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Article Contents

# Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing

• This paper treats the existence of pullback attractors for the non-autonomous 2D Navier--Stokes equations in two different spaces, namely $L^2$ and $H^1$. The non-autonomous forcing term is taken in $L^2_{\rm loc}(\mathbb R;H^{-1})$ and $L^2_{\rm loc}(\mathbb R;L^2)$ respectively for these two results: even in the autonomous case it is not straightforward to show the required asymptotic compactness of the flow with this regularity of the forcing term. Here we prove the asymptotic compactness of the corresponding processes by verifying the flattening property -- also known as Condition (C)". We also show, using the semigroup method, that a little additional regularity -- $f\in L^p_{\rm loc}(\mathbb R;H^{-1})$ or $f\in L^p_{\rm loc}(\mathbb R;L^2)$ for some $p>2$ -- is enough to ensure the existence of a compact pullback absorbing family (not only asymptotic compactness). Even in the autonomous case the existence of a compact absorbing set for this model is new when $f$ has such limited regularity.
Mathematics Subject Classification: 35B41, 35B65, 35Q30.

 Citation:

•  [1] J. M. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc., 352 (2000), 285-310.doi: 10.1090/S0002-9947-99-02528-3. [2] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.doi: 10.1016/j.na.2005.03.111. [3] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.doi: 10.1016/j.crma.2005.12.015. [4] A. N. Carvalho, J. A. Langa and J. C. Robinson, "Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems," Applied Mathematical Sciences, 182, Springer, New York, 2012.doi: 10.1007/978-1-4614-4581-4. [5] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340. [6] S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317.doi: 10.1016/0022-0396(88)90007-1. [7] S.-N. Chow, K. Lu, and G. R. Sell, Smoothness of inertial manifolds, J. Math. Anal. Appl., 169 (1992), 283-312.doi: 10.1016/0022-247X(92)90115-T. [8] C. Foias, O. P. Manley, R. Temam and Y. M. Trève, Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9 (1983), 157-188.doi: 10.1016/0167-2789(83)90297-X. [9] C. Foias, O. Manley and R. Temam, Modelling of the interaction of small and large eddies in two-dimensional turbulent flows, RAIRO Modél. Math. Anal. Numér., 22 (1988), 93-118. [10] C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-353.doi: 10.1016/0022-0396(88)90110-6. [11] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315.doi: 10.1007/BF00276188. [12] J. García-Luengo, P. Marín-Rubio and J. Real, $H^2$-boundedness of the pullback attractors for non-autonomous 2D Navier-Stokes equations in bounded domains, Nonlinear Anal., 74 (2011), 4882-4887.doi: 10.1016/j.na.2011.04.063. [13] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4356.doi: 10.1016/j.jde.2012.01.010. [14] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357. [15] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [16] D. A. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana Univ. Math. J., 42 (1993), 875-887.doi: 10.1512/iumj.1993.42.42039. [17] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181.doi: 10.1098/rspa.2006.1753. [18] P. E. Kloeden, J. A. Langa, and J. Real, Pullback $V$-attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955.doi: 10.3934/cpaa.2007.6.937. [19] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, 1969. [20] Q. Ma, S. Wang, and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.doi: 10.1512/iumj.2002.51.2255. [21] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963.doi: 10.1016/j.na.2009.02.065. [22] P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006.doi: 10.3934/dcds.2010.26.989. [23] J. C. Robinson, "Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.doi: 10.1007/978-94-010-0732-0. [24] 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. [25] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," $2^{nd}$ edition, North Holland, Amsterdam, 1979. [26] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1988.doi: 10.1007/978-1-4684-0313-8.