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July  2017, 22(5): 1835-1855. doi: 10.3934/dcdsb.2017109

Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$

1. 

Institute for Information Transmission Problems, Moscow 127994, Russia

2. 

National Research University Higher School of Economics, Moscow 101000, Russia

3. 

Keldysh Institute of Applied Mathematics, Moscow 125047, Russia

4. 

University of Surrey, Department of Mathematics, Guildford, GU2 7XH, UK

* Corresponding author: V. Chepyzhov

Received  January 2016 Revised  March 2016 Published  March 2017

We consider the damped and driven two-dimensional Euler equations in the plane with weak solutions having finite energy and enstrophy. We show that these (possibly non-unique) solutions satisfy the energy and enstrophy equality. It is shown that this system has a strong global and a strong trajectory attractor in the Sobolev space $H^1$. A similar result on the strong attraction holds in the spaces $H^1\cap\{u:\ \|\text{curl}\, u\|_{L^p}<∞\}$ for $p≥2$.

Citation: Vladimir Chepyzhov, Alexei Ilyin, Sergey Zelik. Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109
References:
[1]

A. V. Babin and M. I. Vishik, Maximal attractors of semigroups corresponding to evolution differential equations, Math. Sb. , 126 (1985), 397-419; English transl. Math USSR Sb. , 54 (1986). Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations North-Holland, Amsterdam, 1992. Google Scholar

[3]

J. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 54 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

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V. BarcilonP. 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

[5]

H. Bessaih and F. Flandoli, Weak attractor for a dissipative Euler equation, J. Dynam. Diff. Eq., 12 (2000), 713-732.  doi: 10.1023/A:1009042520953.  Google Scholar

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F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models Applied Mathematical Sciences, 183. Springer, New York, 2013. Google Scholar

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S. Brull and L. Pareschi, Dissipative hydrodynamic models for the diffusion of impurities in a gas, Appl. Math. Lett., 19 (2006), 516-521.  doi: 10.1016/j.aml.2005.07.008.  Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for dissipative 2D Euler and Navier-Stokes equations, Russian J. Math. Phys., 15 (2008), 156-170.  doi: 10.1134/S1061920808020039.  Google Scholar

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics Amer. Math. Soc. , Providence, 2002. Google Scholar

[10]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar

[11]

V. V. ChepyzhovM. I. Vishik and S. Zelik, A strong trajectory attractor for a dissipative reaction-diffusion system, J. Math. Pures Appl., 96 (2011), 395-407.  doi: 10.1016/j.matpur.2011.04.007.  Google Scholar

[12]

V. V. Chepyzhov and S. Zelik, Infinite energy solutions for dissipative Euler equations in $\mathbb{R}^{2}$, J. Math. Fluid Mech., 17 (2015), 513-532.  doi: 10.1007/s00021-015-0213-x.  Google Scholar

[13]

V. V. Chepyzhov, Trajectory attractors for non-autonomous dissipative 2d Euler equations, Discrete Contin. Dyn. Syst. Series B, 20 (2015), 811-832.  doi: 10.3934/dcdsb.2015.20.811.  Google Scholar

[14]
[15]

R. DiPerna and P. Lions, Ordinary differential equations, Sobolev spaces and transport theory, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[16]

J.-M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Diff. Eq., 110 (1994), 356-359.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[17]

A. A. Ilyin, The Euler equations with dissipation, Sb. Math. , 182 (1991), 1729-1739; English transl. in Math. USSR-Sb. , 74 (1993), 475{485. Google Scholar

[18]

A. A. IlyinA. Miranville and E. S. Titi, Small viscosity sharp estimates for the global attractor of the 2-D damped-driven Navier-Stokes equations, Commun. Math. Sci., 2 (2004), 403-426.  doi: 10.4310/CMS.2004.v2.n3.a4.  Google Scholar

[19]

A. A. IlyinK. Patni and S. V. Zelik, Upper bounds for the attractor dimension of damped Navier--Stokes equations in $\mathbb R^2$, Discrete Contin. Dyn. Syst., 36 (2016), 2085-2102.  doi: 10.3934/dcds.2016.36.2085.  Google Scholar

[20]

A. A. Ilyin and E. S. Titi, Sharp estimates for the number of degrees of freedom of the damped-driven 2-D Navier-Stokes equations, J. Nonlin. Sci., 16 (2006), 233-253.  doi: 10.1007/s00332-005-0720-7.  Google Scholar

[21]

O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions, Int. J. Bifurc. Chaos, 20 (2010), 2723-2734.  doi: 10.1142/S0218127410027313.  Google Scholar

[22]

P. O. Kasyanov, Multivalued dynamics of of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Mat. Zametki, 92 (2012), 225-240; English transl.  doi: 10.1134/S0001434612070231.  Google Scholar

[23]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow Gordon and Breach, New York, 1969. Google Scholar

[24]

J. -L. Lions, Quelques Méthodes de Résolutions des Problémes aux Limites non Linéaires Dunod et Gauthier-Villars, Paris, 1969. Google Scholar

[25]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and generalized differential equations, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[26]

I. MoiseR. 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

[27]

J. Pedlosky, Geophysical Fluid Dynamics Springer, New York, 1979. Google Scholar

[28]

F. Riesz and B. Sz. -Nagy, Functional Analysis Reprint of the 1955 original, Dover Books on Advanced Mathematics. Dover Publications, Inc. , New York, 1990. Google Scholar

[29]

A. Robertson and W. Robertson, Topological Vector Spaces Reprint of the second edition, Cambridge Tracts in Mathematics, 53, Cambridge University Press, Cambridge-New York, 1980. Google Scholar

[30]

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

[31]

J.-C. Saut, Remarks on the damped stationary Euler equations, Diff. Int. Eq., 3 (1990), 801-812.   Google Scholar

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer, New York, 1997. Google Scholar

[33]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam-New York-Oxford, 1977. Google Scholar

[34]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Uspekhi Mat. Nauk, 66 (2011), 3-102; English tarnsl.  doi: 10.1070/RM2011v066n04ABEH004753.  Google Scholar

[35]

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

[36]

V. I. Yudovich, Some bounds for solutions of elliptic equations, Mat. Sb. , (N. S. ) 59 (1962), 229-244. English transl. in Amer. Math. Soc. Transl. , (2) 56 (1966). Google Scholar

[37]

V. I. Yudovich, Non-Stationary flow of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., 3 (1963), 1032-1066.   Google Scholar

[38]

V. I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett., 2 (1995), 27-38.  doi: 10.4310/MRL.1995.v2.n1.a4.  Google Scholar

[39]

V. I. Yudovich, The Linearization Method in Hydrodynamical Stability Theory Translations of Mathematical Monographs, 74. American Mathematical Society, Providence, RI, 1989. Google Scholar

[40]

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

[41]

S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains, Instability in models connected with fluid flows. II, 255-327, Int. Math. Ser. (N. Y. ), 7, Springer, New York, 2008. Google Scholar

[42]

S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $\mathbb{R}^{2}$, J. Math. Fluid Mech., 15 (2013), 717-745.  doi: 10.1007/s00021-013-0144-3.  Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Maximal attractors of semigroups corresponding to evolution differential equations, Math. Sb. , 126 (1985), 397-419; English transl. Math USSR Sb. , 54 (1986). Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations North-Holland, Amsterdam, 1992. Google Scholar

[3]

J. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 54 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[4]

V. BarcilonP. 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

[5]

H. Bessaih and F. Flandoli, Weak attractor for a dissipative Euler equation, J. Dynam. Diff. Eq., 12 (2000), 713-732.  doi: 10.1023/A:1009042520953.  Google Scholar

[6]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models Applied Mathematical Sciences, 183. Springer, New York, 2013. Google Scholar

[7]

S. Brull and L. Pareschi, Dissipative hydrodynamic models for the diffusion of impurities in a gas, Appl. Math. Lett., 19 (2006), 516-521.  doi: 10.1016/j.aml.2005.07.008.  Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for dissipative 2D Euler and Navier-Stokes equations, Russian J. Math. Phys., 15 (2008), 156-170.  doi: 10.1134/S1061920808020039.  Google Scholar

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics Amer. Math. Soc. , Providence, 2002. Google Scholar

[10]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar

[11]

V. V. ChepyzhovM. I. Vishik and S. Zelik, A strong trajectory attractor for a dissipative reaction-diffusion system, J. Math. Pures Appl., 96 (2011), 395-407.  doi: 10.1016/j.matpur.2011.04.007.  Google Scholar

[12]

V. V. Chepyzhov and S. Zelik, Infinite energy solutions for dissipative Euler equations in $\mathbb{R}^{2}$, J. Math. Fluid Mech., 17 (2015), 513-532.  doi: 10.1007/s00021-015-0213-x.  Google Scholar

[13]

V. V. Chepyzhov, Trajectory attractors for non-autonomous dissipative 2d Euler equations, Discrete Contin. Dyn. Syst. Series B, 20 (2015), 811-832.  doi: 10.3934/dcdsb.2015.20.811.  Google Scholar

[14]
[15]

R. DiPerna and P. Lions, Ordinary differential equations, Sobolev spaces and transport theory, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[16]

J.-M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Diff. Eq., 110 (1994), 356-359.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[17]

A. A. Ilyin, The Euler equations with dissipation, Sb. Math. , 182 (1991), 1729-1739; English transl. in Math. USSR-Sb. , 74 (1993), 475{485. Google Scholar

[18]

A. A. IlyinA. Miranville and E. S. Titi, Small viscosity sharp estimates for the global attractor of the 2-D damped-driven Navier-Stokes equations, Commun. Math. Sci., 2 (2004), 403-426.  doi: 10.4310/CMS.2004.v2.n3.a4.  Google Scholar

[19]

A. A. IlyinK. Patni and S. V. Zelik, Upper bounds for the attractor dimension of damped Navier--Stokes equations in $\mathbb R^2$, Discrete Contin. Dyn. Syst., 36 (2016), 2085-2102.  doi: 10.3934/dcds.2016.36.2085.  Google Scholar

[20]

A. A. Ilyin and E. S. Titi, Sharp estimates for the number of degrees of freedom of the damped-driven 2-D Navier-Stokes equations, J. Nonlin. Sci., 16 (2006), 233-253.  doi: 10.1007/s00332-005-0720-7.  Google Scholar

[21]

O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions, Int. J. Bifurc. Chaos, 20 (2010), 2723-2734.  doi: 10.1142/S0218127410027313.  Google Scholar

[22]

P. O. Kasyanov, Multivalued dynamics of of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Mat. Zametki, 92 (2012), 225-240; English transl.  doi: 10.1134/S0001434612070231.  Google Scholar

[23]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow Gordon and Breach, New York, 1969. Google Scholar

[24]

J. -L. Lions, Quelques Méthodes de Résolutions des Problémes aux Limites non Linéaires Dunod et Gauthier-Villars, Paris, 1969. Google Scholar

[25]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and generalized differential equations, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[26]

I. MoiseR. 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

[27]

J. Pedlosky, Geophysical Fluid Dynamics Springer, New York, 1979. Google Scholar

[28]

F. Riesz and B. Sz. -Nagy, Functional Analysis Reprint of the 1955 original, Dover Books on Advanced Mathematics. Dover Publications, Inc. , New York, 1990. Google Scholar

[29]

A. Robertson and W. Robertson, Topological Vector Spaces Reprint of the second edition, Cambridge Tracts in Mathematics, 53, Cambridge University Press, Cambridge-New York, 1980. Google Scholar

[30]

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

[31]

J.-C. Saut, Remarks on the damped stationary Euler equations, Diff. Int. Eq., 3 (1990), 801-812.   Google Scholar

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer, New York, 1997. Google Scholar

[33]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam-New York-Oxford, 1977. Google Scholar

[34]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Uspekhi Mat. Nauk, 66 (2011), 3-102; English tarnsl.  doi: 10.1070/RM2011v066n04ABEH004753.  Google Scholar

[35]

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

[36]

V. I. Yudovich, Some bounds for solutions of elliptic equations, Mat. Sb. , (N. S. ) 59 (1962), 229-244. English transl. in Amer. Math. Soc. Transl. , (2) 56 (1966). Google Scholar

[37]

V. I. Yudovich, Non-Stationary flow of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., 3 (1963), 1032-1066.   Google Scholar

[38]

V. I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett., 2 (1995), 27-38.  doi: 10.4310/MRL.1995.v2.n1.a4.  Google Scholar

[39]

V. I. Yudovich, The Linearization Method in Hydrodynamical Stability Theory Translations of Mathematical Monographs, 74. American Mathematical Society, Providence, RI, 1989. Google Scholar

[40]

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

[41]

S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains, Instability in models connected with fluid flows. II, 255-327, Int. Math. Ser. (N. Y. ), 7, Springer, New York, 2008. Google Scholar

[42]

S. Zelik, Infinite energy solutions for damped Navier-Stokes equations in $\mathbb{R}^{2}$, J. Math. Fluid Mech., 15 (2013), 717-745.  doi: 10.1007/s00021-013-0144-3.  Google Scholar

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