The Inviscid Limit of the Navier-Stokes Equations with Kinematic and Navier Boundary Conditions

2018/12/17 by Chen, Gui-Qiang G., Li, Siran, Qian, Zhongmin
#35Q30 #35Q31 #35Q35 #76D03 #76D05 #76D09 #Analysis of PDEs (math.AP) #Chaotic Dynamics (nlin.CD) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1812.06565

Abstract

We are concerned with the inviscid limit of the Navier-Stokes equations on bounded regular domains in ℝ³ with the kinematic and Navier boundary conditions. We first establish the existence and uniqueness of strong solutions in the class C([0,T_⋆); H^r(Ω; ℝ³)) ∩ C¹([0,T_⋆); H^r-2(Ω;ℝ³)) with some T_⋆>0 for the initial-boundary value problem with the kinematic and Navier boundary conditions on ∂ Ω and divergence-free initial data in the Sobolev space H^r(Ω; ℝ³) for r≥ 2. Then, for the strong solution with H^r+1--regularity in the spatial variables, we establish the inviscid limit in H^r(Ω; ℝ³) uniformly on [0,T_⋆) for r > (5)/(2). This shows that the boundary layers do not develop up to the highest order Sobolev norm in H^r(Ω;ℝ³) in the inviscid limit. Furthermore, we present an intrinsic geometric proof for the failure of the strong inviscid limit under a non-Navier slip-type boundary condition.

Related

• A Study of the Navier-Stokes Equations with the Kinematic and Navier Boundary Conditions
• The 3D inviscid limit problem with data analytic near the boundary
• The inviscid limit for the Navier-Stokes equations with data analytic only near the boundary
• Inviscid limit on L^p-based Sobolev conormal spaces for the 3D Navier-Stokes equations with the Navier boundary conditions
• The inviscid limit of Navier-Stokes with critical Navier-slip boundary\n conditions for analytic data