Author: Terry E. Moschandreou, Teacher: Intermediate Science and Mathematics, TVDSB London Ontario, Canada. e-mail: [email protected]...
Author: Terry E. Moschandreou, Teacher: Intermediate Science and Mathematics, TVDSB London Ontario, Canada. e-mail: [email protected]
Abstract- This article gives a general model using specific periodic special functions, that is degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The component forcing terms consist of a function as part of it's expression that is arbitrarily small in an ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result a significant simplification occurs with a (for all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product . On each of these subspaces is continuous and there exists a linear independent subspace associated with the argument of the function. Here the 3-Torus is built up from each compact segment of length on each of the axes on the 3 principal directions , and . The form of the scaled velocities for non zero scaled is related to the definition of the function such that where depends on and proportional to for infinite time . The ratio is equal to 1 making the limit finite and well defined.Considering - balls where the function set equal to where is such that the forcing is singular at every distance of centres of cubes each containing an -ball. At the centre of the balls the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing are to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time for first and higher derivatives to thereby indicating that there is no finite time blowup. So studying problems with finite time singularity are crucial in developing a theory for no finite time singularities for the Navier-Stokes equations. As observed by J. Heywood in , in principle "it is easy to construct a singular solution of the NS equations that is driven by a singular force. One simply constructs a solenoidal vector field that begins smoothly and evolves to develop a singularity, and then defines the force to be the residual." Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references it has been shown that mathematical tools can be applied to allow us to study asymptotic properties of solutions.
Research articles
- T. E. Moschandreou, Global Journal of Researches in Engineering: I Numerical Methods Volume 23 Issue 1 Version 1.0 Year 2023 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Online ISSN: 2249-4596 and Print ISSN: 0975-5861 "Exploring Finite-Time Singularities and Onsager's Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations, https://globaljournals.org/GJRE_Volume23/4-Exploring-Finite-Time-Singularities.pdf
- J. G. Heywood, "Remarks on the possible global regularity of solutions of the three-dimensional Navier-Stokes equations", in: Progress in theoretical and computational fluid mechanics, Paseky 1993, Pitman Res. Notes Math. Ser., 308 (1994) 1-32.
- T.E. Moschandreou "From Hölder continuous solutions to no-finite time singular solutions of the 3D Incompressible Navier Stokes equation",SSRN Number of pages: 33 Posted: 12 Jul 2024, https://papers.ssrn.com/sol3/results.cfm
Short Biography
Dr. Terry E. Moschandreou is a professor in mathematics at the University of Western Ontario in the School of Mathematical and Statistical Sciences where he has taught for several years. He received his PhD degree in Applied Mathematics from the University of Western Ontario in 1996. The greater part of his professional life has been spent at the University of Western Ontario(teaching 8 years mathematics and fluid dynamics courses) and Fanshawe College in London (where he has taught physics courses), Ontario, Canada. Dr. Moschandreou is also currently working for Thames Valley District School Board in London Ontario Canada, where he teaches students Mathematics and Science. For a short period, he worked at the National Technical University of Athens, Greece. Dr. Moschandreou is the author of several research articles in hemodynamics and oxygen transport in the microcirculation, general fluid dynamics, and theory of differential equations. Also, he has contributed to the field of finite element modeling of the upper airways in sleep apnea as well as surgical brain deformation modeling. More recently, he has been working with the partial differential equations of multiphase flow and level set methods as used in fluid dynamics. Finally he proposes to have a solution to the Millennium Prize Problem(main findings submitted in the period 2018-2024) and recently for the existence of solutions to the Periodic Navier Stokes equations on the 3-Torus.
