2020/06/25 by Kenta Oishi, Oishi, Kenta
Chemical Engineering · Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Rheology and Fluid Dynamics Studies
paper · pdf · doi:10.48550/arxiv.2006.14196
openalex publication_date 2020/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathscrR-boundedness of the solution operator families with resolvent parameter λ varying in a sector Σε,γ0 for any γ0>0 and 0γ0 \. 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 \mathscrR-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.