On the maximal $L_p$-$L_q$ regularity of the Stokes problem with first order boundary condition; model problems

Abstract

In this paper, we proved the generalized resolvent estimate and the maximal $L_p$-$L_q$ regularity of the Stokes equation with first order boundary condition in the half-space, which arises in the mathematical study of the motion of a viscous incompressible one phase fluid flow with free surface. The core of our approach is to prove the $\mathcal{R}$ boundedness of solution operators defined in a sector $\Sigma_{\epsilon,\gamma_0} = \{\lambda \in C \: \backslash \: \{0\} \: | \: \left|\mathrm{arg}\lambda\right| \leq \pi - \epsilon, \left|\lambda\right| \geq \gamma_0 \}$ with $0 < \epsilon < \pi / 2$ and $\gamma_0 \geq 0$. This $\mathcal{R}$ boundedness implies the resolvent estimate of the Stokes operator and the combination of this $\mathcal{R}$ boundedness with the operator valued Fourier multiplier theorem of L. Weis implies the maximal $L_p$-$L_q$ regularity of the non-stationary Stokes. For a densely defined closed operator $A$, we know that what $A$ has maximal $L_p$ regularity implies that the resolvent estimate of $A$ in $\lambda \in \Sigma_{\epsilon,\gamma_0}$, but the opposite direction is not true in general (cf. Kalton and Lancien [19]). However, in this paper using the $\mathcal{R}$ boundedness of the operator family in the sector $\Sigma_{\epsilon,\gamma_0}$, we derive a systematic way to prove the resolvent estimate and the maximal $L_p$ regularity at the same time.