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Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping

Dedicated to Professor Peter Kloeden on his 70th birthday

This work has been partially supported by Spanish Ministry of Economy and Competitiveness and FEDER, projects MTM2015-63723-P and MTM2016-74921-P, and by Junta de Andalucía (Spain), project P12-FQM-1492

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  • In this paper we obtain the existence of global attractors for the dynamicalsystems generated by weak solution of the three-dimensional Navier-Stokesequations with damping. We consider two cases, depending on the values of the parameter β controlling the damping term. First, we prove that for β≥4 weaksolutions are unique and establish the existence of the global attractor forthe corresponding semigroup. Second, for 3≤β<4 we define amultivalued dynamical systems and prove the existence of the global attractoras well. Finally, some numerical simulations are performed.

    Mathematics Subject Classification: 35B40, 35B41, 35K55, 35Q30, 37B25, 58C06.

    Citation:

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  • Figure 1.  Flow domain

    Figure 2.  Flow velocity $u$ for the steady state in the $xy$-section at $z = 0$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0}$ is identically zero. Left panel parameters: $\alpha = 0.2;\, \beta = 1$. Right panel parameters: $\alpha = 0.5;\, \beta = 1$

    Figure 3.  Flow velocity $u$ for the steady state in the $xy$-section at $z = 0$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0}$ is identically zero. Left panel parameters: $\alpha = 0.2;\, \beta = 2$. Right panel parameters: $\alpha = 0.5;\, \beta = 2$

    Figure 4.  Flow velocity $u$ for the steady state in the $xy$-section at $z = 0$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0}$ is identically zero. Left panel parameters: $\alpha = 0.2;\, \beta = 4$. Right panel parameters: $\alpha = 0.5;\, \beta = 4$

    Figure 5.  Flow velocity $u$ in the $xy$-section at $z = 0$ for $\alpha = 0.2$ and $\beta = 1$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0} = (1, 0, 0)$. Left panel: state when $t = 0.1$. Right panel: steady state ($t$ large enough)

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