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

II. M aterials and M ethods Consider the incompressible 3D Navier Stokes equations defined on the three-Torus 3 = ℝ 3 ℤ 3 � . The periodic Navier Stokes system is, (P ) � ∂ − Δ + ⋅ ∇ = −∇ + div = 0 =0 = 0 . where = ( , , , ) is velocity, = ( , , , ) is pressure and = ( , , , ) is forcing vector. Here = � , , �, where , , and denote respectively the , and components of velocity. Introducing Poisson's Equation (see [2], [3] and [5]), the second derivative is set equal to the second derivative obtained in the 1 expression further below, as part of , and = −2 ∇ 2 − � ∂ ∂ � 2 + 1 ∂ ∂ � ∂ ∂ + ∂ ∂ � − ∂ 2 ∂ ∂ − ∂ 2 ∂ ∂ + � ∂ ∂ � 2 + 2 ∂ ∂ ∂ ∂ + � ∂ ∂ � 2 where the last three terms on rhs can be shown to be equal to −( + � . [4] Along with Equations below the continuity equation in Cartesian coordinates, is ∇ = 0 . The one parameter group of transformations on a critical space of PNS is given as, Let = ∗ ; = ∗ ; = ∗ ; = ∗ 2 = ∗ ; = ∗ ; = ∗ ; = ∗ 2 , = −1 ∗ ; = −1 ∗ ; = −1 ∗ , = −2 ∗ . Furthermore the right hand side of the one parameter group of transformations are next mapped to variable terms, (note that and are not assumed to be arbitrarily small, they can be at most order one), ∗ = 1 , ∗ = 1 2 , ∗ = , ∗ = 2 , = 1,2,3. The double transformation is used for notational clarity. Note that the original Navier Stokes equations are preserved and simply rearranged in the following forms and Navier Stokes Equations become, ( ) = ( ) 1 + ( ) 2 + ( ) 3 + ( ) 4 = 0 where ( ) 2 = 3 6 � ∂ 3 ∂ 3 � ∂ 3 ∂ + ( 3 ) 2 6 ∂ 2 3 ∂ 3 ∂ + 2 � ∂ 1 ∂ � 3 ∂ 3 ∂ 1 + 2 � ∂ 2 ∂ � 3 ∂ 3 ∂ 2 + 2 � ∂ 3 ∂ � 3 ∂ 3 ∂ 3 6 ( ) 3 = 1 3 × ��  � 1 32 ∇ 1 2 + 1 ⃗ 1 3 ∂ ∂ 3 � ⋅ �⃗ − �  Ω  ∥∥∥∂ 3 ∂ �⃗ ⋅ � �⃗ ⊗∇ 3 � ∥∥∥ ∥ �⃗ ∥ � possible to maintain when the initial conditions and boundary conditions are posed properly for (PNS) ([5]). The endpoint regularity in Onsager’s conjecture is addressed, and it is found that conservation of energy occurs when the Ḧ lder regularity is exactly 1/3. Finally it is proposed that the periodic Liutex new equations[1] (The new equations referred to previously) do not exhibit finite time blow up. This is the focus of the ongoing work of the author to be presented in the near future. © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 47 Year 2023 ( ) I Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations ( ) 1 = 1 6 ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ ( −1 − 1) � ∂ 3 ∂ � 2 + � ∂ 3 ∂ � � ∂ 2 3 ∂ 1 2 + ∂ 2 3 ∂ 2 2 + ∂ 2 3 ∂ 3 2 � + ( −1 − 1) � ∂ 3 ∂ � ∂ ∂ 3 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ (1 − δ −1 )