August  2022, 15(8): 2275-2288. doi: 10.3934/dcdss.2022051

Trajectory attractors for 3D damped Euler equations and their approximation

1. 

Keldysh Institute of Applied Mathematics, Moscow, Russia

2. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

3. 

Imperial College, London SW7 2AZ, UK

4. 

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

Received  December 2021 Published  August 2022 Early access  March 2022

Fund Project: This work was supported by Moscow Center for Fundamental and Applied Mathematics, Agreement with the Ministry of Science and Higher Education of the Russian Federation, No. 075-15-2019-1623 and by the Russian Science Foundation grant No.19-71-30004 (sections 2-4). The second author was partially supported by the Leverhulme grant No. RPG-2021-072 (United Kingdom)

We study the global attractors for the damped 3D Euler–Bardina equations with the regularization parameter $ \alpha>0 $ and Ekman damping coefficient $ \gamma>0 $ endowed with periodic boundary conditions as well as their damped Euler limit $ \alpha\to0 $. We prove that despite the possible non-uniqueness of solutions of the limit Euler system and even the non-existence of such solutions in the distributional sense, the limit dynamics of the corresponding dissipative solutions introduced by P. Lions can be described in terms of attractors of the properly constructed trajectory dynamical system. Moreover, the convergence of the attractors $ \mathcal A(\alpha) $ of the regularized system to the limit trajectory attractor $ \mathcal A(0) $ as $ \alpha\to0 $ is also established in terms of the upper semicontinuity in the properly defined functional space.

Citation: Alexei Ilyin, Anna Kostianko, Sergey Zelik. Trajectory attractors for 3D damped Euler equations and their approximation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2275-2288. doi: 10.3934/dcdss.2022051
References:
[1]

A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol 25. North-Holland Publishing Co., Amsterdam, 1992.

[2]

J. Bardina, J. Ferziger and W. Reynolds, Improved Subgrid Scale Models for Large Eddy Simulation, in Proceedings of the 13th AIAA Conference on Fluid and Plasma Dynamics, 1980. doi: 10.2514/6.1980-1357.

[3]

K. Bardos and E. Titi, Euler equations for incompressible ideal fluids, Uspekhi Mat. Nauk, 62 (2007), 5-46.  doi: 10.1070/RM2007v062n03ABEH004410.

[4]

Y. CaoE. Lunasin and E. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[5]

V. ChepyzhovA. Ilyin and S. Zelik, Vanishing viscosity limit for global attractors for the damped Navier–Stokes system with stress free boundary conditions, Physica D, 376–377 (2018), 31-38.  doi: 10.1016/j.physd.2017.08.005.

[6]

V. ChepyzhovA. Ilyin and S. Zelik, Strong trajectory and global $W^{1, p}$-attractors for the damped-driven Euler system in ${\mathbb R}^2$, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1835-1855.  doi: 10.3934/dcdsb.2017109.

[7]

V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., 49, Providence, RI: Amer. Math. Soc., 2002.

[8]

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

[9]

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.

[10]

R. DiPerna and A. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), 667-689.  doi: 10.1007/BF01214424.

[11]

C. Fefferman, Existence and smoothness of the Navier-Stokes equation, In: Millennium Prize Problems, Clay Math. Inst., Cambridge, MA, (2006), 57–67.

[12]

C. FoiasD. Holm and E. Titi, The three dimensional viscous Camassa–Holm equations, and their relation to the Navier–Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.

[13] C. FoiasO. ManelyR. Rosa and R. Temam, Navier–Stokes Equations and Turbulence, Cambridge Univ. Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.
[14] U. Frisch, Turbulence. The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1995.  doi: 10.1063/1.881555.
[15]

M. HolstE. Lunasin and G. Tsogtgerel, Analysis of a general family of regularized Navier–Stokes and MHD models, J. Nonlinear Sci., 20 (2010), 523-567.  doi: 10.1007/s00332-010-9066-x.

[16]

A. Il'in, The Euler equations with dissipation, Mat. Sb., 74 (1993), 475-485.  doi: 10.1070/SM1993V074N02ABEH003357.

[17]

A. Ilyin, A. Kostianko and S. Zelik, Sharp upper and lower bounds of the attractor dimension for 3D damped Euler–Bardina equations, Physica D, 432 (2022), Paper No. 133156. doi: 10.1016/j.physd.2022.133156.

[18]

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

[19]

A. Ilyin and E. Titi, Attractors to the two-dimensional Navier–Stokes-$\alpha$ models: An $\alpha$-dependence study, J. Dynam. Differential Equations, 15 (2003), 751-778.  doi: 10.1023/B:JODY.0000010064.06851.ff.

[20]

A. Ilyin and S. Zelik, Sharp dimension estimates of the attractor of the damped 2D Euler-Bardina equations, In book: Partial Differential Equations, Spectral Theory, and Mathematical Physics, EMS Series of Congress Reports, EMS Press, Berlin, (2021), 209–229. doi: 10.4171/ECR/18-1/12.

[21]

V. Kalantarov and E. Titi, Global attractors and determining modes for the 3D Navier–Stokes–Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.

[22] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.
[23]

W. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Continuous Dyn. Sys. B, 6 (2006), 111-128.  doi: 10.3934/dcdsb.2006.6.111.

[24]

J. Leray, Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.

[25]

J.-L. Lions, Quelques Méthodes des Problèmes aux Limites non Linéaires, Doud, Paris, 1969.

[26]

P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford Lecture Series in Mathematics and Its Applications, 1996.

[27]

M. Lopes FilhoH. Nussenzveig LopesE. Titi and A. Zang, Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: Indifference to boundary layers, Phys. D, 292-293 (2015), 51-61.  doi: 10.1016/j.physd.2014.11.001.

[28]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In: Handbook of Differential Equations: Evolutionary Equations. Vol. IV, 103–200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[29]

E. Olson and E. Titi, Viscosity versus vorticity stretching: Global well-posedness for a family of Navier-Stokes-$\alpha$-like models, Nonlinear Anal., 66 (2007), 2427-2458.  doi: 10.1016/j.na.2006.03.030.

[30]

A. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem. LOMI, 38 (1973), 98-136. 

[31]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1987. doi: 10.1007/978-1-4612-4650-3.

[32]

T. Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation, J. Amer. Math. Soc., 29 (2016), 601-674.  doi: 10.1090/jams/838.

[33]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Springer-Verlag, New York 1997. doi: 10.1007/978-1-4612-0645-3.

[34]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, vol. 66, Siam, 1995. doi: 10.1137/1.9781611970050.

[35]

E. Wiedemann, Existence of weak solutions for the incompressible Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 727-730.  doi: 10.1016/j.anihpc.2011.05.002.

[36]

V. 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.

[37]

S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Disc. Cont. Dyn. Sys., 11 (2004), 351-392.  doi: 10.3934/dcds.2004.11.351.

[38]

S. ZelikA. Ilyin and A. Kostianko, Sharp dimension estimates for the attractors of the regularized damped Euler system, Doklady Mathematics, 104 (2021), 169-172.  doi: 10.1134/S1064562421040165.

show all references

References:
[1]

A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol 25. North-Holland Publishing Co., Amsterdam, 1992.

[2]

J. Bardina, J. Ferziger and W. Reynolds, Improved Subgrid Scale Models for Large Eddy Simulation, in Proceedings of the 13th AIAA Conference on Fluid and Plasma Dynamics, 1980. doi: 10.2514/6.1980-1357.

[3]

K. Bardos and E. Titi, Euler equations for incompressible ideal fluids, Uspekhi Mat. Nauk, 62 (2007), 5-46.  doi: 10.1070/RM2007v062n03ABEH004410.

[4]

Y. CaoE. Lunasin and E. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[5]

V. ChepyzhovA. Ilyin and S. Zelik, Vanishing viscosity limit for global attractors for the damped Navier–Stokes system with stress free boundary conditions, Physica D, 376–377 (2018), 31-38.  doi: 10.1016/j.physd.2017.08.005.

[6]

V. ChepyzhovA. Ilyin and S. Zelik, Strong trajectory and global $W^{1, p}$-attractors for the damped-driven Euler system in ${\mathbb R}^2$, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1835-1855.  doi: 10.3934/dcdsb.2017109.

[7]

V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., 49, Providence, RI: Amer. Math. Soc., 2002.

[8]

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

[9]

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.

[10]

R. DiPerna and A. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), 667-689.  doi: 10.1007/BF01214424.

[11]

C. Fefferman, Existence and smoothness of the Navier-Stokes equation, In: Millennium Prize Problems, Clay Math. Inst., Cambridge, MA, (2006), 57–67.

[12]

C. FoiasD. Holm and E. Titi, The three dimensional viscous Camassa–Holm equations, and their relation to the Navier–Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.

[13] C. FoiasO. ManelyR. Rosa and R. Temam, Navier–Stokes Equations and Turbulence, Cambridge Univ. Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.
[14] U. Frisch, Turbulence. The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1995.  doi: 10.1063/1.881555.
[15]

M. HolstE. Lunasin and G. Tsogtgerel, Analysis of a general family of regularized Navier–Stokes and MHD models, J. Nonlinear Sci., 20 (2010), 523-567.  doi: 10.1007/s00332-010-9066-x.

[16]

A. Il'in, The Euler equations with dissipation, Mat. Sb., 74 (1993), 475-485.  doi: 10.1070/SM1993V074N02ABEH003357.

[17]

A. Ilyin, A. Kostianko and S. Zelik, Sharp upper and lower bounds of the attractor dimension for 3D damped Euler–Bardina equations, Physica D, 432 (2022), Paper No. 133156. doi: 10.1016/j.physd.2022.133156.

[18]

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

[19]

A. Ilyin and E. Titi, Attractors to the two-dimensional Navier–Stokes-$\alpha$ models: An $\alpha$-dependence study, J. Dynam. Differential Equations, 15 (2003), 751-778.  doi: 10.1023/B:JODY.0000010064.06851.ff.

[20]

A. Ilyin and S. Zelik, Sharp dimension estimates of the attractor of the damped 2D Euler-Bardina equations, In book: Partial Differential Equations, Spectral Theory, and Mathematical Physics, EMS Series of Congress Reports, EMS Press, Berlin, (2021), 209–229. doi: 10.4171/ECR/18-1/12.

[21]

V. Kalantarov and E. Titi, Global attractors and determining modes for the 3D Navier–Stokes–Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.

[22] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.
[23]

W. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Continuous Dyn. Sys. B, 6 (2006), 111-128.  doi: 10.3934/dcdsb.2006.6.111.

[24]

J. Leray, Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.

[25]

J.-L. Lions, Quelques Méthodes des Problèmes aux Limites non Linéaires, Doud, Paris, 1969.

[26]

P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford Lecture Series in Mathematics and Its Applications, 1996.

[27]

M. Lopes FilhoH. Nussenzveig LopesE. Titi and A. Zang, Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: Indifference to boundary layers, Phys. D, 292-293 (2015), 51-61.  doi: 10.1016/j.physd.2014.11.001.

[28]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In: Handbook of Differential Equations: Evolutionary Equations. Vol. IV, 103–200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[29]

E. Olson and E. Titi, Viscosity versus vorticity stretching: Global well-posedness for a family of Navier-Stokes-$\alpha$-like models, Nonlinear Anal., 66 (2007), 2427-2458.  doi: 10.1016/j.na.2006.03.030.

[30]

A. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem. LOMI, 38 (1973), 98-136. 

[31]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1987. doi: 10.1007/978-1-4612-4650-3.

[32]

T. Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation, J. Amer. Math. Soc., 29 (2016), 601-674.  doi: 10.1090/jams/838.

[33]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Springer-Verlag, New York 1997. doi: 10.1007/978-1-4612-0645-3.

[34]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, vol. 66, Siam, 1995. doi: 10.1137/1.9781611970050.

[35]

E. Wiedemann, Existence of weak solutions for the incompressible Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 727-730.  doi: 10.1016/j.anihpc.2011.05.002.

[36]

V. 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.

[37]

S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Disc. Cont. Dyn. Sys., 11 (2004), 351-392.  doi: 10.3934/dcds.2004.11.351.

[38]

S. ZelikA. Ilyin and A. Kostianko, Sharp dimension estimates for the attractors of the regularized damped Euler system, Doklady Mathematics, 104 (2021), 169-172.  doi: 10.1134/S1064562421040165.

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