Observations on the vanishing viscosity limit

Whether, in the presence of a boundary, solutions of the Navier-Stokes equations converge to a solution to the Euler equations in the vanishing viscosity limit is unknown. In a seminal 1983 paper, Tosio Kato showed that the vanishing viscosity limit is equivalent to having sufficient control of the gradient of the Navier-Stokes velocity in a boundary layer of width proportional to the viscosity. In a 2008 paper, the present author showed that the vanishing viscosity limit is equivalent to the formation of a vortex sheet on the boundary. We present here several observations that follow on from these two papers. Compiled on Sunday 20 March 2016 1. Definitions and past results 3 2. A 3D version of vorticity accumulation on the boundary 6 3. L-norms of the vorticity blow up for p > 1 7 4. Improved convergence when vorticity bounded in L 8 5. Some kind of convergence always happens 9 6. Width of the boundary layer: 2D 10 7. Optimal convergence rate: 2D 12 8. A condition on the boundary equivalent to (V V ): 2D 14 9. Examples where condition on the boundary holds: 2D 17 10. On a result of Bardos and Titi: 2D 22 Appendix A. A Trace Lemma 23 Acknowledgements 25 References 26 The Navier-Stokes equations for a viscous incompressible fluid in a domain Ω ⊆ Rd, d ≥ 2, with no-slip boundary conditions can be written, (NS)  ∂tu+ u · ∇u+∇p = ν∆u+ f in Ω, div u = 0 in Ω, u = 0 on Γ := ∂Ω. Date: (compiled on Sunday 20 March 2016). 2010 Mathematics Subject Classification. Primary 76D05, 76B99, 76D10.