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 2020Assigned to Editor

27 Jun 2020Submission Checks Completed

27 Jun 2020Reviewer(s) Assigned

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

10 Sep 2020Editorial Decision: Revise Major