Navier–Stokes Equations on R3 × [0, T‪]‬

In this monograph, leading researchers in
the world of numerical analysis, partial differential equations, and hard
computational problems study the properties of solutions of the Navier–Stokes partial
differential equations on (x, y, z, t) ∈ ℝ3× [0, T]. Initially converting the PDE to a
system of integral equations, the authors then describe spaces A of analytic functions that house
solutions of this equation, and show that these spaces of analytic functions
are dense in the spaces S of rapidly
decreasing and infinitely differentiable functions. This method benefits from
the following advantages:

The
functions of S are nearly always conceptual rather than explicit
Initial
and boundary conditions of solutions of PDE are usually drawn from the
applied sciences, and as such, they are nearly always piece-wise analytic,
and in this case, the solutions have the same properties
When
methods of approximation are applied to functions of A they converge at an exponential rate, whereas methods of
approximation applied to the functions of S converge only at a polynomial rate
Enables
sharper bounds on the solution enabling easier existence proofs, and a
more accurate and more efficient method of solution, including accurate
error bounds

Following the proofs of denseness, the
authors prove the existence of a solution of the integral equations in the
space of functions A ∩ ℝ3× [0, T], and provide an explicit novel algorithm based on Sinc
approximation and Picard–like iteration for computing the solution.
Additionally, the authors include appendices that provide a custom Mathematica
program for computing solutions based on the explicit algorithmic approximation
procedure, and which supply explicit illustrations of these computed solutions.