M. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.
doi: 10.1080/03605308608820443. |
[36] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[37] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[38] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1977. |
[39] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Second edition. Johann Ambrosius Barth, Heidelberg, 1995. |
[40] |
M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on $R^n$, J. London Math. Soc., (2) 35 (1987), 303-313.
doi: 10.1112/jlms/s2-35.2.303. |
[41] |
G. Wolansky, Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Comm. Pure Appl. Math., 41 (1988), 19-46.
doi: 10.1002/cpa.3160410104. |
[42] |
S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip, Glasg. Math. J., 49 (2007), 525-588.
doi: 10.1017/S0017089507003849. |
[43] |
S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $\mathbbR^2$, J. Math. Fluid Mech., 15 (2013), 717-745.
doi: 10.1007/s00021-013-0144-3. |
show all references
References:
[1] |
P. Anthony and S. Zelik, Infinite-energy solutions for the Navier-Stokes equations in a strip revisited, Commun. Pure Appl. Anal., 13 (2014), 1361-1393.
doi: 10.3934/cpaa.2014.13.1361. |
[2] |
A. Babin, The attractor of a Navier-Stokes system in unbounded channel-like domain, Jour. Dyn. Diff. Eqns., 4 (1992), 555-584.
doi: 10.1007/BF01048260. |
[3] |
A. Babin and M. Vishik, Attractors of evolution partial differential equations and estimates of their dimension, Uspekhi Mat. Nauk, 38 (1983), 133-187; English transl. Russian Math. Surveys., 38 (1983). |
[4] |
A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992. |
[5] |
J. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[6] |
V. Barcilon, P. Constantin and E. S. Titi, Existence of solutions to the Stommel-Charney model of the gulf stream, SIAM J. Math. Anal., 19 (1988), 1355-1364.
doi: 10.1137/0519099. |
[7] |
M. Bartuccelli, J. Deane and S. Zelik, Asymptotic expansions and extremals for the critical Sobolev and Gagliardo-Nirenberg inequalities on a torus, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 445-482.
doi: 10.1017/S0308210511000473. |
[8] |
V. Chepyzhov and A. Ilyin, On the fractal dimension of invariant sets: Applications to Navier-Stokes equations, Discrete Contin. Dyn. Syst., 10 (2004), 117-135. |
[9] |
V. Chepyzhov and A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Anal. Theory, Methods & Applications, 44 (2001), 811-819.
doi: 10.1016/S0362-546X(99)00309-0. |
[10] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Providence, RI: Amer. Math. Soc., 2002. (Amer. Math. Soc. Colloq. Publ. V.49.) |
[11] |
V. Chepyzhov, M. Vishik and S. Zelik, Strong trajectory attractors for dissipative Euler equations, J. Math. Pures Appl., 96 (2011), 395-407.
doi: 10.1016/j.matpur.2011.04.007. |
[12] |
V. Chepyzhov and S. Zelik, Infinite energy solutions for dissipative Euler equations in $R^2$, J. Math. Fluid Mech., 17 (2015), 513-532, arXiv:1501.00684.
doi: 10.1007/s00021-015-0213-x. |
[13] |
P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for the 2D Navier-Stokes equations, Comm. Pure Appl. Math., 38 (1985), 1-27.
doi: 10.1002/cpa.3160380102. |
[14] |
P. Constantin and C. Foias, Navier-Stokes Equations, Univ. of Chicago Press, Chicago, 1988. |
[15] |
P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Physica D, 30 (1988), 284-296.
doi: 10.1016/0167-2789(88)90022-X. |
[16] |
P. Constantin and F. Ramos, Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbb R^2$, Comm. Math. Phys., 275 (2007), 529-551.
doi: 10.1007/s00220-007-0310-7. |
[17] |
C. Doering and J. Gibbon, Note on the Constantin-Foias-Temam attractor dimension estimate for two-dimensional turbulence, Phys. D, 48 (1991), 471-480.
doi: 10.1016/0167-2789(91)90098-T. |
[18] |
J. Dolbeault, A. Laptev and M. Loss, Lieb-Thirring inequalities with improved constants, J. European Math. Soc., 10 (2008), 1121-1126.
doi: 10.4171/JEMS/142. |
[19] |
C. Foias, O. Manely, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge Univ. Press, Cambridge, 2001.
doi: 10.1017/CBO9780511546754. |
[20] |
Th. Gallay, Infinite energy solutions of the two-dimensional Navier-Stokes equations,, , ().
|
[21] |
Th. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math. Phys., 255 (2005), 97-129.
doi: 10.1007/s00220-004-1254-9. |
[22] |
A. A. Ilyin, Euler equations with dissipation, Mat. Sbornik, 182 (1991), 1729-1739; English transl. in Math. USSR, Sbornik, 74 (1993), 475-485. |
[23] |
A. Ilyin, On the spectrum of the Stokes operator, Funktsional. Anal. i Prilozhen, 43 (2009), 14-25; English transl. in Funct. Anal. Appl., 43 (2009), 254-263.
doi: 10.1007/s10688-009-0034-x. |
[24] |
A. Ilyin, A. Miranville and E. Titi, Small viscosity sharp estimate for the global attractor of the 2-D damped-driven Navier-Stokes equations, Commun. Math. Sciences, 2 (2004), 403-426.
doi: 10.4310/CMS.2004.v2.n3.a4. |
[25] |
A. A. Ilyin, Lieb-Thirring inequalities on some manifolds, J. Spectr. Theory, 2 (2012), 57-78.
doi: 10.4171/JST/21. |
[26] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[27] |
E. Lieb, On characteristic exponents in turbulence, Comm. Math. Phys., 92 (1984), 473-480.
doi: 10.1007/BF01215277. |
[28] |
V. X. Liu, A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier-Stokes equations, Comm. Math. Phys., 158 (1993), 327-339.
doi: 10.1007/BF02108078. |
[29] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of differential equations: Evolutionary equations. Vol. IV, Handb. Differ. Equ., Elsevier North-Holland, Amsterdam, (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[30] |
I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393.
doi: 10.1088/0951-7715/11/5/012. |
[31] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. |
[32] |
J. Pennant, A finite dimensional global attractor for infinite energy solutions of the damped Navier-Stokes equations in $\mathbb R^2$,, submitted., ().
|
[33] |
J. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011. |
[34] |
R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), 71-85.
doi: 10.1016/S0362-546X(97)00453-7. |
[35] |
M. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.
doi: 10.1080/03605308608820443. |
[36] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[37] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[38] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1977. |
[39] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Second edition. Johann Ambrosius Barth, Heidelberg, 1995. |
[40] |
M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on $R^n$, J. London Math. Soc., (2) 35 (1987), 303-313.
doi: 10.1112/jlms/s2-35.2.303. |
[41] |
G. Wolansky, Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Comm. Pure Appl. Math., 41 (1988), 19-46.
doi: 10.1002/cpa.3160410104. |
[42] |
S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip, Glasg. Math. J., 49 (2007), 525-588.
doi: 10.1017/S0017089507003849. |
[43] |
S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $\mathbbR^2$, J. Math. Fluid Mech., 15 (2013), 717-745.
doi: 10.1007/s00021-013-0144-3. |
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