Navier-Stokes equations on a flat cylinder with vorticity production on the boundary

We study a singular version of the incompressible two-dimensional Navier-Stokes (NS) system on a flat cylinder C, with Neumann conditions for the vorticity and a vorticity production term on the boundary partial derivative C to restore the no-slip boundary condition for the velocity u|partial derivative C = 0. The problem is formulated as an infinite system of coupled ordinary differential equations (ODEs) for the Neumann Fourier modes. For a general class of initial data we prove existence and uniqueness of the solution, and equivalence to the usual NS system. The main tool in the proofs is a suitable decay of the modes, obtained by the explicit form of the ODEs. We finally show that the resulting expansions of the velocity u and of its first and second space derivatives converge and define continuous functions up to the boundary.