Vanishing Viscous Limits for 3D Navier–Stokes Equations with a Navier-Slip Boundary Condition

Abstract

In this paper, we investigate the vanishing viscosity limit for solutions to the Navier-Stokes equations with a Navier slip boundary condition on general compact and smooth domains in R 3 . We first obtain the higher order regularity estimates for the solutions to Prandtl's equation boundary layers. Furthermore, we prove that the strong solution to Navier-Stokes equations converges to the Eulerian one in C([0, T ]; H 1 (Ω)) and L ∞ ((0, T ) × Ω), where T is independent of the viscosity, provided that initial velocity is regular enough. Furthermore, rates of convergence are obtained also.

Gevrey regularity for Navier–Stokes equations under Lions boundary conditions

“…The study of Navier boundary conditions have been addressed by many authors in the last years, either because in some situations they may be more realistic than no-slip boundary conditions or because they are more appropriate in finding a solution for the Euler system as a limit of solutions for the Navier-Stoles system as ν goes to zero (cf. [XX07,WXZ12], [Kel06, Section 8]), or even the possibility to recover the solution under no-slip boundary conditions as a limit of solutions under Navier boundary conditions (cf. [JM01], and conversely (cf.…”

Initial–boundary layer associated with the nonlinear Darcy–Brinkman system

“…There is an abundant literature on boundary layer associated with incompressible flows and the related question of vanishing viscosity (see for instance [2,7,47,48,10,40,14,22,46,19,20,8,27,71,31,3,4,32,29,68,1,69,21,57,58,53,55,56,62,23,9,24,18,25,18,25,38,6,15,33,62,39,36,35,12,13,67,61] among many others). We will refrain from surveying the literature here, but emphasize that the boundary layer problem associated with the Navier-Stokes equation is still open and that there is a need to develop tools and methods to tackle it.…”

Vorticity layers of the 2D Navier–Stokes equations with a slip type boundary condition

“…Concerning the slip boundary condition (1.5) (or its 3D version; see (1.11) below), in a 2 or 3D domain with flat boundaries, the convergence in higher order Sobolev norms is well-studied in, e.g., [1,8] or [21]; for the case of a 3D curved domain, see, e.g., [2] or [20]. From the boundary layer analysis point of view, in [8], the authors study the boundary layers of the barotropic quasigeostrophic equations (6.1) in a 2D channel domain, which is essentially the vorticity formulation of the 2D Navier-Stokes equations with the slip boundary condition; see Section 6 for more details.…”

“…In a recent work, [20], the authors considered the Navier-Stokes equations (1.1) in a 3D general smooth domain supplemented with the 3D version of (1.5):…”

“…By taking curl on the asymptotic expansion (1.12) and analyzing the vorticity formulations of the Navier-Stokes and Euler equations, they prove that curl u ε − curl u 0 − √ ε curl(v − v| Γ ) L ∞ (0,T ;L 2 (Ω)) Cε 1/2 ; (1.13) see (6.30) in [20]. Then, from (1.13) and lemmas in [20], (1.10) follows in the end. In addition, the uniform convergence in time and space is also obtained with optimal rate ε 1/2 , but only for (1.11).…”