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

III. O n the E ndpoint R egularity in O nsager’s C onjecture In order to obtain the solution previously shown as 3 ( , ) we let epsilon approach zero for solutions 3 ( , 3 , ) in the space ℑ( 3 , ) . In this space a ball � 3 ; � exists with > 0 . Here is defined as a measure of how close one is to the center of a given cell in the lattice of the 3-Torus. Due to the definition of the space ℑ( 3 , ) , the set {1, 1 , 1 2 } is linearly independent, implying that all the constants are zero in the solution except 3 and 4 associated with variables 3 , respectively. The constants ranging from = 1. .6 in the solution of the Euler Equation (I) appear in the solution and in particular as an argument of the Lambert W function and is expressed as the following linear sum in spatial and time variables, = 1 1 + 2 2 + 3 3 + 4 + 5 Note that the solution can be obtained by solving Eq.(I) when 3 � ∂ 3 ∂ 3 � 2 − � ∂ 3 ∂ 3 � 2 ≈ 3 � ∂ 3 ∂ 3 � 2 , that is for ≫ 100 3 . It is found that an exact solution is given by Maple 2023 software when this approximation is made for large enough density. It is also worthy to note that for lower densities when we retain both terms in the previous approximation, that for the locally Ḧ lder continuous functions in time , with Ḧ lder constant equal to exactly 1/3, the product term � ∂ 3 ∂ 3 � 2 ∂ 3 ∂ in Eq.(I) becomes independent of time and is only dependent on the spatial variables. The Onsager conjecture suggested the value = 1/3 for the case of the Euler equations but the conjecture was mainly considering only the Ḧ lder regularity with respect to the space variables. Here we consider a combination of velocity-time conditions ( , ) , which depend precisely on the Ḧ lder exponent. As outlined in the introduction, P. Isett’s proof shows that if < 1/3 (strictly less than) then conservation of energy fails. The works of Eyink[7,8] and Constantin, E, Titi [9] on the Onsager conjecture describe results in a Fourier setting and in a space called a Besov space (slightly larger than Ḧ lder spaces), respectively. A well known result is that if the velocity is a weak solution to the Euler equations such that, ∈ 3 (0, ; 3 ,∞ ( 3 ))⋂ (0, ; 2 ( 3 )) with > 1/3, (strictly greater than) then, ‖ ( )‖ = ‖ 0 ‖, for all ∈ [0, ]. This result is also true in Ḧ lder spaces which was the setting that L. Onsager stated his conjecture rather than Besov spaces. Ḧ lder continuous functions, as defined in Berselli [10] with a focus on space-time properties of functions with “homogeneous behavior”, that is the one of the Ḧ lder semi-norm [. ] (to be defined) and denote by ̇ the space of measurable functions such that this quantity is bounded. We say that, ∈ (0, ;̇ ( 3 )) , if there exists : [0, ] → ℝ + such that 1) | ( , ) − ( , )| ≤ ( )| − | , ∀ , ∈ 3 , for a.e. ∈ [0, ], 2) ∫ ( ) < ∞ 0 and ( ) = [ ( )] for almost all ∈ [0, ]. The space is endowed with the semi-norm ‖ ‖ (0, ;̇ � 3 �) ≔ �� ( ) 0 � 1/ Finally [ ] ≔ ≠ | ( ) − ( )| | − | In Berselli [10], (see Theorem 4.2 there) it is proven that if is a weak solution to the Euler equation (in usual form), such that ∈ 1/ (0, ;̇ ( 3 )) with ∈ � 1 3 , 1� (where ̇ ( 3 ) ⊂ ( 3 ) is the slightly smaller space defined through the norm ‖ ‖ = max ∈ 3 ���� | ( )| + [ ] , Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 54 Year 2023 © 2023 Global Journals ( ) I Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations