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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 [4] 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.  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. Chepyzhov, M. 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. Ilyin, A. 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. Ilyin, K. 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. 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.  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. 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.  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. Chepyzhov, M. 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. Ilyin, A. 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. Ilyin, K. 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. 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.  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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