From Hölder Continuous Solutions of 3D Incompressible Navier-Stokes Equations to No-Finite Time Blowup on [ 0 , ∞ ]

Abstract

This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The y i component forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v 3 ( v i for all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ = J x , t × J y , t × J z , t . On each of these subspaces v i , i = 1 , 2 , 3 is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2 R on each of the axes on the 3 principal directions x , y , and z . The form of the scaled velocities for non zero scaled δ is related to the definition of the W function such that e − W ( ξ ) = W ( ξ ) ξ where ξ depends on t and proportional to δ → 0 for infinite time t . The ratio W ξ is equal to 1, making the limit δ → 0 finite and well defined. Considering r balls where the function F = ( x − a i ) 2 + ( y − b i ) 2 + ( z − c i ) 2 − η set equal to − 1 e + r where r > 0 . is such that the forcing is singular at every distance r of centres of cubes each containing an r -ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time t i ∗ for first and higher derivatives to t = ∞ thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.