Talk:Navier–Stokes existence and smoothness

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On the 2d case

About"Partial results

The Navier–Stokes problem in two dimension has already been solved positively since the 60's: there exist smooth and globally defined solutions.[2]"

I scanned the book [2]

O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows", 2nd edition, Gordon and Breach, 1969.

and did not find this result. Anyone can specify the pages of this result. Thank you!

The only paper I know about the 2d case is one of Weigant, VA and Kazhikhov, AV On the existence of global solutions to two-dimensional Navier-Stokes equations of compressible viscous fluids

Siberian Math. J,36 1108--1141,1995

zhangvszhang 18:22 27th July 2009(UTC)

Spherical points

This would easily be solved by counting all the points of the sphere from which its circle has been defined according to three static singular points defining the perfect curve of that circle.

The (non-)compressible fluid-theories could then be computed if those points have been expressed.

Conjecture: there are three successive points on the curve of the geometric circle, what is their mathematical expression?

zionion blogpot com 102.176.162.48 (talk) 14:02, 18 September 2021 (UTC)[reply]

Importance of Tao's work

In response to LouScheffer (talk · contribs) and recent edits. The book "The Navier-Stokes problem in the 21st century" by Pierre Lemarié-Rieusset is a very qualified and useful source. Tao's work takes up about one page total (out of 700), and is clearly contextualized as being important for showing that the particular nonlinear structure is important, and that function space bounds and energy estimates are insufficient for regularity. That information, i.e. that certain kinds of approaches will not work, is certainly valuable for some practitioners, but there doesn't seem to be any reason to think it compares to many more significant works on Navier-Stokes (Hopf, Serrin, Caffarelli−Kohn−Nirenberg, Buckmaster−Vicol, Sverak etc). More to the point, I have not seen any source which suggests that Tao's paper is very important and not just an interesting paper. Gumshoe2 (talk) 02:23, 6 December 2021 (UTC)[reply]

Although citations are an imperfect metric, in some form they indicate what other mathematicians believe important. In 5 years, Tao's work has accumulated about half the cites (173) that the "classic" papers in the field (those by Scheffer (no relation)) and Shnirelman (300-400 cites each) that are cited in the Clay official problem description. Looking at these citing documents briefly, it looks like most references to Tao's paper are about the Navier-Stokes problem, as opposed to applying his techniques to other problems (another typical source of cites). On the other hand, another review of the current situation (by Robinson) also only mentions Tao's work in passing. My preference is to leave it in, if for no other reason than to indicate the field is not dead (the other cites are from 1934 and 1969). LouScheffer (talk) 00:53, 7 December 2021 (UTC)[reply]

If such citations are the metric, then there's at least 100 articles that warrant mention. But wikipedia is not the place for an up-to-date academic review, so I think there's no need to resort to such imperfect criteria anyway. There's no reason to mention Tao's work when there are several high-quality papers from the last 20 years whose importance is widely acknowledged in the literature. Gumshoe2 (talk) 04:05, 7 December 2021 (UTC)[reply]

I'll write a version of the section, then maybe we can discuss specifics in a more useful way. Gumshoe2 (talk) 04:24, 7 December 2021 (UTC)[reply]

A new solution proposed by Hua-Shu Dou

https://doi.org/10.3390/e24030339 Entropy | Free Full-Text | No Existence and Smoothness of Solution of the Navier-Stokes Equation (mdpi.com) — Preceding unsigned comment added by Un11imig (talk • contribs) 12:22, 30 November 2023 (UTC)[reply]

on the equations of the 2D case

There are 4 equations in the text. Eq1 and Eq2 are the components x and y of the NS equations, Eq3 is the continuity equation... and what is the Eq 4?? JosuAguirrebeitia (talk) 14:01, 16 May 2024 (UTC)[reply]