On the $\mathcal{R}$-Boundedness of Solution Operators for the Stokes Equations with Free Boundary Condition

Abstract

In this paper, we consider the boundary value problem of Stokes operators with variable viscosity in the case of free boundary condition in a uniform $W^{2-1/r}_r$ domain of $N$-dimensional Euclidean space $\mathbb R^N$ ($N\geq 2$, $N < r < \infty$). We prove the $\mathcal R$-boundedness of solution operators with spectral parameter $\lambda$ varying in a sector $\Sigma_{\epsilon, \lambda_0} = \{\lambda \in \mathbf C : | \arg\lambda| \leq \pi-\epsilon, \,\, | \lambda| \geq \lambda_0\}$, from which we can deduce the $L_p$-$L_q$ maximal regularity as well as the generation of analytic semigroup for the time dependent problem. The problem of this type arises in the mathematical study of the incompressible viscous fluid flow with free surface. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely, it is assumed the unique existence of solution $p \in \mathcal W^1_q(\Omega)$ to the variational problem: $(\nabla p, \nabla\varphi)_\Omega = (f, \nabla\varphi)_\Omega$ for any $\varphi \in \mathcal W^1_{q'}(\Omega)$ with $1 < q < \infty$ and $q' = q/(q-1)$, where $\mathcal W^1_q(\Omega)$ is a closed subspace of $\hat W^1_{q,\Gamma}(\Omega) = \{p \in L_{q, {\rm loc}} (\Omega) : \nabla p \in L_q(\Omega)^N, \,\, p|_\Gamma = 0\}$ with respect to gradient norm $\|\nabla\cdot\|_{L_q(\Omega)}$ that contains a space $W^1_{q,\Gamma}(\Omega) = \{p \in W^1_q(\Omega) : p|_\Gamma = 0\}$, and $\Gamma$ is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in $(\lambda_0, \infty)$, which was proved in Shibata [28]. Our assumption is satisfied for any $q \in (1, \infty)$ by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer with $\mathcal W^1_q(\Omega) = \hat W^1_{q, \Gamma}(\Omega)$, and by exterior domains with $\mathcal W^1_{q,\Gamma}(\Omega)$ $=$ the closure of $W^1_{q, \Gamma}(\Omega)$ with respect to the gradient norm. Combining the result in this paper with that in a forthcoming paper about the nonlinear problems, we can conclude that the unique existence of solutions to weak Dirichlet-Neumann problem implies a local in time unique existence theorem of strong solutions to the free boundary problem without surface tension taken into account for the Navier-Stokes equations in a uniform $W^{2-1/r}_r$ domain.