Trajectory attractors method for dissipative partial differential equations with small parameter

V. V. Chepyzhov

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 32 (2024) no. 6, pp. 858-877.

Résumé

The purpose of this work is to study the limit behaviour of trajectory attractors for some equations and systems from mathematical physics depending on a small parameter when this small parameter approaches zero. The main attention is given to the cases when, for the limit equation, the uniqueness theorem for a solution of the corresponding initial-value problem does not hold or is not proved. The following problems are considered: approximation of the 3D Navier–Stokes system using the Leray $\alpha$-model, homogenization of the complex Ginzburg–Landau equation in a domain with dense perforation, and zero viscosity limit of 2D Navier–Stokes system with Ekman friction. Methods. In this paper, the method of trajectory dynamical systems and trajectory attractors is used that is especially effective in the study of complicated partial differential equations for which the uniqueness theorem for a solution of the corresponding initial-value problem does not hold or is not proved. Results. For all problems under the consideration, we obtain the limit equations and prove the Hausdorff convergence for trajectory attractors of the initial equations to the trajectory attractors of the limit equations in the appropriate topology when the small parameter tends to zero. Conclusion. In the work, we demonstrate that the method of trajectory attractors is highly effective in the study of dissipative equations of mathematical physics with small parameter. We succeed to find the limit equations and to prove the convergence of trajectory attractors of the considered equations to the trajectory attractors of the limit (homogenized) equations in the corresponding topology as small parameter is vanishes.

Détail

Keywords: global attractors, trajectory attractors, small parameter, convergence of attractors

Bibliographie

[1] Babin A. V., Vishik M. I., Attraktory evolyutsionnykh uravnenii, Nauka, M., 1989, 296 pp. | MR

[2] Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, v. 68, Applied Mathematical Sciences, 2, Springer-Verlag, New York, 1997, 650 pp. | DOI | MR | Zbl

[3] Vishik M. I., Chepyzhov V. V., Attractors for Equations of Mathematical Physics, v. 49, American Mathematical Society, American Mathematical Society Colloquium Publications, Providence, 2002, 364 pp. | DOI | MR | Zbl

[4] Sell G. R., “Global attractors for thethree-dimensional Navier–Stokes equations”, J. Dyn.Diff. Eq, 8:1 (1996), 1–33 | DOI | MR | Zbl

[5] Chepyzhov V. V., Conti M., Pata V., “Aminimal approach to the theory of global attractors”, Discrete andContinuous Dyn. Sys, 32 (2012), 2079–2088 | DOI | MR | Zbl

[6] Chepyzhov V. V., Vishik M. I., “Trajectory attractors for evolution equations”, C. R. Acad. Sci. Paris, 321(Série I) (1995), 1309–1314 | MR | Zbl

[7] Chepyzhov V. V., Vishik M. I., “Evolutionequations and their trajectory attractors”, J. Math.PuresAppl., 76:10 (1997), 913–964 | DOI | Zbl

[8] Vishik M. I., Chepyzhov V. V., “Traektornyeattraktory uravnenii matematicheskoi fiziki”, UMN, 66:4 (2011), 3–102 | DOI | MR | Zbl

[9] Lions J.-L., Quelques Méthodes de Résolutions des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969, 554 pp. | MR

[10] Albritton D., Brue E., Colombo M., “Gluing non-unique Navier-Stokes solutions”, Ann. PDE, 9:17 (2023) | DOI | MR | Zbl

[11] Cheskidov A., Holm D. D., Olson E., Titi E. S., “On Leray-$\alpha $ model of turbulence”, Proceedings of the Royal Society a Mathematical Physical and Engineering Sciences, 461 (2005), 629–649 | DOI | MR | Zbl

[12] Chepyzhov V. V., Titi E. S., Vishik M. I., “On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier–Stokes system”, Discrete and Continuous Dyn. Sys, 17:3 (2007), 33–52 | MR

[13] Chepyzhov V. V., “Ob approksimatsii traektornogo attraktora 3D sistemy Nave–Stoksa razlichnymi $\alpha$-modelyami gidrodinamiki”, Matem. sb, 207:4 (2016), 143–172 | DOI | Zbl

[14] Bekmaganbetov K.A., Chepyzhov V.V., Chechkin G.A., “Ob attraktorakh uravnenii reaktsii–diffuzii v poristoi ortotropnoi srede”, Dokl. RAN. Matem., inform., prots. upr, 498 (2021), 10–15 | DOI | MR | Zbl

[15] Pedlosky J., Geophysical Fluid Dynamics, Springer, New York, 1979 | DOI | Zbl

[16] Ilyin A. A., Patni K., Zelik S. V., “Upper bounds for the attractor dimension of damped Navier–Stokes equations in $\mathbb{R}^2$”, Discrete and Continuous Dyn. Sys, 36 (2016), 2085–2102 | DOI | MR | Zbl

[17] Rosa R., “The global attractor for the 2D Navier–Stokes flow on some unbounded domains”, Nonlinear Anal, 32 (1998), 71–85 | DOI | MR | Zbl

[18] DiPerna R., Lions P., “Ordinary differential equations, Sobolev spaces and transport theory”, Invent. Math., 98 (1989), 511–547 | DOI | MR | Zbl

[19] Boyer F., Fabrie P., Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, v. 183, Applied Mathematical Sciences, Springer, New York, 2013, 526 pp. | DOI | MR | Zbl

[20] Chepyzhov V. V., Ilyin A. A., Zelik S. V., “Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb{R}^{2}$”, Discrete Contin. Dyn. Syst. B, 22 (2017), 123–155 | DOI | MR

[21] Yudovich V. I., “Nestatsionarnye techeniya idealnoi neszhimaemoi zhidkosti”, Zh. Vych. Mat. Fiz., 3 (1963), 1032–1066 | Zbl

[22] Ilin A. A., Chepyzhov V. V., “O silnoi skhodimosti attraktorov uravnenii Nave–Stoksa v predele ischezayuschei vyazkosti”, Matem. zametki, 101:4 (2017), 635–639 | DOI | Zbl

[23] Chepyzhov V. V., Ilyin A. A., Zelik S. V., “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 | MR | Zbl