My initial take on: “Ill-posedness of Leray solutions for the ipodissipative Navier-Stokes equations “

Last year, 2017 (Happy New Year! Holla!) a friend of mine pointed out this relatively new results on archive using convex integration.

“ILL-POSEDNESS OF LERAY SOLUTIONS FOR THE IPODISSIPATIVE NAVIER–STOKES EQUATIONS” by MARIA COLOMBO, CAMILLO DE LELLIS, AND LUIGI DE ROSA

Where these talented mathematicians show an early result on the failure of highly touted Leray solutions for a variant of the navier-stokes problem.

I wrote my friend back a disgustingly long “hot take” of the paper after a quick skim. I wanted to share what I wrote here for future reference. Please pardon, my lackadaisical deference. I genuinely have a great respect for all these researchers, but when I write fast e-mails this doesn’t always come across.

P.S. If you’re interested in more reading on the basics of Convex Integration please check out this other page I threw together with some references: Convex Integration

Cool find! I didn’t see this, so thanks for sharing it!

I haven’t read it in detail, but from skimming it, I have some initial thoughts.

It’s definitely convex integration and the second author, Camillo DeLellis, is one of the original people to use convex integration for fluids and lately he’s been doing a lot of work with this method. What’s kind of odd is that I don’t recognize the other two authors as people who have ever done anything with this method, but they work in Zurich, the same city as DeLellis, so I guess that’s how they got into this.

Off the bat, they’re working on 3-D periodic domain, Navier-Stokes where the dissipation is fractional. Ever since the Euler equations were done in 3-d (the resolved 3-d Onsager conjecture) I think everyone has been looking toward Navier-Stokes, and this might be the easiest, most natural next problem to look at.

It seems like they prove the non-uniqueness of a weak “Leray” solution when the fractional power of the dissipation is small enough. Here “Leray” means the weak solution satisfies some special energy bounds (equations 2 & 3 in the intro).

What’s kind of cool is that the non-uniqueness is unexpected. Typically, dissipation has a helpful “smoothing” effect, so that when one produces a weak solution one would expect to also get uniqueness by using that dissipation term (in maybe a DeGiorgi argument or something). But this is not the case! I guess, Sverak coauthored a couple papers conjecturing the non-uniqueness of weak Leray solutions too, but since he didn’t prove it I’m guessing it must be hard to prove this result by standard methods.

Otherwise, the setup of the paper looks like pretty standard convex integration. Their main theorem follows the usual outline where given any energy profile, e(t), depending on time, they 1. construct a weak solution and 2. show the weak solution has the given energy profile. (Usually at this point we say that by the ability to choose any energy profile, we can construct two different functions which are both weak solutions and therefore weak solutions are non-unique).

What’s different in this paper is that they need to also show that the Leray energy inequalities are satisfied by their constructed solution and they make a second claim i.e. thm 1.2 and thm 1.3, that a specially choosen initial data can lead to infinitely many weak solutions. What this amounts to in the proof is the authors doing some extra book-keeping to track the error terms in the iteration of the convex integration.

I also see some notational and technical choices used that are kinda like DeLellis’s “calling card” when he writes convex integration papers lol. Also, he doesn’t use the newest “gluing technique” in this one (I hear it’s a total b*tch to layout the machinery for the gluing technique, but after it’s set up it gives improved estimates. I’m guessing the better estimates are overkill for what these authors are trying to prove, so that’s why they didn’t use it).

Sorry for the super long e-mail! But I really liked this paper you sent! I’ll probably even mention it to Phil when I get the chance. I hope your summer is going well too!

What are you up to these days??

Thanks again!
Andy

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