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

DOI10.1002/MMA.6999zbMATH Open1475.35239arXiv2006.14196OpenAlexW3109218713MaRDI QIDQ5004152

Publication date: 30 July 2021

Published in: Mathematical Methods in the Applied Sciences (Search for Journal in Brave)

Abstract: 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

m a t h s c r R

-boundedness of the solution operator families with resolvent parameter

l a m b d a

varying in a sector

S i g m a_{v a r e p s i l o n, g a m m a_{0}}

for any

g a m m a_{0} > 0

and

0 < v a r e p s i l o n < p i / 2

, where

S i g m a_{v a r e p s i l o n, g a m m a_{0}} = l a m b d a i n m a t h b b C s e t m i n u s 0 m i d | a r g l a m b d a | l e q p i - v a r e p s i l o n, | l a m b d a | > g a m m a_{0}

. As applications, we obtain the maximal

L_{p}

-

L_{q}

regularity for the nonstationary Stokes problem and then establish the well-posedness locally in time of the nonlinear free boundary problem mentioned above in

L_{p}

-

L_{q}

setting. We make full use of the solution formula to take

g a m m a_{0} > 0

arbitrarily, while in general domains we only know the

m a t h s c r R

-boundedness for

g a m m a_{0} g g 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.

Full work available at URL: https://arxiv.org/abs/2006.14196

zbMATH Keywords

free boundary problem maximal regularity \(\mathcal{R}\)-boundedness infinite layer Stokes resolvent problem

Mathematics Subject Classification ID