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

: ( , ) = 1 6 : $ S + S S + : S −2 $: − 2 $ S S − 2 $ S S − 2 $ : S − 2 S : − 2 S : S + e ¸ ƒ] ¸ ¹ ¸ º −exp $ + S : + e ¸ ƒ ] ¸ ¹ ¸ º where is the Lambert W function. We replaced by – +large shifts and found that the solution for 3 for large (example = 600 ), the solution is locally Ḧ lder continuous with Ḧ lder constant 1/3 at arbitrary large values of . (specifically in plot shown, = 10000) . In this analysis there is no restriction on the largeness of the data, thereby proving that the solution is admissible for arbitrary large data. The solution as seen in Figure 2 is not smooth from the first and higher derivatives in of . This is discussed further in the chapter as it pertains to the Onsager regularity problem particularly the endpoint regularity problem. See the following Figure 2, where the dashed line is the solution for 3 and the non-dashed line is the Ḧ lder solution, given for example as (-0.52+ �10000 − ) 1 3 � Figure 2: Locally Ḧ lder continuous functions. 3 = −0.052 4 = 0.05 © 2023 Global Journals Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 53 Year 2023 ( ) I Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations