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 (Formula presented.) -boundedness of the solution operator families with resolvent parameter λ varying in a sector (Formula presented.) for any γ0 > 0 and 0 < ε < π/2, where (Formula presented.). As applications, we obtain the maximal Lp-Lq regularity for the nonstationary Stokes problem and then establish the well-posedness locally in time of the nonlinear free boundary problem mentioned above in Lp-Lq setting. We make full use of the solution formula to take γ0 > 0 arbitrarily, while in general domains, we only know the (Formula presented.) -boundedness for γ0 ≫ 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.
ASJC Scopus subject areas
- 数学 (全般)