On the convergence of periodic Navier-Stokes flows
  The 3D spatially periodic Navier-Stokes equation is posed as a nonlinear matrix differential equation. When the flow is assumed to be a time series having unknown wavenumber coefficients, then the matrix in this periodic Navier-Stokes matrix differential equation becomes a time series of matrices. Posed in this way, any flow's unknown wavenumber coefficients can be solved for recursively beginning with the zeroth-order coefficient representing any initial flow. When all matrices in the time series commute, a flow's unknown coefficients also represent the Taylor expansion of a stable matrix exponential product operating on the initial flow. This paper argues that a solution's coefficients converge to these bounded Taylor coefficients when these bounds are evaluated with a general solution's noncommutative matrices. 
David T. Purvance