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

where the differential has been transformed to spherical coordinates in 3D. Substituting this into the main statement of the theorem, we obtain, Δ 3 = 3 4 3 4 5 15 1 2 � 2 3 1 2 + 2 3 2 2 + 2 3 3 2 � + ( 3 ) = 2 10 Δ 3 + + ( 3 ) Finally we take the limit, lim →0 10 2 Δ 3 = lim →0 [Δ 3 + ( )] = Δ 3 In Eq.(II) the Laplacian is differentiated wrt to 3 . Using Fet theory, where we integrate Δ 3 on an epsilon ball centered at zero and generalized to the center of any cell center of the lattice of the 3-Torus, we obtain the following PDE for large density: 1/6 �3 3 ∂ 2 3 ∂ 3 2 + 3 � ∂ 3 ∂ 3 � 2 + ∂ 2 3 ∂ 3 ∂ 1 + ∂ 2 3 ∂ 3 ∂ 2 � ∂ 3 ∂ + ( − 1) 3 3 = 0 (III) with solution: 3 = (1/3 − 4 1 − 4 2 + (−(6 1 1 + 6 2 2 + 6 3 3 + 6 4 + 6 5 ) 3 42 5 + 6 3 42 6 − 18( 1 1 + 2 + 3 + 4 + 5 ) 2 3 4 + 1 2 42 + 2 1 2 42 + 22 42 ) 1/2 �/( 3 4 ) 3 = 0.052, 4 = 0.05 for = 1000, = 10000 , the following result follows in Figure 5. Figure 5: Locally Ḧ lder continuous functions in where , i=1…6 are constants. Using the same initial condition in terms of as in the first part in Eq(I), we can determine 6 and on the space ℑ( 3 , ), all constants are zero and the only constants that survive are 3 , 4 and arbitrarily large . When the two constants are as follows, © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 57 Year 2023 ( ) I Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations