Analytical Solution of the Unsimplified Incompressible Navier-Stokes Equations for Arbitrary Cauchy Data by Second-Curl Poisson Self-Similarity and Fixed-Point Triquartic Decoupling Amid Modified Cole-Hopf Transformation

A novel analytical solution to the incompressible Navier-Stokes equations for arbitrary flow geometries and Cauchy data is introduced to establish a mathematical theory of turbulence through a direct attack on the Millennium Problem from first principles. All nine advective terms are left fully nonlinear. The velocity field is first separated into rotational and irrotational parts in a Helmholtz decomposition. The irrotational velocity is found by assembling standard vector calculus identities in potential flow while Leray projections are used to carefully handle irrotational and rotational velocity Cauchy data. The rotational velocity derivation is begun by taking the curl of the momentum equations twice, effectively replacing pressure with a Poisson integral of Cauchy data and the forcing terms with incompressibility preventing vorticity entanglement. The resultant pseudo-depressurized momentum equations are addressed by a heavily generalized integral transformation similar to the Cole-Hopf transformation, which captures all nine nonlinear terms in a coupled yet solvable triquadratic algebraic system in the velocity components whose coefficients satisfy heat equations. When said system is solved for the rotational velocity in terms of quartic polynomial roots, the root cause of turbulence is identified as multiplicity collapse of quartic root pairs. Cauchy data is reconciled between the heat equations and velocity components using the method of characteristics. The irrotational and rotational velocities are substituted into the Helmholtz decomposition and its Leray projections, finally resulting in velocity closure. Pressure is recovered when the velocity field is substituted into the standard pressure-Poisson equation. Existence, uniqueness, differentiability class, and kinetic boundedness are analyzed in the Millennium Problem context. Practical implementation is expected to shift CFD paradigms since HPC jobs requiring days would require mere minutes.