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

[ ] , ≔ ≠ | ( )− ( )| (| − |)| − | , with : ℝ + → ℝ + a non-decreasing function such that lim →0 + ( ) = 0. ) then conserves the energy. In our proof of the endpoint regularity of Onsager’s conjecture we are considering the Ḧ lder continuous functions in the space ( 3 ). ∬ 2 ∫ 32 3 =− ( , 3 , ) d 3 1 2 = ∭ (�⃗; ) 32 (0) = ∭ (�⃗; ) ( + (| 1 | 2 + | 2 | 2 + | 3 | 2 )) 2 The integrals are carried out over a cube ( ⃗; ) = [− , ] 3 , centered about ⃗ . For = 1/2 the scaled solutions and hence graphs are shown in Figure 3 and 4. It is seen that in either step in both figures that energy is conserved thereby proving the endpoint regularity in Onsager’s Conjecture. In Figure 3 and 4, the thicker part of curves hides the energy (E) at = 0, behind the solution curve. For > 0 there are two curves coinciding and the same is true for < 0 . The key empirical fact underlying the Onsager theory is the non-vanishing of turbulent energy dissipation in the zero-viscosity limit. The requirement for a non-vanishing limit of dissipation is that space-gradients of velocity must diverge. It is observed in experiment that when integrated over small balls or cubes in space the high-Reynolds limit of the the kinetic energy dissipation rate defines a positive measure with multifractal scaling. The solution for Euler’s equation given in this paper agrees with this fact that gradient of 3 with respect to spatial position 3 does in fact diverge. This is a short-distance/ultraviolet (UV) divergence in the language of quantum field-theory, or what Onsager himself termed a “violet catastrophe” [12]. Since the fluid equations of motion (I.1) contain diverging gradients, they become ill-defined in the limit. In order to develop a dynamical description which can be valid even as ν → 0, some regularization of this divergence must be introduced. Figure 3: Energy of PNS system for arbitrarily large and positive data There are two steps here. First we set 3 ( 3 , ) equal to the variable appearing in the initial condition when solving for the unknown constant 6 where > 0 , and recall that the initial condition for 3 is given as the sum of arbitrarily large data and sums of reciprocal degenerate Weierstrass P functions in the three directions for small . (By reciprocal we mean that unity is divided by the Weierstrass P functions with a bounded periodic result.) In the second step we solve for 3 ( , 3 , ) for arbitrarily large negative data < 0 . In both steps separately we keep 3 ∈ � 3 ; � and integrate the square associated with energy of solution 3 ( , 3 , ) , that is we will show that our solution satisfies conservation of energy, (for all times ∈ [0, )) . © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 55 Year 2023 ( ) I Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations