TY - JOUR

T1 - A solution formula and the ℛ-boundedness for the generalized Stokes resolvent problem in an infinite layer with Neumann boundary condition

AU - Oishi, Kenta

N1 - Funding Information:
I am very grateful to Professor Yoshihiro Shibata for bringing the problem studied in this paper to my attention and for many useful suggestions. I also thank Professor Hirokazu Saito for many stimulating discussions. This work does not have any conflicts of interest. There are no funders to report for this submission.
Publisher Copyright:
© 2020 John Wiley & Sons, Ltd.

PY - 2021/3/30

Y1 - 2021/3/30

N2 - We consider the generalized Stokes resolvent problem in an infinite layer with Neumann boundary conditions. This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without surface tension in an infinite layer bounded both from above and from below by free surfaces. We derive a new exact solution formula to the generalized Stokes resolvent problem and prove the (Formula presented.) -boundedness of the solution operator families with resolvent parameter λ varying in a sector (Formula presented.) for any γ0 > 0 and 0 < ε < π/2, where (Formula presented.). As applications, we obtain the maximal Lp-Lq regularity for the nonstationary Stokes problem and then establish the well-posedness locally in time of the nonlinear free boundary problem mentioned above in Lp-Lq setting. We make full use of the solution formula to take γ0 > 0 arbitrarily, while in general domains, we only know the (Formula presented.) -boundedness for γ0 ≫ 1 from the result by Shibata. As compared with the case of Neumann-Dirichlet boundary condition studied by Saito, analysis is even harder on account of higher singularity of the symbols in the solution formula.

AB - We consider the generalized Stokes resolvent problem in an infinite layer with Neumann boundary conditions. This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without surface tension in an infinite layer bounded both from above and from below by free surfaces. We derive a new exact solution formula to the generalized Stokes resolvent problem and prove the (Formula presented.) -boundedness of the solution operator families with resolvent parameter λ varying in a sector (Formula presented.) for any γ0 > 0 and 0 < ε < π/2, where (Formula presented.). As applications, we obtain the maximal Lp-Lq regularity for the nonstationary Stokes problem and then establish the well-posedness locally in time of the nonlinear free boundary problem mentioned above in Lp-Lq setting. We make full use of the solution formula to take γ0 > 0 arbitrarily, while in general domains, we only know the (Formula presented.) -boundedness for γ0 ≫ 1 from the result by Shibata. As compared with the case of Neumann-Dirichlet boundary condition studied by Saito, analysis is even harder on account of higher singularity of the symbols in the solution formula.

KW - free boundary problem

KW - infinite layer

KW - maximal regularity

KW - Stokes resolvent problem

KW - ℛ-boundedness

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U2 - 10.1002/mma.6999

DO - 10.1002/mma.6999

M3 - Article

AN - SCOPUS:85096714489

SN - 0170-4214

VL - 44

SP - 3925

EP - 3959

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

IS - 5

ER -