Navier–Stokes equations in 3D thin domains with Navier friction boundary condition

Abstract

In this article we study the 3D Navier–Stokes equations with Navier friction boundary condition in thin domains. We prove the global existence of strong solutions to the 3D Navier–Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. We generalize the techniques developed to study the 3D Navier–Stokes equations in thin domains, see [G. Raugel, G. Sell, Navier–Stokes equations on thin 3D domains I: Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993) 503–568; G. Raugel, G. Sell, Navier–Stokes equations on thin 3D domains II: Global regularity of spatially periodic conditions, in: Nonlinear Partial Differential Equations and Their Application, College de France Seminar, vol. XI, Longman, Harlow, 1994, pp. 205–247; R. Temam, M. Ziane, Navier–Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996) 499–546; R. Temam, M. Ziane, Navier–Stokes equations in thin spherical shells, in: Optimization Methods in Partial Differential Equations, in: Contemp. Math., vol. 209, Amer. Math. Soc., Providence, RI, 1996, pp. 281–314], to the Navier friction boundary condition by introducing a new average operator M ε in the thin direction according to the spectral decomposition of the Stokes operator A ε . Our analysis hinges on the refined investigation of the eigenvalue problem corresponding to the Stokes operator A ε with Navier friction boundary condition.