Optimal Korn’s inequality for solenoidal vector fields on a periodic slab

Teramoto, Yoshikuni; Tomoeda, Kyoko

doi:10.3792/pjaa.88.168

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Cited by 2 publications

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“…These terms can be treated by the estimate of J 8 and (5. 19) and (5.20) in Lemma 5.6. Therefore, we obtain K 13 ≤ (ǫ + C ǫ Ẽ2 ) Ẽm .…”

mentioning

confidence: 86%

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Uniform estimates for the flow of a viscous incompressible fluid down an inclined plane in the thin film regime

Ueno

Shiraishi

Iguchi

2016

Journal of Mathematical Analysis and Applications

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We consider a two-dimensional motion of a thin film flowing down an inclined plane under the influence of the gravity and the surface tension. In order to investigate the stability of such flow, it is hard to treat the Navier-Stokes equations directly, so that a thin film approximation is often used. It is an approximation obtained by the perturbation expansion with respect to the aspect ratio δ of the film under the thin film regime δ ≪ 1. Our purpose is to give a mathematically rigorous justification of the thin film approximation by establishing an error estimate between the solution of the Navier-Stokes equations and those of approximate equations. To this end, in this paper we derive a uniform estimate for the solution of the Navier-Stokes equations with respect to δ under appropriate assumptions.

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“…These terms can be treated by the estimate of J 8 and (5. 19) and (5.20) in Lemma 5.6. Therefore, we obtain K 13 ≤ (ǫ + C ǫ Ẽ2 ) Ẽm .…”

mentioning

confidence: 86%

“…Remark 3.1. Teramoto and Tomoeda [19] proved that the best constant of K is 3. Note that in the case of δ = 1, this inequality is well-known.…”

Section: Energy Estimatesmentioning

confidence: 99%

Uniform estimates for the flow of a viscous incompressible fluid down an inclined plane in the thin film regime

Ueno

Shiraishi

Iguchi

2016

Journal of Mathematical Analysis and Applications

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On a linearized system arising in the study of viscous surface flow down an inclined plane

Teramoto

Tomoeda

2014

Ann Univ Ferrara

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Optimal Korn’s inequality for solenoidal vector fields on a periodic slab

Cited by 2 publications

References 9 publications

Uniform estimates for the flow of a viscous incompressible fluid down an inclined plane in the thin film regime

Uniform estimates for the flow of a viscous incompressible fluid down an inclined plane in the thin film regime

On a linearized system arising in the study of viscous surface flow down an inclined plane

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