2018/11/24 by Tatsu‐Hiko Miura, Miura, Tatsu-Hiko
Computer Science · Engineering · Mathematics · #35Q30 #76A20 #76D03 #76D05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Primary: 35B25 #Secondary: 35R01 #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1811.09816
openalex publication_date 2018/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the three-dimensional incompressible Navier--Stokes equations in a curved thin domain with Navier's slip boundary conditions. The curved thin domain is defined as a region between two closed surfaces which are very close to each other and degenerates into a given closed surface as its width tends to zero. We establish the global-in-time existence and uniform estimates of a strong solution for large data when the width of the thin domain is very small. Moreover, we study a singular limit problem as the thickness of the thin domain tends to zero and rigorously derive limit equations on the limit surface, which are the damped and weighted Navier--Stokes equations on a surface with viscous term involving the Gaussian curvature of the surface. We prove the weak convergence of the average in the thin direction of a strong solution to the bulk Navier--Stokes equations and characterize the weak limit as a weak solution to the limit equations as well as provide estimates for the difference between solutions to the bulk and limit equations. To deal with the weighted surface divergence-free condition of the limit equations we also derive the weighted Helmholtz--Leray decomposition of a tangential vector field on a closed surface.