# 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 - A, 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.  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.  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.   Google Scholar [4] A. Babin and M. Vishik, Attractors of Evolution Equations,, Studies in Mathematics and its Applications, (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.  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.  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.  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.   Google Scholar [9] V. Chepyzhov and A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems,, Nonlinear Anal. Theory, 44 (2001), 811.  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, (2002).   Google Scholar [11] V. Chepyzhov, M. Vishik and S. Zelik, Strong trajectory attractors for dissipative Euler equations,, J. Math. Pures Appl., 96 (2011), 395.  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.  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.  doi: 10.1002/cpa.3160380102.  Google Scholar [14] P. Constantin and C. Foias, Navier-Stokes Equations,, Univ. of Chicago Press, (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.  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.  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.  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.  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, (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.  doi: 10.1007/s00220-004-1254-9.  Google Scholar [22] A. A. Ilyin, Euler equations with dissipation,, Mat. Sbornik, 182 (1991), 1729.   Google Scholar [23] A. Ilyin, On the spectrum of the Stokes operator,, Funktsional. Anal. i Prilozhen, 43 (2009), 14.  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.  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.  doi: 10.4171/JST/21.  Google Scholar [26] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Leizioni Lincei, (1991).  doi: 10.1017/CBO9780511569418.  Google Scholar [27] E. Lieb, On characteristic exponents in turbulence,, Comm. Math. Phys., 92 (1984), 473.  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.  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, (2008), 103.  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.  doi: 10.1088/0951-7715/11/5/012.  Google Scholar [31] J. Pedlosky, Geophysical Fluid Dynamics,, Springer, (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, (2011).   Google Scholar [34] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains,, Nonlinear Anal., 32 (1998), 71.  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.  doi: 10.1080/03605308608820443.  Google Scholar [36] G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (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, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar [38] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, North-Holland, (1977).   Google Scholar [39] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, Second edition. Johann Ambrosius Barth, (1995).   Google Scholar [40] M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on $R^n$,, J. London Math. Soc., 35 (1987), 303.  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.  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.  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.  doi: 10.1007/s00021-013-0144-3.  Google Scholar

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.  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.  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.   Google Scholar [4] A. Babin and M. Vishik, Attractors of Evolution Equations,, Studies in Mathematics and its Applications, (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.  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.  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.  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.   Google Scholar [9] V. Chepyzhov and A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems,, Nonlinear Anal. Theory, 44 (2001), 811.  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, (2002).   Google Scholar [11] V. Chepyzhov, M. Vishik and S. Zelik, Strong trajectory attractors for dissipative Euler equations,, J. Math. Pures Appl., 96 (2011), 395.  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.  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.  doi: 10.1002/cpa.3160380102.  Google Scholar [14] P. Constantin and C. Foias, Navier-Stokes Equations,, Univ. of Chicago Press, (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.  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.  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.  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.  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, (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.  doi: 10.1007/s00220-004-1254-9.  Google Scholar [22] A. A. Ilyin, Euler equations with dissipation,, Mat. Sbornik, 182 (1991), 1729.   Google Scholar [23] A. Ilyin, On the spectrum of the Stokes operator,, Funktsional. Anal. i Prilozhen, 43 (2009), 14.  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.  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.  doi: 10.4171/JST/21.  Google Scholar [26] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Leizioni Lincei, (1991).  doi: 10.1017/CBO9780511569418.  Google Scholar [27] E. Lieb, On characteristic exponents in turbulence,, Comm. Math. Phys., 92 (1984), 473.  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.  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, (2008), 103.  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.  doi: 10.1088/0951-7715/11/5/012.  Google Scholar [31] J. Pedlosky, Geophysical Fluid Dynamics,, Springer, (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, (2011).   Google Scholar [34] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains,, Nonlinear Anal., 32 (1998), 71.  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.  doi: 10.1080/03605308608820443.  Google Scholar [36] G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (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, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar [38] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, North-Holland, (1977).   Google Scholar [39] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, Second edition. Johann Ambrosius Barth, (1995).   Google Scholar [40] M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on $R^n$,, J. London Math. Soc., 35 (1987), 303.  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.  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.  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.  doi: 10.1007/s00021-013-0144-3.  Google Scholar
 [1] Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408 [2] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [3] Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001 [4] Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352 [5] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [6] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [7] Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142 [8] Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039 [9] Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 [10] Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189 [11] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [12] Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, , () : -. doi: 10.3934/era.2020128 [13] 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 [14] Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268 [15] Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020394 [16] Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141 [17] Duy Phan. Approximate controllability for Navier–Stokes equations in $\rm3D$ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062 [18] Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073 [19] Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283 [20] Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053

2019 Impact Factor: 1.338