Incompressible Navier Stokes Equations: Finite Time Blowup for a Special Class of Initial Conditions, Wave Structure Subject to Special Auxiliary Equations

Abstract

A rigorous proof of no finite time blowup of the 3D Incompressible Navier Stokes equations in \(\mathbb{R}\)3 / \(\mathbb{Z}\)3 has been shown by corresponding author of the present work [1]. The first study objective is to recall the smooth solutions for the z-component momentum equation \({u}\)z assuming the \({x}\) and \({y}\) component equations have vortex smooth solutions with constant vorticity in [1], and to extend the analysis to non-constant vorticity. The idea is to show that Geometric Algebra can be applied to all three momentum equations by adding any two of the three equations and thus combinatorially producing either \({u}\)x, \({u}\)y or \({u}\)z as smooth solutions at a time. It was shown that using the Gagliardo-Nirenberg and PrékopaLeindler inequalities together with Debreu’s theorem and some auxiliary theorems proven in [1] that there is no finite time blowup for 3D Navier Strokes equations for a constant vorticity in the \({z}\) direction. The second study objective is to show that using Hardy’s inequality for \({u^2_z}\) term in the Navier Stokes Equations that a resulting PDE emerges which can be coupled to auxiliary pde’s which give us wave equations in each of the three principal directions of flow. The present work is extended to all spatial directions of flow for the most general flow conditions. Finally the last objective of this study is to show that the full system of 3D Incompressible Navier Stokes equations without the above mentioned coupling may consist of non-smooth solutions at \({t}\) = 0 when the initial condition is smooth and subsequent times may lead to a non-blowup. This would have to be proven here. It has been demonstrated [14] that a non-smooth solution exists for the Navier Stokes equation at \({t}\) = 0. In particular if \({u}\)x, \({u}\)y satisfy a non-constant z- vorticity for 3D vorticity \({^\rightarrow_\varpi}\), then if these initial conditions are not fulfilled, it is proposed, that higher order derivatives blowup in finite time but \({u}\)z remains regular. A specific time dependent vorticity is considered for possible finite-time velocity blowup as well as the Modified-Navier-Stokes equations are proposed with smooth solutions appearing to not possess finite time singularities.