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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
References:
[1]

P. S. Alexandrov, Introduction to Set Theory and General Topology,, Nauka, (1977).

[2]

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

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989).

[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. doi: 10.1137/0519099.

[5]

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

[6]

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

[7]

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

[8]

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

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS Colloquium Publications, (2002).

[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. doi: 10.1134/S1061920808020039.

[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. doi: 10.1016/j.matpur.2011.04.007.

[12]

P. Constantin and C. Foias, Navier-Stokes Equations,, The University of Chicago Press, (1989).

[13]

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

[14]

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

[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. doi: 10.4310/CMS.2004.v2.n3.a4.

[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. doi: 10.1007/s00332-005-0720-7.

[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.

[18]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Gordon and Breach, (1969).

[19]

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

[20]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1979).

[21]

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

[22]

R. Temam, Navier-Stokes Equations,, Theory and Numerical Analysis, (1984).

[23]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). doi: 10.1007/978-1-4612-0645-3.

[24]

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

[25]

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

[26]

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

[27]

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

show all references

References:
[1]

P. S. Alexandrov, Introduction to Set Theory and General Topology,, Nauka, (1977).

[2]

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

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989).

[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. doi: 10.1137/0519099.

[5]

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

[6]

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

[7]

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

[8]

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

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS Colloquium Publications, (2002).

[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. doi: 10.1134/S1061920808020039.

[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. doi: 10.1016/j.matpur.2011.04.007.

[12]

P. Constantin and C. Foias, Navier-Stokes Equations,, The University of Chicago Press, (1989).

[13]

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

[14]

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

[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. doi: 10.4310/CMS.2004.v2.n3.a4.

[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. doi: 10.1007/s00332-005-0720-7.

[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.

[18]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Gordon and Breach, (1969).

[19]

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

[20]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1979).

[21]

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

[22]

R. Temam, Navier-Stokes Equations,, Theory and Numerical Analysis, (1984).

[23]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). doi: 10.1007/978-1-4612-0645-3.

[24]

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

[25]

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

[26]

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

[27]

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

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