Navier-Stokes Equations on R3 × [0, T] (PDF)

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 n 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.