Micropolar meets Newtonian in 3D. The Rayleigh–Bénard problem for large Prandtl numbers*
Piotr Kalita, Grzegorz Łukaszewicz
Abstract
Abstract We consider the Rayleigh–Bénard problem for the three-dimensional Boussinesq system for the micropolar fluid. We introduce the notion of the multivalued eventual semiflow and prove the existence of the two-space global attractor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="script">A</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:msup> </mml:math> corresponding to weak solutions, for every micropolar parameter K ⩾ 0 denoting the deviation of the considered system from the classical Rayleigh–Bénard problem for the Newtonian fluid. We prove that for every K the attractor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="script">A</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:msup> </mml:math> is the smallest compact, attracting, and invariant set. Moreover, the semiflow restricted to this attractor is single-valued and governed by strong solutions. Further, we prove that the global attractors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="script">A</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:msup> </mml:math> converge to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="script">A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> </mml:math> upper semicontinuously as K → 0, and that the projection of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="script">A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> </mml:math> on the restricted phase space corresponding to the classical Rayleigh–Bénard problem is the global attractor for the latter problem, having the invariance property. These results are established under the assumption that the Prandtl number is relatively large with respect to the Rayleigh number.