On the Convergence of Periodic Navier-Stokes Flows

David Purvance — Thu, 28 Aug 2008 21:26:32 +0000

It’s not that the 3D spatially periodic, incompressible Navier-Stokes equation can be posed as a nonlinear matrix differential equation. It’s not that this matrix differential equation has a smooth and bounded matrix exponential solution, found when the flow in this differential equation is Taylor expanded in time and the resulting system matrices are all commutative. And, it’s not that this time expanded matrix differential equation also has a more general time series solution. It is called the noncommutative solution because its coefficients involve noncommutative matrix polynomials. This matrix differential equation and its commutative and noncommutative solutions have been recognized by the world-renown expert Terence Tao. Read Terence’s 22 March 2008 comments under his blog Why global regularity for the Navier-Stokes is so hard.

The question is: Do the noncommutative solutions converge?

In my most recent work I have boiled the question of convergence down to a question of linear independence.

Comments are welcome.

Solutions to the Periodic Navier-Stokes Equation

On the Convergence of Periodic Navier-Stokes Flows