Repetitive curling of the incompressible viscid Navier–Stokes differential equation leads to a higher-order diffusion equation. Substituting this equation into the Navier–Stokes differential equation transposes the latter into the Korteweg–De Vries–Burgers-equation with the Weierstrass p-function as the soliton solution. However, a higher-order derivative of the studied variable produces the so-called N-soliton solution, which is comparable with the N-soliton solution of the Kadomtsev–Petviashvili equation. Experiments have made it clear that the system behaves like a coupled (an)harmonic oscillator on a discrete collapsed-state level. The streamlines obtained are derivatives of the error function as a function of the obtained Lax functional of the particle filaments dynamics induced by the (hypothetical) Calogero–Moser many-body system with elliptical potential and are the so-called Hermite functions. Hermite tried to introduce doubly periodic Hermite functions (the so-called Hermite problem) using coefficients related to the Weierstrass p-function. A solution-sensitive analysis of the incompressible viscid Navier–Stokes equation is performed using the Lamb vector. Cases with a meaningful potential-energy contribution require a particle interaction model with an N-soliton solution using a hierarchy-like solution of the Kadomtsev–Petviashvili equation. A three-soliton solution is emulated for the cylinder-wake problem. Our analytical results are put in perspective by comparison with two well-studied benchmark cases of fluid dynamics: the cylinder-wake problem and the driven-lid problem. The time-average velocity distribution (limit of streamline patterns) is consistent with published results and is enclosed in an asymmetrical lemniscate.
In this manuscript, a proof for the age-old Riemann hypothesis is delivered, interpreting the Riemann Zeta function as an analytical signal, and using a signal analyzing affine model used in radar technology to match the warped Riemann Zeta function on the time domain with its conjugate pair on the warped frequency domain (a Dirichlet series), through a scale invariant composite Mellin transform. As an application of above, since the Navier Stokes system solution's Dirichlet transforms are also Dirichlet series, a minimal general solution of the 3d Navier Stokes differential equation for viscid incompressible flows is constructed through a fractional derivative Fourier transform of the found begin-solutions preserving the geometric properties of the 2d version assuming that the solution is an analytic solution that suffices the Laplace equation in cylindrical coordinates, which is the momentum equation for both the 2d and the 3d Navier Stokes systems of differential equations.
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