Solution formula for generalized two-phase Stokes equations and its applications to maximal regularity: Model problems

Naoto Kajiwara; Naoto Kajiwara

doi:10.3934/math.2024888

AIMS Mathematics

2024, Volume 9, Issue 7: 18186-18210. doi: 10.3934/math.2024888

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Solution formula for generalized two-phase Stokes equations and its applications to maximal regularity: Model problems

Naoto Kajiwara ^,

Applied Physics Course, Department of Electrical, Electronic and Computer Engineering, Gifu University, Yanagido 1-1, Gifu, Gifu 501-1193, Japan

Received: 11 February 2024 Revised: 13 May 2024 Accepted: 15 May 2024 Published: 29 May 2024
MSC : 35D35, 35B45, 35K51

In this paper, we give a solution formula for the two-phase Stokes equations with and without surface tension and gravity over the whole space with a flat interface. The solution formula has already been considered by Shibata and Shimizu. However, we have reconstructed the formula so that we are able to easily prove resolvent and maximal regularity estimates. The previous work required the assumption of additional conditions on normal components. Here, although we consider normal components, the assumption is weaker than before. The method is based on an $ H^\infty $-calculus which has already been applied for the Stokes problems with various boundary conditions in the half-space.
- solution formula,
- resolvent estimate,
- maximal regularity,
- two-phase Stokes equations
Citation: Naoto Kajiwara. Solution formula for generalized two-phase Stokes equations and its applications to maximal regularity: Model problems[J]. AIMS Mathematics, 2024, 9(7): 18186-18210. doi: 10.3934/math.2024888

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Abstract

In this paper, we give a solution formula for the two-phase Stokes equations with and without surface tension and gravity over the whole space with a flat interface. The solution formula has already been considered by Shibata and Shimizu. However, we have reconstructed the formula so that we are able to easily prove resolvent and maximal regularity estimates. The previous work required the assumption of additional conditions on normal components. Here, although we consider normal components, the assumption is weaker than before. The method is based on an $ H^\infty $-calculus which has already been applied for the Stokes problems with various boundary conditions in the half-space.

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