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

Figure 4: Energy of PNS system for arbitrarily large and negative data In the book “Theory of unitary symmetry” by Rumer and Fet[12], the Laplacian is defined an integration over a 3D-ball, in particular an epsilon ball. Therefore Eq(I) becomes: (II) � ∂ 3 3 ∂ 3 3 + ∂ 3 3 ∂ 3 ∂ 2 2 + ∂ 3 3 ∂ 3 ∂ 1 2 � ( − 1) + 1/6 �3 3 ∂ 2 3 ∂ 3 2 + 3 � ∂ 3 ∂ 3 � 2 − � ∂ 3 ∂ 3 � 2 + ∂ 2 3 ∂ 3 ∂ 1 + ∂ 2 3 ∂ 3 ∂ 2 � ∂ 3 ∂ = 0 Equation (II) is integrated over an epsilon ball so we solve Eq.(II) in a neighborhood of epsilon =0 that is near the center of each cell of the lattice in the space ℑ( 3 , ) . So we integrate Eq. (II) over an epsilon ball first and then take limit. We use the Fet theory on writing the Laplacian as an integral over an epsilon ball. Here we know that there is an operator Δ 3 = 3 4 3 ∫ 3 ( ) − 3 (0) such that in the limit as epsilon approaches zero, 10 2 Δ 3 = Δ 3 . Integral is over epsilon ball centered at ⃗ = ( , , ). Proof: We take the Taylor expansion around 0 (or center ⃗ to second order, which gives terms proportional to 1 , 1 2 and 12 , however due to the symmetry of the 1 , 1 2 related terms these integrate to zero over the ball and thus we have that, Δ 3 = 3 4 3 � 1 2 2 3 12 ∫ 12 + 1 2 2 3 22 ∫ 22 + 1 2 2 3 32 ∫ 32 � + ( 3 ) where all derivatives are evaluated at the center ⃗ . The integrals all give the same value, ∫ 12 = 1 3 ∫ 12 + 22 + 32 = 4 3 ∫ 4 0 = 4 5 15 The viscous solution when is non-zero is subject to a rewriting of Eq (I) and to use this result first we integrate Eq.(I) over an − ball, centered at each center of cells of the lattice of 3-Torus. Next using the divergence theorem for the term of Eq(I), that is specifically the expression 3 � ∂ 2 3 ∂ 3 2 + ∂ 2 3 ∂ 2 2 + ∂ 2 3 ∂ 1 2 � , gives |∇ 3 | 2 ∈ ( ) ∫ = 0 where the surface integral is zero and since we are integrating a positive expression on an epsilon ball, at epsilon =0 the integral is zero. Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 56 Year 2023 © 2023 Global Journals ( ) I Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations