Navier–Stokes Equations on R3 × (0, T)

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In this monograph, leading researchers in numerical analysis, partial differential equations, and complex computational problems explore the properties of solutions to the Navier–Stokes partial differential equations in the domain (x, y, z, t) ∈ ℝ³ × [0, T]. The authors begin by transforming the PDE into a system of integral equations and identifying spaces A of analytic functions that contain the solutions. They demonstrate that these spaces are dense within the spaces S of rapidly decreasing and infinitely differentiable functions. This approach offers several advantages: functions in S are often conceptual rather than explicit, initial and boundary conditions from applied sciences are typically piece-wise analytic, allowing solutions to retain similar properties. Furthermore, approximation methods applied to functions in A converge exponentially, while those for functions in S converge at a polynomial rate. This leads to sharper bounds on solutions, facilitating existence proofs and providing a more accurate, efficient solution method with precise error bounds. After establishing denseness, the authors prove the existence of solutions in the space A ∩ ℝ³ × [0, T] and introduce a novel algorithm based on Sinc approximation and Picard-like iteration for computing these solutions. Appendices include a custom Mathematica program for implementing the algorithm and visualizing the computed solutions.