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The sequential regularization method is a reformulation of the unsteady Navier–Stokes equations from the viewpoint of constrained dynamical systems or the approximate Helmholtz–Hodge projection. In this paper we study the long time behavior of the sequential regularization formulation. We give a uniform-in-time estimate between the solution of the reformulated system and that of the Navier–Stokes equations. We also conduct an error analysis for the temporal discrete system and show that the error bound is independent of time. A couple of long time flow examples are computed to demonstrate this method.

This project was sponsored through the Schiff Fellowship program of Brandeis University. This project involved using the power series method to construct a third order nonlinear ordinary differential equation, a Schwarzian equation, for each of the "genus zero" modular functions, described in the Conway-Norton paper. We first use the Borcherd recursion formuli to generate, in each case, a modular function up to whatever degree we desire, and then use the fact that there is a Schwarzian equation, determined by a single rational function we call a Q-value. By similar power series methods, we compute the coefficients of our rational function, and hence have all the necessary data to create a Schwarzian differential equation for each modular function. This equation can, in turn, be used to recover the modular function itself.