In this short note, we study the local well-posedness of a 3D model for
incompressible Navier-Stokes equations with partial viscosity. This model was
originally proposed by Hou-Lei in \cite{HouLei09a}. In a recent paper, we prove
that this 3D model with partial viscosity will develop a finite time
singularity for a class of initial condition using a mixed Dirichlet Robin
boundary condition. The local well-posedness analysis of this initial boundary
value problem is more subtle than the corresponding well-posedness analysis
using a standard boundary condition because the Robin boundary condition we
consider is non-dissipative. We establish the local well-posedness of this
initial boundary value problem by designing a Picard iteration in a Banach
space and proving the convergence of the Picard iteration by studying the
well-posedness property of the heat equation with the same Dirichlet Robin
boundary condition
