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

Theorem 1 + 2 + 4 = 0 if and only if Ξ 1 is continuous on the epsilon ball ; . Proof: Apply (V) to Ξ 1 IV. C onclusion Satisfying a divergence free vector field and periodic boundary conditions respectively with a general spatio- temporal forcing term which is smooth and spatially periodic, the existence of solutions which blowup in finite time for PNS can occur starting with the first derivative and higher with respect to time. P. Isett (2016) (see [13]) has shown that the conservation of energy fails for the 3D incompressible Euler flows with Ḧ lder regularity below 1/3. (Onsager’s second conjecture) The endpoint regularity in Onsager’s conjecture has been addressed, and it is found that conservation of energy occurs when the Ḧ lder regularity is exactly 1/3. 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 as L. Onsager proposed. Finally very recent developed new governing equations of fluid mechanics are proposed to have no finite time singularities. This is the focus of the ongoing work of the author to be presented in the near future. Finally future work to conclude the nature of flows in a non-epsilon or arbitrary small ball for the 3-Torus will be carried out. A cknowledgement I thank both reviewers for help with their insightful and valuable comments which were taken into consideration. B ibliography 1. C. Liu and Z Liu, New governing equations for fluid dynamics. AIP Advances 11, 115025 (2021), 115025 1-11 doi.org/10.1063/5.0074615 2. T. E. Moschandreou and K. C. Afas, Existence of incompressible vortex-class phenomena and variational formulation of Rayleigh Plesset cavitation dynamics, Applied Mechanics. (2021), 2(3):613-629. https://doi.org/10.3390/applmech2030035 3. T. E. Moschandreou, No Finite Time Blowup for 3D Incompressible Navier Stokes Equations via Scaling Invariance. Mathematics and Statistics 2021, 9(3), 386-393. 4. R, Poisson Equation for Pressure, www.thevisualroom.com/poisson for pressure.html poisson-equation-for- pressure 5. T. E. Moschandreou and K. Afas, Periodic Navier Stokes Equations for a 3D Incompressible Fluid with Liutex Vortex Identification Method, Intech Open, 2023, doi: 10.5772/intechopen.110206, 1-22. 6. J-Y Chemin, Isabelle Gallagher and Ping Zhang, Some remarks about the possible blow-up for the Navier- Stokes equations, Communications in Partial Differential Equations, Volume 44, 2019-Issue 12, Pages 1387- 1405. 7. G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Phys. D, 78(3-4): 222-240, 1994. 8. G. L. Eyink, Besov spaces and the multifractal hypothesis. J. Statist. Phys., 78(1-2):353-375, 1995. Papers dedicated to the memory of Lars Onsager. 9. P. Constantin, W. E, and E. S. Titi. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm. Math. Phys., 165(1):207-209, 1994. 10. L. C. Berselli, Energy conservation for weak solutions of incompressible fluid equations: the Ḧ lder case and connections with Onsager's conjecture, Journal of Differential Equations, 2023, 368:350, DOI: 10.1016/ j.jde.2023.06.00 11. Lars Onsager, “The distribution of energy in turbulence [abstract],” in Minutes of the Meeting of the Metropolitan Section held at Columbia Physical Review, Vol. 68 (American Physical Society, College Park, MD, 1945) pp. 286–286. 12. Yu B, Rumer, A I Fet, Theory of Unitary Symmetry (In Russian-Published in Moscow), 1970-01-01. 13. P. Isett, A proof of Onsager’s conjecture. Ann. Of Math. (2), 188(3): 871-963, 2018. © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 59 Year 2023 ( ) I Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations