On the dimension of the attractor for a perturbed 3D Ladyzhenskaya model. (English) Zbl 1358.35130
Summary: We consider the so-called Ladyzhenskaya model of incompressible fluid, with an additional artificial smoothing term \(\varepsilon\Delta^3\). We establish the global existence, uniqueness, and regularity of solutions. Finally, we show that there exists an exponential attractor, whose dimension we estimate in terms of the relevant physical quantities, independently of \(\varepsilon >0\).
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |
| 76A05 | Non-Newtonian fluids |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35B25 | Singular perturbations in context of PDEs |
| 35B41 | Attractors |
| 35B65 | Smoothness and regularity of solutions to PDEs |
References:
| [1] | Ball J.M., Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 1997, 7(5), 475-502 http://dx.doi.org/10.1007/s003329900037; · Zbl 0903.58020 |
| [2] | Bulíček M., Ettwein F., Kaplický P., Pražák D., The dimension of the attractor for the 3D flow of a non-Newtonian fluid, Commun. Pure Appl. Anal., 2009, 8(5), 1503-1520 http://dx.doi.org/10.3934/cpaa.2009.8.1503; · Zbl 1200.37074 |
| [3] | Bulíček M., Ettwein F., Kaplický P., Pražák D., On uniqueness and time regularity of flows of power-law like non-Newtonian fluids, Math. Methods Appl. Sci., 2010, 33(16), 1995-2010; · Zbl 1202.35152 |
| [4] | Constantin P., Foias C., Navier-Stokes Equations, Chicago Lectures in Math., The University of Chicago Press, Chicago, 1988; · Zbl 0687.35071 |
| [5] | Feireisl E., Pražák D., Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS Ser. Appl. Math., 4, American Institute of Mathematical Sciences, Springfield, 2010; · Zbl 1198.37002 |
| [6] | Ladyzhenskaya O.A., New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems, Proc. Steklov Inst. Math., 1967, 102, 95-118; |
| [7] | Lions J.-L., Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969; · Zbl 0189.40603 |
| [8] | Málek J., Nečas J., Rokyta M., Růžička M., Weak and Measure-Valued Solutions to Evolutionary PDEs, Appl. Math. Math. Comput., 13, Chapman & Hall, London, 1996; · Zbl 0851.35002 |
| [9] | de Rham G., Variétés Différentiables, 3rd ed., Publications de l’Institut de mathématique de l’Université de Nancago, 3, Hermann, Paris, 1973; · Zbl 0284.58001 |
| [10] | Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Appl. Math. Sci., 68, Springer, New York, 1997; · Zbl 0871.35001 |
| [11] | Temam R., Navier-Stokes Equations, American Mathematical Society Chelsea, Providence, 2001; · Zbl 0981.35001 |
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