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

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