# Looking for an interesting result on the Navier-Stokes equations

I am in my second year of master in Mathematics and one of my courses consists of a reading of Navier-Stokes Equations by Roger Temam. We have proven the existence for the weak Stokes and Navier-Stokes (defined on a bounded domain) in a stationary and non-stationary regime using a Galerkin method.

Now I have to do a 2 hours presentation on the subject of my choice (still related to hydrodynamics) and I would like to know if one of you had an idea of something I could talk about. I looked on the internet for an article that could match my critera but all I found was way too advance or too long for what I am looking for. I am looking for a paper that could be presented in two hours, that is accessible to a master student (who has read Temam's book) and that is an important result in the modern understanding of the Navier-Stokes equations (a paper by Leray, Ladyzhenskaya, Fujita-Kato, Lions, ... for example). Could one of you, who has a better knowledge of what has been done in hydrodynamics, give me some advice on this subject?

I first asked the question on Stackexchange Mathematics but I don't know which site is best for what I'm looking for. You may find here my question on Stackexchange Mathematics.

• No one should listen to a 2-hr presentation, so this sounds like a poor assignment. I doubt the other students can benefit from listening to student presentations of longer than 15-20 minutes. It sounds like your professor is shirking their teaching duties. May 8 at 8:05
• This Quanta Magazine article could help: Mathematicians Coax Fluid Equations Into Nonphysical Solutions. May 8 at 17:14
• @MattF. On the other hand, a student can benefit a lot from (preparing and) giving a two-hour-long presentation (I speak from experience here). May 8 at 17:27

Beale Kato Majda criteria: Beale, J. Thomas, Tosio Kato, and Andrew Majda. "Remarks on the breakdown of smooth solutions for the 3-D Euler equations." Communications in Mathematical Physics 94.1 (1984): 61-66.

Provides a necessary condition for blowup, used frequently in modern research on NS (1500+ citations) and is only 6 pages long (!).

What about a talk on uniqueness theorems for classical solutions? This problem was settled by Emanuele Foà (1929) and David Dolidze (1954), who succeded independently in proving uniqueness for bounded domains (Serrin gives a brief description of their work in [1], p. 251, footnote 1 and [2] p. 271): their proof is nothing more than a clever use of Grönwall's inequality. Later Dario Graffi succeded in extendig their theorem to unbounded domains, under various supplementary conditions. The Wikipedia entries linked above and the main references listed below could give you a basic bibliography: finally, regarding Graffi's contribution, the main reference is [1] which is in Italian, but the topic is also dealt extensively in the monograph [2] (precisely chapter 4), written in English, and which also deals with the results for bounded domains.

Edit

In his comment Calvin Khor has pointed out that a digitized version of [2] is available from the borrowing service of the Internet Archive: I have embedded the link he found into reference [2] and gratefully thank him for the notification.

References

[1] Dario Graffi, "Sul teorema di unicità nella dinamica dei fluidi" [On the uniqueness theorem in fluid mechanics], Annali di Matematica Pura ed Applicata, IV Serie (in Italian), 50: 379–387, (1960), DOI: 10.1007/BF02414524, MR0122198, Zbl 0102.41103.

[2] Dario Graffi, Nonlinear partial differential equations in physical problems, Research Notes in Mathematics, vol. 42, Boston–London–Melbourne: Pitman Advanced Publishing Program, pp. IV+105, ISBN 978-0-273-08474-7, MR0580946, Zbl 0453.35001.

[3] James Serrin, "Mathematical principles of classical fluid mechanics", in Flügge, Siegfried; Truesdell, Clifford A. (eds.), Fluid Dynamics I/Strömungsmechanik I, Handbuch der Physik (Encyclopedia of Physics), vol. VIII/1, Berlin–Heidelberg–New York: Springer-Verlag, pp. 125–263, (1959), DOI: 10.1007/978-3-642-45914-6_2, MR0108116, Zbl 0102.40503.

[4] James Serrin, "On the Uniqueness of Compressible Fluid Motions", Archive for Rational Mechanics and Analysis, 3 (1): pp. 271–288, (1959b), DOI: 10.1007/BF00284180, MR0106646, Zbl 0089.19103.

You could consider talking about sufficient conditions for energy conservation. There are two I know that are 'easy'. Firstly there is Shinbrot's [1] condition (Leray-Hopf solution $$u\in L^p_t L^q_x$$ for $$\frac2p+\frac2q=1$$ conserves energy). The paper is written well (Shinbrot in general is a good writer), and a recent short paper [2] extended this result with elementary techniques.

Secondly, in the ideal case $$\nu=0$$, there is the 'rigid' part of the Onsager conjecture, which is a theorem of Constantin, E, and Titi [3]: it asserts that even distributional solutions conserve energy if they belong to $$L^3 B^{\alpha}_{3,\infty}$$ for some $$\alpha>1/3$$ (In particular if they belong to $$L^3 C^{0,\alpha}$$.)

The above mentioned papers are all quite short. The 'flexible' part of Onsager's conjecture was recently [4] solved (more or less, but this is still actively researched) but IMO would be too technical and need more than 2 hours. Still, you might want to consult e.g. the survey [5].

[1] Shinbrot, Marvin, The energy equation for the Navier-Stokes system, SIAM J. Math. Anal. 5(1974), 948-954 (1975). ZBL0316.76011.

[2] Beirão da Veiga, Hugo; Yang, Jiaqi, On the Shinbrot’s criteria for energy equality to Newtonian fluids: a simplified proof, and an extension of the range of application, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 196, Article ID 111809, 4 p. (2020). ZBL1437.35525.

[3] Constantin, Peter; E, Weinan; Titi, Edriss S., Onsager’s conjecture on the energy conservation for solutions of Euler’s equation, Commun. Math. Phys. 165, No. 1, 207-209 (1994). ZBL0818.35085.

[4] Isett, Philip, A proof of Onsager’s conjecture, Ann. Math. (2) 188, No. 3, 871-963 (2018). ZBL1416.35194.

[5] Buckmaster, Tristan; Vicol, Vlad, Convex integration and phenomenologies in turbulence, EMS Surv. Math. Sci. 6, No. 1-2, 173-263 (2019). ZBL1440.35231.

You could choose a paper by the same author as the book you have studied. A particular short and nice one concerns the incompressible Euler equations:

The methods you learned will make it accessible to you and your audience.

A motivation for choosing this paper could be further research interest in the direction of so-called inviscid limit problems: do solutions to Navier-Stokes equations converge to solutions to an Euler equation when the viscosity parameter tends to zero(especially in bounded domains this is very interesting, keyword: boundary layers).

Maybe the following papers will be helpful: Some open questions in hydrodynamics and Between Hydrodynamics and Elasticity Theory: The First Five Births of the Navier-Stokes Equation.

Another option would be to talk about Arnold's result that solutions to Euler's equations correspond to geodesics on the infinite-dimensional Riemannian manifold of volume preserving diffeomorphisms: Geometric Hydrodynamics: from Euler, to Poincare, to Arnold.