On the regularity of the Navier–Stokes equation in a thin periodic domain (original) (raw)
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Regularity for the stationary Navier-Stokes equations in bounded domain
Archive for Rational Mechanics and Analysis, 1994
Let u, p be a weak solution of the stationary Navier-Stokes equations in a bounded domain f~ _~ IR N, 5 ~< N < oo. If u, p satisfy the additional conditions (B) fVuV(ujdx<__f(~+p)u.Vydx+ff.uTdx V7 ~ C~(a), y __> 0, they become regular. Moreover, it is proved that every weak solution u, p satisfying (A) with q = oe is regular. The existence of such solutions for N = 5 has been established in a former paper [3].
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arXiv: Analysis of PDEs, 2021
A rigorous proof of no finite time blowup of the 3D Incompressible Navier Stokes equations in R 3 /T 3 is hereby shown as well as results on the velocity-pressure distribution. 1 Introduction The global regularity of the Navier-Stokes equations remains to be an outstanding unsolved problem in fluid mechanics. The Clay Institute is offering a significant prize for those who are successful in solving either one of four proposed problems, that is either a periodic or non-periodic regular or finite time blowup problem for the full 3D Navier Stokes equations.[1] The motivation behind the author's present contribution and work was to pursue the periodic Navier Stokes equations problem and it was shown by corresponding author [2](using Geometric Algebra) and then the authors in [3] (also see references contained therein) who confirmed in an applied setting that there is no finite time blowup for the special case of the theoretical class of solutions G being multiplicative between spatial variable y and coordinates (x, y, t). (see[2] and Eq.(13,15) of [3]) In addition the plots shown in [3] (Figure 1 there) were obtained for the special case of F (x, y) = 1 in [3](Eq13); Saddle surface generation and saddle orbits were obtained where the initial condition was a min-max type function with a min-max point at the center of a typical cube in the periodic Lattice of R 3 /T 3. This was plotted in [3] in Fig 1 (a,c) there. These saddle orbits indicate that there are instabilities leading to turbulence or possibly even STC. [4] Also the form of the solution obtained showed no finite time blowup. However the problem remained unsolved as to the general class of the form of solutions G due to the arbitrariness of the function itself. It is the purpose of the following article to provide a rigorous mathematical proof that G cannot be a blowup itself with respect to t except at a finite number of points in any given bounded subset of R 3 /T 3 if and only if in the initial data the energy is infinite. It is hypothesized that energy "cascades" from large-scale structures to smaller scale structures by an inertial and inviscid mechanism. This process does continue, and there is the creation of smaller and smaller structures which produce a hierarchy of eddies. It is inevitable that this process produces structures that are so small that molecular diffusion becomes important and viscous dissipation of energy eventually takes place. The scale at which this happens is the Kolmogorov length scale. [5] At these fine scales there is potential finite time blowup. This is also addressed above and beyond what was done in [2] and [3]. The issue of stability of solutions to the forced Navier-Stokes and damped Euler systems in periodic boxes has been studied in [6].There it is shown that for large, but fixed, Grashoff (Reynolds) number the turbulent behavior of all Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equations, in a periodic box, is suppressed, when viewed in the right frame of reference, by large enough average flow of the initial data. In the present work there exists such a frame of reference in the z direction where the average flow of the initial data is arbitrarily large. For the form of solution G which is multiplicative, a unique time-periodic solution exists when the average flow of the initial data is large or a fast oscillating forcing term occurs, with no blowup. Here there is a suppression of turbulence. However for a general form of G, it is shown that there exists infinite type singularities on sets of measure zero corresponding to the centres of periodic cells of the