# American Institute of Mathematical Sciences

April  2016, 36(4): 2085-2102. doi: 10.3934/dcds.2016.36.2085

## 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

Received  March 2015 Revised  June 2015 Published  September 2015

We consider finite energy solutions for the damped and driven two-dimensional Navier--Stokes equations in the plane and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension when the forcing term belongs to the whole scale of homogeneous Sobolev spaces from $-1$ to $1$.
Citation: Alexei Ilyin, Kavita Patni, Sergey Zelik. Upper bounds for the attractor dimension of damped Navier-Stokes equations in $\mathbb R^2$. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2085-2102. doi: 10.3934/dcds.2016.36.2085
##### 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.  Google Scholar [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.  Google Scholar [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).  Google Scholar [4] A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.)  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] P. Constantin and C. Foias, Navier-Stokes Equations, Univ. of Chicago Press, Chicago, 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] C. Foias, O. Manely, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge Univ. Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.  Google Scholar [20] Th. Gallay, Infinite energy solutions of the two-dimensional Navier-Stokes equations,, , ().   Google Scholar [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.  Google Scholar [22] A. A. Ilyin, Euler equations with dissipation, Mat. Sbornik, 182 (1991), 1729-1739; English transl. in Math. USSR, Sbornik, 74 (1993), 475-485.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] A. A. Ilyin, Lieb-Thirring inequalities on some manifolds, J. Spectr. Theory, 2 (2012), 57-78. doi: 10.4171/JST/21.  Google Scholar [26] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar [27] E. Lieb, On characteristic exponents in turbulence, Comm. Math. Phys., 92 (1984), 473-480. doi: 10.1007/BF01215277.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. Google Scholar [32] J. Pennant, A finite dimensional global attractor for infinite energy solutions of the damped Navier-Stokes equations in $\mathbb R^2$,, submitted., ().   Google Scholar [33] J. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar [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.  Google Scholar [38] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1977.  Google Scholar [39] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Second edition. Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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).  Google Scholar [4] A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.)  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] P. Constantin and C. Foias, Navier-Stokes Equations, Univ. of Chicago Press, Chicago, 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] C. Foias, O. Manely, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge Univ. Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.  Google Scholar [20] Th. Gallay, Infinite energy solutions of the two-dimensional Navier-Stokes equations,, , ().   Google Scholar [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.  Google Scholar [22] A. A. Ilyin, Euler equations with dissipation, Mat. Sbornik, 182 (1991), 1729-1739; English transl. in Math. USSR, Sbornik, 74 (1993), 475-485.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] A. A. Ilyin, Lieb-Thirring inequalities on some manifolds, J. Spectr. Theory, 2 (2012), 57-78. doi: 10.4171/JST/21.  Google Scholar [26] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar [27] E. Lieb, On characteristic exponents in turbulence, Comm. Math. Phys., 92 (1984), 473-480. doi: 10.1007/BF01215277.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. Google Scholar [32] J. Pennant, A finite dimensional global attractor for infinite energy solutions of the damped Navier-Stokes equations in $\mathbb R^2$,, submitted., ().   Google Scholar [33] J. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar [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.  Google Scholar [38] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1977.  Google Scholar [39] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Second edition. Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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