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Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic
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 |
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 $p≥2$.
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). |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations North-Holland, Amsterdam, 1992. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics Amer. Math. Soc. , Providence, 2002. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow Gordon and Breach, New York, 1969. |
[24] |
J. -L. Lions, Quelques Méthodes de Résolutions des Problémes aux Limites non Linéaires Dunod et Gauthier-Villars, Paris, 1969. |
[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. |
[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. |
[27] |
J. Pedlosky,
Geophysical Fluid Dynamics Springer, New York, 1979. |
[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. |
[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. |
[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. |
[31] |
J.-C. Saut,
Remarks on the damped stationary Euler equations, Diff. Int. Eq., 3 (1990), 801-812.
|
[32] |
R. Temam,
Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer, New York, 1997. |
[33] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam-New York-Oxford, 1977. |
[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. |
[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. |
[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). |
[37] |
V. I. Yudovich,
Non-Stationary flow of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., 3 (1963), 1032-1066.
|
[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. |
[39] |
V. I. Yudovich,
The Linearization Method in Hydrodynamical Stability Theory Translations of Mathematical Monographs, 74. American Mathematical Society, Providence, RI, 1989. |
[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. |
[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. |
[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. |
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). |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations North-Holland, Amsterdam, 1992. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics Amer. Math. Soc. , Providence, 2002. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow Gordon and Breach, New York, 1969. |
[24] |
J. -L. Lions, Quelques Méthodes de Résolutions des Problémes aux Limites non Linéaires Dunod et Gauthier-Villars, Paris, 1969. |
[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. |
[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. |
[27] |
J. Pedlosky,
Geophysical Fluid Dynamics Springer, New York, 1979. |
[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. |
[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. |
[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. |
[31] |
J.-C. Saut,
Remarks on the damped stationary Euler equations, Diff. Int. Eq., 3 (1990), 801-812.
|
[32] |
R. Temam,
Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer, New York, 1997. |
[33] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam-New York-Oxford, 1977. |
[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. |
[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. |
[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). |
[37] |
V. I. Yudovich,
Non-Stationary flow of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., 3 (1963), 1032-1066.
|
[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. |
[39] |
V. I. Yudovich,
The Linearization Method in Hydrodynamical Stability Theory Translations of Mathematical Monographs, 74. American Mathematical Society, Providence, RI, 1989. |
[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. |
[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. |
[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. |
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