Global Journal of Researches in Engineering, I: Numerical Methods, Volume 23 Issue 1

3 1 − 3 2 = 1 3 − 2 3 − ( 1 + 2 ) Multiplying both sides of this equation by 12 + 22 + 32 = 2 and letting approach zero gives, 2 2 3 − 2 1 3 − 2 3 ( 2 − 1 ) = 0 so 3 = − 3 ( 2 − 1 ) 1 − 2 Introduce the following shifts, ( 1 − 1 , 2 − 2 , 3 − 3 ) ranging over all the centers of cells in the expanding lattice, and we set: 3 − 3 = ( 1 − 1 ) − ( 2 − 2 ) Cancellation occurs between 3 and 1 − 2 terms leaving us with, 3 = −( 2 − 1 ) Here we see clearly that we have an isotropic condition on the finite time blowup of the velocities. If the first derivatives and higher of the third component of velocity blows up then so do the corresponding derivatives of 1 and 2 respectively. The third component of vorticity is calculated as twice the third component of angular velocity, �2 ( ⃗ × ⃗) 3 1 2 + 2 2 + 3 2 � = 2 − 1 2 + 2 1 1 2 + 2 2 + 3 2 3 = 1 2 − 2 1 = 2 − 1 2 + 2 1 1 2 + 2 2 + 3 2 Substitute 2 = −2 1 1 into previous PDE, 1 2 + 2 1 + 2 1 1 1 = 2 (−2 12 − 2 ) 2 1 where the sphere of radius is introduced, at the center of each cell of the lattice. Solving PDE, gives, for arbitrary function 1 , 1 = 1 −1− − ( 1 ) 2 + 2 2 1 � −ln ( 1 ) 2 + 2 , 3 , � − 12 2 − ln ( 1 ) 2 4 2 A particular maximal class of solutions is obtained by setting, which is in the required form of the general function and where is an arbitrary function to be determined. Back substituting 1 into the solution for 1 , gives, 1 = −2 2 2 − 12 − 22 2 ( 3 , ) Here 1 is Gaussian. Substituting 1 into 2 = −2 1 1 , gives, 2 = −2 1 −2 2 2 − 12 − 22 2 ( 3 , ) which is double sided Gaussian. Near the center of each cell of the lattice, the solutions are non singular in spatial variables. However ( 3 , ) , is yet to be determined and related to 3 solution since 3 = −( 2 − 1 ) . Now the general form was reduced to a particular maximal class of solutions since as 1 → 0 , 1 → 0 , which is inadmissible according to a theorem of J.Y Chemin [6] (“Some remarks about the possible blowup for the Navier Stokes equations”) If there is finite time blowup then it is impossible for one component of velocity to approach zero Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 50 Year 2023 © 2023 Global Journals ( ) I Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations 1 = ln ( 1 )−2 2 � ln( 1 ) 2� − 2 � � 2 ( 3 , )