## Abstract and Applied Analysis

### On the Convergence of the Uniform Attractor for the 2D Leray-α Model

Gabriel Deugoué

#### Abstract

We consider a nonautonomous 2D Leray-$\alpha$ model of fluid turbulence. We prove the existence of the uniform attractor ${\mathcal{A}}^{\alpha }$. We also study the convergence of ${\mathcal{A}}^{\alpha }$ as $\alpha$ goes to zero. More precisely, we prove that the uniform attractor ${\mathcal{A}}^{\alpha }$ converges to the uniform attractor of the 2D Navier-Stokes system as $\alpha$ tends to zero.

#### Article information

Source
Abstr. Appl. Anal. Volume 2017 (2017), Article ID 1681857, 11 pages.

Dates
Revised: 30 March 2017
Accepted: 16 April 2017
First available in Project Euclid: 16 June 2017

https://projecteuclid.org/euclid.aaa/1497578543

Digital Object Identifier
doi:10.1155/2017/1681857

#### Citation

Deugoué, Gabriel. On the Convergence of the Uniform Attractor for the 2D Leray- α Model. Abstr. Appl. Anal. 2017 (2017), Article ID 1681857, 11 pages. doi:10.1155/2017/1681857. https://projecteuclid.org/euclid.aaa/1497578543

#### References

• T. Tachim Medjo, “A non-autonomous two-phase flow model with oscillating external force and its global attractor,” Nonlinear Analysis, vol. 75, no. 1, pp. 226–243, 2012.
• V. V. Chepyzhov, V. Pata, and M. I. Vishik, “Averaging of 2D Navier-Stokes equations with singularly oscillating forces,” Nonlinearity, vol. 22, no. 2, pp. 351–370, 2009.
• V. V. Chepyzhov and M. I. Vishik, “Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor,” Journal of Dynamics and Differential Equations, vol. 19, no. 3, pp. 655–684, 2007.
• S. Lu, “Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces,” Journal of Differential Equations, vol. 230, no. 1, pp. 196–212, 2006.
• S. Lu, H. Wu, and C. Zhong, “Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,” Discrete and Continuous Dynamical Systems. Series A, vol. 13, no. 3, pp. 701–719, 2005.
• H. Song, S. Ma, and C. Zhong, “Attractors of non-autonomous reaction-diffusion equations,” Nonlinearity, vol. 22, no. 3, pp. 667–681, 2009.
• P. E. Kloeden and B. Schmalfuss, “Non-autonomous systems, cocycle attractors and variable time-step discretization,” Numerical Algorithms, vol. 14, no. 1–3, pp. 141–152, 1997.
• G. Yue and C. Zhong, “On the convergence of the uniform attractor of the 2D NS-$\alpha$ model to the uniform attractor of the 2D NS system,” Journal of Computational and Applied Mathematics, vol. 233, no. 8, pp. 1879–1887, 2010.
• A. Haraux, “Systémes dynamiques dissipatifs et applications,” in Recherches en Mathématiques Appliqueés, vol. 17, Mason, Paris, 1991.
• V. V. Chepyzhov and M. I. Vishik, in Attractors for equations of mathematical physics, vol. 49, American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 2002.
• R. Temam, Infinite Dynamical Dimensional Dynamical Systems in Mechanics and Physics, vol. 68, Springer-Verlag, New York, NY, USA, 2nd edition, 1988.
• A. V. Babin and M. I. Vishik, “Attractors of evolutions equations,” in Studies in Mathematics and Its Applications, vol. 25, North-Holland, Publishing Co, Amsterdam, The Netherlands, 1992.
• E. Lunasin, S. Kurien, and E. S. Titi, “Spectral scaling of the Leray-$\alpha$ model for the two-dimensional turbulence,” Journal of Physics. A. Mathematical and Theoretical, vol. 41, Article ID 344014, 2008.
• A. Cheskidov, D. D. Holm, E. Olson, and E. S. Titi, “On a Leray-$\alpha$ model of turbulence,” Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, vol. 461, pp. 629–649, 2005.
• V. V. Chepyzhov, E. S. Titi, and M. I. Vishik, “On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-stokes system,” Discrete and Continuous Dynamical Systems, vol. 17, no. 3, pp. 481–500, 2007.
• V. V. Chepyzhov, E. S. Titi, and M. I. Vishik, “On convergence of trajectory attractors of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ approaches,” Sb. Math, vol. 198, pp. 1703–1736, 2007.
• H. Bessaih and P. A. Razafimandimby, “On the rate of convergence of the 2D stochastic Leray-$\alpha$ model to the 2D stochastic Navier-Stokes equations with multiplicative noise,” Applied Mathematics and Optimization, vol. 74, no. 1, pp. 1–25, 2016.
• Y. Cao and E. S. Titi, “On the rate of convergence of the two-dimensional $\alpha$-models of turbulence to the Navier-Stokes equations,” Numerical Functional Analysis and Optimization, vol. 30, no. 11-12, pp. 1231–1271, 2009.
• A. A. Ilyin and E. S. Titi, “Attractors for the two-dimensional Navier-Stokes-$\alpha$ model: an $\alpha$-dependence study,” Journal of Dynamics and Differential Equations, vol. 14, pp. 751–778, 2003.
• E. Zeidler, Nonlinear Functional Analysis and Its applications II/A: Linear Monotone Operators, Springer–Verlag, New York, NY, USA, 1990.
• J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, Farnce, 1969. \endinput