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An upper bound on the attractor dimension of a 2D turbulent shear flow in lubrication theory. (English) Zbl 1133.37340

Summary: We consider a two-dimensional Navier-Stokes shear flow. There exists a unique global-in-time solution of the considered problem as well as the global attractor for the associated semigroup. Our aim is to estimate from above the dimension of the attractor in terms of given data and geometry of the domain of the flow. First we obtain a Kolmogorov-type bound on the time-averaged energy dissipation rate, independent of viscosity at large Reynolds numbers. Then we establish a version of the Lieb-Thirring inequality for a class of functions defined on the considered non-rectangular flow domain.
This research is motivated by a problem from lubrication theory.

MSC:

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B41 Attractors
76D08 Lubrication theory
76F10 Shear flows and turbulence
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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References:

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