Regularity criteria for a two dimensional Erying-Powell fluid flowing in a MHD porous medium

José Luis Díaz Palencia; Saeed Ur Rahman; Saman Hanif; José Luis Díaz Palencia; Saeed Ur Rahman; Saman Hanif

doi:10.3934/era.2022201

Electronic Research Archive

2022, Volume 30, Issue 11: 3949-3976. doi: 10.3934/era.2022201

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Regularity criteria for a two dimensional Erying-Powell fluid flowing in a MHD porous medium

1.
Department of Mathematics and Education, Universidad a Distancia de Madrid, Madrid 28400, Spain
2.
Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus 22060, Pakistan

Academic Editor: Baowei Feng

Received: 28 July 2022 Revised: 23 August 2022 Accepted: 25 August 2022 Published: 05 September 2022

The intention and novelty in the presented study were to develop the regularity analysis for a parabolic equation describing a type of Eyring-Powell fluid flow in two dimensions. We proved that, under certain general conditions involving the space of bounded mean oscillation ($ BMO $) and the Lebesgue space $ L^2 $, there exist bounded and regular velocity solutions under the $ L^{2} $ space scope. This conclusion was additionally supplemented by the condition of a finite square integrable initial data (also some of the obtained expressions involved the gradient and the laplacian of the initial velocity distribution). To make our results further general, the proposed analysis was extended to cover regularity results in $ L^{p}\left(p > 2\right) $ spaces. As a remarkable conclusion, we highlight that the solutions to the two dimensional Eyring-Powell fluid flow did not exhibit blow up behaviour.
- regularity criteria,
- porous medium,
- Powell-Fluid flow,
- two dimensional fluid,
- evolutionary flow
Citation: José Luis Díaz Palencia, Saeed Ur Rahman, Saman Hanif. Regularity criteria for a two dimensional Erying-Powell fluid flowing in a MHD porous medium[J]. Electronic Research Archive, 2022, 30(11): 3949-3976. doi: 10.3934/era.2022201

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Abstract

The intention and novelty in the presented study were to develop the regularity analysis for a parabolic equation describing a type of Eyring-Powell fluid flow in two dimensions. We proved that, under certain general conditions involving the space of bounded mean oscillation ($ BMO $) and the Lebesgue space $ L^2 $, there exist bounded and regular velocity solutions under the $ L^{2} $ space scope. This conclusion was additionally supplemented by the condition of a finite square integrable initial data (also some of the obtained expressions involved the gradient and the laplacian of the initial velocity distribution). To make our results further general, the proposed analysis was extended to cover regularity results in $ L^{p}\left(p > 2\right) $ spaces. As a remarkable conclusion, we highlight that the solutions to the two dimensional Eyring-Powell fluid flow did not exhibit blow up behaviour.

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