Incompressible Navier Stokes Equations: Finite Time Blowup for a Special Class of Initial Conditions, Wave Structure Subject to Special Auxiliary Equations | Chapter 14 | Recent Recent Advances in Mathematical Research and Computer Science Vol. 8

 The corresponding author of the current study [1] gave a rigorous proof of no finite time blowup of the 3D Incompressible Navier Stokes equations in R3/Z3. The primary goal of this research is to recollect the smooth solutions for the z-component momentum equation u z, assuming that the x and y component equations have vortex smooth solutions with constant vorticity as shown in [1], and to extend the analysis to non-constant vorticity. The goal is to show that Geometric Algebra can be used to solve all three momentum equations by combining any two of them and yielding either u x, u y, or u z as smooth solutions at the same time. It was demonstrated that there is no finite time blowup for 3D Navier Strokes equations for a constant vorticity in the z direction using the Gagliardo-Nirenberg and PrékopaLeindler inequalities, Debreu's theorem, and some auxiliary theorems proven in [1]. The second goal of the research is to show that by employing Hardy's inequality for the u z2 term in the Navier Stokes Equations, a PDE emerges that can be linked to auxiliary pde's to produce wave equations in each of the three major directions of flow. For the most general flow conditions, the current work is extended to all spatial directions of flow. Finally, the study's final goal is to prove that while the initial condition is smooth, the whole system of 3D Incompressible Navier Stokes equations without the above-mentioned coupling may contain non-smooth solutions at t=0, and future times may result in a non-blowup. This would have to be demonstrated in this case. At t=0, a non-smooth solution for the Navier Stokes equation has been demonstrated [14]. If these initial conditions are not met, it is hypothesised that higher order derivatives blow up in finite time but u z remains regular if u x,u y satisfy a non-constant z - vorticity for 3D vorticity. The Modified-Navier-Stokes equations are proposed with smooth solutions that appear to not have finite time singularities, and a specific time dependent vorticity is examined for possible finite-time velocity blowup.

Author(S) Details

Terry E. Moschandreou
Mathematics, Science Senior Division, Thames Valley District School Board, 1250 Dundas Street, London, N5W 5P2, Ontario, Canada.

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