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May  2015, 20(3): 811-832. doi: 10.3934/dcdsb.2015.20.811

Trajectory attractors for non-autonomous dissipative 2d Euler equations

1. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127994, GSP-4, Russian Federation

Received  November 2013 Revised  May 2014 Published  January 2015

We construct the trajectory attractor $\mathfrak{A}_{\Sigma }$ for the non-autonomous dissipative 2d Euler systems with periodic boundary conditions that contain time dependent dissipation terms $-r(t)u$ such that $0<\alpha \le r(t)\le \beta$, for $t\ge 0$. External forces $g(x,t),x\in \mathbb{T}^{2},t\ge 0,$ also depend on time. The corresponding non-autonomous dissipative 2d Navier--Stokes systems with the same terms $-r(t)u$ and $g(x,t)$ and with viscosity $\nu >0$ also have the trajectory attractor $\mathfrak{A}_{\Sigma }^{\nu }.$ Such systems model large-scale geophysical processes in atmosphere and ocean. We prove that $\mathfrak{A}_{\Sigma }^{\nu }\rightarrow \mathfrak{A}_{\Sigma }$ as viscosity $\nu \rightarrow 0+$ in the corresponding metric space. Moreover, we establish the existence of the minimal limit $\mathfrak{A}_{\Sigma }^{\min }\subseteq \mathfrak{A}_{\Sigma }$ of the trajectory attractors $\mathfrak{A}_{\Sigma }^{\nu }$ as $\nu \rightarrow 0+.$ Every set $\mathfrak{A}_{\Sigma }^{\nu }$ is connected. We prove that $\mathfrak{A}_{\Sigma }^{\min }$ is a connected invariant subset of $\mathfrak{A}_{\Sigma }.$ The problem of the connectedness of the trajectory attractor $\mathfrak{A}_{\Sigma }$ itself remains open.
Citation: Vladimir V. Chepyzhov. Trajectory attractors for non-autonomous dissipative 2d Euler equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 811-832. doi: 10.3934/dcdsb.2015.20.811
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P. S. Alexandrov, Introduction to Set Theory and General Topology, Nauka, Moscow, 1977.  Google Scholar

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J. P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044.  Google Scholar

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A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989; North Holland, Amsterdam, 1992.  Google Scholar

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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

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V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for 2D Navier-Stokes systems and some generalizations, Top. Meth. Nonlin. Anal., J. Julius Schauder Center, 8 (1996), 217-243.  Google Scholar

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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

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V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS Colloquium Publications, Vol. 49, AMS, Providence, 2002.  Google Scholar

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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

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V. V. Chepyzhov, M. I. Vishik and S. V. Zelik, Strong trajectory attractors for dissipative Euler equations, J. Math. Pures Appl., 96 (2011), 395-407. doi: 10.1016/j.matpur.2011.04.007.  Google Scholar

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P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago and London, 1989.  Google Scholar

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Yu. A. Dubinskiĭ, Weak convergence in nonlinear elliptic and parabolic equations, Sb. Math., 67 (1965), 609-642.  Google Scholar

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A. A. Ilyin, The Euler equations with dissipation, Sb. Math., 74 (1993), 475-485.  Google Scholar

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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

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

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A. A. Ilyin, Lieb-Thirring integral inequalities and sharp bounds for the dimension of the attractor of the Navier-Stokes equations with friction, Proc. Steklov Inst. Math., 255 (2006), 136-149.  Google Scholar

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

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J.-L. Lions, Quelques Méthodes de Résolutions Des Problèmes Aux Limites Non-linéaires, Dunod et Gauthier-Villars, Paris, 1969.  Google Scholar

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

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

[22]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.  Google Scholar

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R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[24]

M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193. doi: 10.1023/A:1014190629738.  Google Scholar

[25]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Math. Surveys., 66 (2011), 637-731. doi: 10.1070/RM2011v066n04ABEH004753.  Google Scholar

[26]

V. I. Yudovich, Non stationary flow of an ideal incompressible liquid, J. Vych. Mat. i Mat. Fiz., 3 (1963), 1407-1456. doi: 10.1016/0041-5553(63)90247-7.  Google Scholar

[27]

V. I. Yudovich, Some bounds for solutions of elliptic equations, Sb. Math., 59 (1962), 229-244.  Google Scholar

show all references

References:
[1]

P. S. Alexandrov, Introduction to Set Theory and General Topology, Nauka, Moscow, 1977.  Google Scholar

[2]

J. P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989; North Holland, Amsterdam, 1992.  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]

C. Bardos, Éxistence et unicité de la solution de l'equation d'Euler en dimensions deux, J. Math. Anal. Appl., 40 (1972), 769-790. doi: 10.1016/0022-247X(72)90019-4.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C. R. Acad. Sci., Paris, Series I, 321 (1995), 1309-1314. doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar

[7]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for 2D Navier-Stokes systems and some generalizations, Top. Meth. Nonlin. Anal., J. Julius Schauder Center, 8 (1996), 217-243.  Google Scholar

[8]

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

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS Colloquium Publications, Vol. 49, AMS, Providence, 2002.  Google Scholar

[10]

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

[11]

V. V. Chepyzhov, M. I. Vishik and S. V. Zelik, Strong trajectory attractors for dissipative Euler equations, J. Math. Pures Appl., 96 (2011), 395-407. doi: 10.1016/j.matpur.2011.04.007.  Google Scholar

[12]

P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago and London, 1989.  Google Scholar

[13]

Yu. A. Dubinskiĭ, Weak convergence in nonlinear elliptic and parabolic equations, Sb. Math., 67 (1965), 609-642.  Google Scholar

[14]

A. A. Ilyin, The Euler equations with dissipation, Sb. Math., 74 (1993), 475-485.  Google Scholar

[15]

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

[16]

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

[17]

A. A. Ilyin, Lieb-Thirring integral inequalities and sharp bounds for the dimension of the attractor of the Navier-Stokes equations with friction, Proc. Steklov Inst. Math., 255 (2006), 136-149.  Google Scholar

[18]

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

[19]

J.-L. Lions, Quelques Méthodes de Résolutions Des Problèmes Aux Limites Non-linéaires, Dunod et Gauthier-Villars, Paris, 1969.  Google Scholar

[20]

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

[21]

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

[22]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.  Google Scholar

[23]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[24]

M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193. doi: 10.1023/A:1014190629738.  Google Scholar

[25]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Math. Surveys., 66 (2011), 637-731. doi: 10.1070/RM2011v066n04ABEH004753.  Google Scholar

[26]

V. I. Yudovich, Non stationary flow of an ideal incompressible liquid, J. Vych. Mat. i Mat. Fiz., 3 (1963), 1407-1456. doi: 10.1016/0041-5553(63)90247-7.  Google Scholar

[27]

V. I. Yudovich, Some bounds for solutions of elliptic equations, Sb. Math., 59 (1962), 229-244.  Google Scholar

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