Upper bounds for the attractor dimension of damped Navier-Stokes equations in \mathbb R^2

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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 $\mathbb R^2$

Alexei Ilyin ^1, , Kavita Patni ^2, and Sergey Zelik ^3,

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

Abstract
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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 $1$.

Keywords: unbounded domains, attractors, Ekman damping, box-counting dimension., Navier-Stokes equations.

Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35Q30, 76D0.

Citation: Alexei Ilyin, Kavita Patni, Sergey Zelik. Upper bounds for the attractor dimension of damped Navier-Stokes equations in $\mathbb R^2$. Discrete and 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.
[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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