# 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] Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239 [2] Reinhard Farwig, Yasushi Taniuchi. Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1215-1224. doi: 10.3934/dcdss.2013.6.1215 [3] Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279 [4] Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-26. doi: 10.3934/dcdsb.2019172 [5] 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 [6] Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675 [7] V. V. Chepyzhov, A. A. Ilyin. On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 117-135. doi: 10.3934/dcds.2004.10.117 [8] Alain Miranville, Xiaoming Wang. Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 95-110. doi: 10.3934/dcds.1996.2.95 [9] Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355 [10] 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 [11] Reinhard Farwig, Paul Felix Riechwald. Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 157-172. doi: 10.3934/dcdss.2016.9.157 [12] Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779 [13] Fang Li, Bo You, Yao Xu. Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4267-4284. doi: 10.3934/dcdsb.2018137 [14] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [15] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [16] Bixiang Wang, Xiaoling Gao. Random attractors for wave equations on unbounded domains. Conference Publications, 2009, 2009 (Special) : 800-809. doi: 10.3934/proc.2009.2009.800 [17] Yutaka Tsuzuki. Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains. Evolution Equations & Control Theory, 2014, 3 (1) : 191-206. doi: 10.3934/eect.2014.3.191 [18] Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 [19] Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613 [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

2018 Impact Factor: 1.143