Remarks about the inviscid limit of the Navier-Stokes system. Nader MASMOUDI ∗ Courant Institute of Mathematical Sciences 251 Mercer Street New York, NY 10012-1185, USA masmoudi@cims.nyu.edu September 19, 2006 Abstract In this paper we prove two results about the inviscid limit of the Navier- Stokes system. The first one concerns the convergence in H s of a sequence of solutions to the Navier-Stokes system when the viscosity goes to zero and the initial data is in H s . The second result deals with the best rate of convergence for vortex patch initial data in 2 and 3 dimensions. We present here a simple proof which also works in the 3D case. The 3D case is new. 1 The inviscid limit The Navier-Stokes system is the basic mathematical model for viscous in- compressible flows. In a bounded domain, it reads    ∂ t u + u.∇u − ν Δu + ∇p =0, div(u)=0, u =0 on ∂ Ω, (1) where u is the velocity, p is the pressure and ν is the kinematic viscosity. We can define a typical length scale L and a typical velocity U . The dimensionless * Courant Institute, New York University 251 Mercer St, New York, NY 10012 1