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

too fast. So we will show further that 3 is not smooth. Thus 1 , 2 blow up at the center of cells of lattice if we can conclude that ( ) = lim 3 →0 ( 3 , ) has finite time blowup. Again recall that 3 = −( 2 − 1 ) , where in ℑ( 3 , ) 3 = −(−2 1 1 − 1 ) = (2 1 + 1) 1 ≠ 0 at the centers of cells of ℝ 3 ℤ 3 � since 2 1 + 1 ≠ 0 there and 1 is also not zero there. Define ( ) = (0, ) = ∫ ( ) , where (0, ) = lim 3 →0 ( 3 , ) and ( ) is the solution associated with 3 in the − ball as → 0. − 2 + 2 = �−2 1 1 1 2 1 3 − 2� ( ) The pressure gradient is oscillatory, that is it is written as a product of reciprocals of degenerate Weierstrass P functions added to a constant as is the forcing. Finally the surface S given by 3 = ±( 1 2 + 1 + ) , plotted in ℝ 3 is such that by shifting and sweeping through 1 values and heights along 3 axis we can find intersection points between surface S and points or centers of cells ( , , ). Equation (1) together with = 0 gives the following PDE which has viscosity in it and where in Eq.(8.21) we have condensed the PDE by collecting the terms that contribute to the Laplacian. Also the divergence theorem is applied to the volume integral of Eq(I) for the term with Laplacian multiplied by 3 . The calculations are taking into account that density is large, (fluids like water and higher densities.) � ∂ 3 3 ∂ 3 3 + ∂ 3 3 ∂ 3 ∂ 2 2 + ∂ 3 3 ∂ 3 ∂ 1 2 � + 2/3( 3 � ∂ 2 3 ∂ 3 2 + ∂ 2 3 ∂ 2 2 + ∂ 2 3 ∂ 1 2 � + 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 (I) Finally the solutions for 1 , 2 satisfy the 1 , 2 momentum equations for PNS when − 1 + 1 = � 1 1 1 2 1 3 + 1� ( ) , for > 0 arbitrarily small and where 1 , 2 are the forcing terms associated with the 1 , 2 momentum equations. It remains to prove that the derivatives of ( ) blowup in finite time. © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 51 Year 2023 ( ) I Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations In Equation (I) it is understood that in the top line with two expressions appearing there, that these both include a product of ( − 1 ) � ∂ 3 ∂ � 2 which has been set to a constant. Solving this implies that 3 is a linear function in . As → 1 , 3 approaches infinity from the right of a potential blowup point = 0 . See Figure (1c) below,