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

- Kenta Oishi

## 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 $\mathscr{R}$-boundedness of the solution
operator families with resolvent parameter $\lambda$
varying in a sector
$\Sigma_{\varepsilon,\gamma_0}$
for any $\gamma_0>0$ and
$0<\varepsilon<\pi/2$,
where
$\Sigma_{\varepsilon,\gamma_0}
=\{
\lambda\in\mathbb{C}\setminus\{0\}
\mid
|\arg\lambda|\leq\pi-\varepsilon,
\
|\lambda|>\gamma_0
\}$. 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
$\gamma_0>0$ arbitrarily, while in
general domains we only know the
$\mathscr{R}$-boundedness for
$\gamma_0\gg1$ 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.

24 Jun 2020Submitted to *Mathematical Methods in the Applied Sciences* 27 Jun 2020Submission Checks Completed

27 Jun 2020Assigned to Editor

27 Jun 2020Reviewer(s) Assigned

10 Sep 2020Review(s) Completed, Editorial Evaluation Pending

10 Sep 2020Editorial Decision: Revise Major

13 Oct 20201st Revision Received

13 Oct 2020Submission Checks Completed

13 Oct 2020Assigned to Editor

13 Oct 2020Reviewer(s) Assigned

14 Oct 2020Review(s) Completed, Editorial Evaluation Pending

15 Oct 2020Editorial Decision: Accept

25 Nov 2020Published in Mathematical Methods in the Applied Sciences. 10.1002/mma.6999