Please look into for desired answer

H. Berestycki and Y. Porneau (eds), Nonlinear PDE's in Condensed Matter and Reactive Flows, Springer, 2002.

on the comment of P. K. Karmakar = the sols and theory by

Székelyhidi jr+de lellis on, at first, the incEE (and after, other models of flu mech), date from c2007 (excepted mistakes of my part for all this text), not before, and it is of a hight level of theorical maths (a talk at the bourbaki seminar was decided and made on it) one of the most important results on incEE for the last eg 20 years. the book you said is earlier, ie 2002 and their +- 40 authors didn't produce ever (ie 2018 included) anything on the theory by Székelyhidi jr+de lellis on flu mech models but if you think that an answer to my questions on SD theory, is in this book, tell me exactly where. thanks. i wait for your answer.

--on solutions "convex optimized" by l. Székelyhidi jr. and c de lellis (and descendants) of nd (or 2d, 3d if there is a difference for LS-CD frames and if so, then how ?) incompressible euler equations, could anyone tell what is the best x-reg (so better than C ^ s, s about 1/3) that LSjr+CD or their descendants, 1 / have already, 2 / could be expected and what are the obstacles getting (or not) a better x-reg ? (all with the same question in t-reg and mixed (x, t) -regs), this regardless of their first motivations, that is, breaking the uniqueness (more generally realistic up to C ^ s for all s

21/7/18=

thanks for your partial answers on a part of my questions.

--on your remarks here= what blocks in the absolute, to go beyond 1/3 or between 1/3 and 1? I repeat, my questions are not concerned at all with the non-uniqueness or the onsager / conservation-of-energy (which, I imagine, are also very different from each other, but it is not either at all my question). What does this machinery have or hope to have as better space regularity for existing solutions (whether the solutions are or not unique or conserving or not the energy = it is not at all my problem or my question right here). --if I say that non-uniqueness must probably go up to C ^ s, s

The scheme consists of high-frequency oscillations (in the form of Beltrami or Mikado flows etc.) that are successively added to a "subsolution", i.e. a solution to the Euler-Reynolds system. There is a competition between strong convergence (favouring very high frequencies) and regularity (favouring low frequencies), and it is a very subtle issue to "balance" the choice of frequencies to achieve the maximum possible regularity while still converging to a solution of Euler. To see where 1/3 comes about in the construction, one has to study the method in detail.

Regarding your other questions: Since convex integration has only produced fractionally regular solutions (and is likely not to do any better), the 2D solutions of type Youdovitch, DiPerna-Majda, Delort are out of reach (as they have one full derivative in some sense). Bute there is an interesting example of Szekelyhidi (CRAS 2012) of solutions with initial data in the Delort class, which instantaneusly lose the Delort regularity however. I believe these solutions remain pressure-free though.

There is no overlap between "wild" solutions and classical ones; one way to see this is the weak-strong uniqueness property: Solutions with one bounded derivative remain unique, whereas convex integration always produces non-unique solutions.

There exist examples of 2D wild solutions. Rotational/helical symmetry not yet, but it's a good question. I would conjecture that e.g. 3D axisymmetric "wild" solutions should exist.

* 23/7/18 = to the 2nd response of emil Wiedemann (who is one of the first specialists and descendants of the subject here, with the 2 1st, d. Lellis, l.Szekelyhidi jr, then buckmaster, isett etc) = if 1/3 means nothing for nonunicity (which should in absolute terms go to s

9/8/18 = emil wiedemann via rg = I would try to specify later, but already, it may be first of all your very enlightened opinion that I ask you on some qualitative facts (I still specify that, at the time, I had read and worked seriously a number of works, some of you, on this theory and euler singular) = are you really sure we can not exceed 1/3 with the even improved mechanics of DS and if we can not, for what reasons / fundamental obstacles "philosophical" or it is only a technical question (fine) but can be crossed (and to go beyond 1/3 is possible in existence of solutions)? and is it a stroke of luck (or coincidence) extraordinary that 1/3 is also precisely the critical threshold (it is very trivial and very simple) of onsager whereas the mechanics of DS seems much more general than onsager (except maybe the function e(t)) even, moreover, that DS etal first attacked the non-uniqueness (which completely laughs at the threshold 1/3) to do only then and a few years later, Onsager (unless they knew, from the beginning, that there was Onsager) ? --other remarks etc later on this and also on the nilpotents.

9/8/18/ii= other small + - easy questions (.. but for any DS experts, it must be very easy to answer ), eg = I / Can the DS solutions with e (t) constant have or can they have other better properties of all orders (eg in x-reg!) than those with e (t) variable, and which ones? II / can we have 1 / non-uniqueness or 2 / solutions with different x-regs, in the cases: a / for a same e (t), b / for a same constant e (t)? III/ Can DS give non-unique solutions in Cs, s> 1/3? IV / the pressure worth in DS, p = cst lul² = p_ds, and in regular theory, p = p_tr = som_ij d_id_j (Lap ^ -1) (u_iu_j) and thus the integral operator singular non-local u -> som_ij d_id_j (Lap ^ -1) (u_iu_j) becomes punctual in DS and is there a link between p_ds and p_tr and how to explain or eclaircize all this in a few words and some remarkable algebraic properties eg in terms of harmonic analysis and functional? and also in a more general way, non-local properties (and linked to singular integrals) that are basic and ubiquitous in TR (regular theory) that become better in DS, eg become punctual or anything else better and what?

10/8/18 =even if the DS frame was made in fact, especially for onsager at first and only, it is very surprising that this frame is just optimal in x-reg and cannot absolutely give extra x-regs, ie x-Cs existing solutions with s>1/3 = what do DS experts think about it ?

.15/8/18 = to the reactualized public answer this day by emil wiedemann (all, always and earlier on this/my public rg-"question")=  "Most of the current theory is 3D, as the construction using Mikado flows is genuinely 3D. The optimal regularity (in time and space) is Hölder 1/3. See the recent papers of Isett and Buckmaster-De Lellis-Szekelyhidi-Vicol. For Hölder larger than 1/3 weak solutions conserve energy (Constantin-E-Titi '94) and it seems unlikely at the moment that convex integration methods could produce solutions that necessarily conserve energy (there are examples of convex integration solutions conserving energy, but in these cases the scheme produces dissipative solutions as well). It is completely open whether non-unique Hölder solutions exist between 1/3 and 1. The questions of uniqueness and of energy conservation are very different in this regard."  = --------------

questions by p. serfati=

 --my questions are not if we have or not uniqueness or energy conservation or if (non-)uniqueness or energy (non-)conservation are different or not = my 1st question is = Hölder solutions exist or not between 1/3 and 1 eg by DS technics  = only existence matters to me here ?

--2nd question = is that not extraordinary that DS AND ONSAGER are BOTH optimal then at s=1/3, DS solving onsager ?? 

--is this for an unsurpassable "philosophical" reason and what is this reason ? 

--or is this for momentary (expectably non-optimazed) technical reasons and we can hope future technical refinements and adaptations of DS technics and proofs and go to existence for at least one s between 1/3 and 1 ??

--in term of history of maths, how to explain that DS had non-uniqueness (where DS is never optimal) and only few years later had onsager where DS appears to be here really more adapted and optimal at least in practical terms (and even = optimal, definitively or momentary), than on its 1st "victory" on non-uniqueness ??

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of course existing DS sols in Cs, s in )1/3,1(, for eg dense sets of many initial data (dense sets as in DS for onsager and nonuniqueness) have to be dense sets of many IRREGULAR initial data... "irregular" means here eg Cs, s in )1/3,1(, but not C1or C^(1+e), e>0, where we have already (classical regular) sols.

To your actualized questions:

1) It is actually possible to get solutions with more derivatives in space via convex integration, but you then need to give up a lot on their summability. Check the recent paper of Buckmaster and Vicol on Navier-Stokes: they actually get solutions which are C_t W^{1,1}_x. Although their construction is for Navier-Stokes, it works for Euler as well. Note however that, although solutions in C_t W^{1,1} have one full spatial derivative, the regularity is nonetheless not enough to guarantee the energy identity, cf. my comment 3) below.

2) Yes, to me it is certainly extraordinary that we get up to 1/3 with what we started more than ten years ago: at the time it certainly looked as a "long shot". There is however a simple computation that we did very very early and which showed the plausibility of this threshold for the convex integration approach. You can find an explanation at Section 7.4 of the following expository paper:

http://www.math.uzh.ch/fileadmin/user/delellis/publikation/Nash_Bull_15.pdf

This fact is also explained in some of Laszlo's lecture notes.

3) The point of Emil was that, at least according to our experience so far, any time that we can apply convex integration methods we can violate the energy identity. This does not prove that it is impossible to construct, say, C^{1/2} solutions, but it is more than just a technical obstruction: we don't really violate the energy identity because we want to... it always comes as a rather easy byproduct of the proof itself. Hence, so far, if you have a space where you can prove energy conservation, it sounds very unlikely that you can produce solutions via convex integration in that space. Anyway, if I learned something in this business is "never say never" :-)

4) In terms of history of math, our very first paper was aimed at giving a short proof of Scheffer's nonuniqueness theorem on weak solutions of Euler. The possibility of tackling Onsager's question came to our mind only after we got that... and the reason is that we noticed a similarity with a problem in geometry which was better understood. The expository paper that I linked above explains this clearly.

.22/8/18 = in answer to the answer of Camillo De Lellis = +-google translate of my text in french = if I summarize (with some remarks and some (new) questions/(see the "?")) = again, my questions are never interested here to the energy identity violated or not violated, what interests me is the pure existence, by DS (improved), of solutions C ^ s, s in] 1 / 3,1 [even if I imagine, those next, they will conserve the energy (except for an "ultra" weak writing of the equations, which does not allow any more to show it), existence very improbable by the frame-DS according to your 2 + 3 / (it is indeed? the short section 7.4. Beltrami flows p28 where it does not seem to have (this said) any manipulations and estimates of x-regularities ..? =((=it is in fact section 7.3 as CDL said to me))

1 / in Buckmaster and Vicol, a / we have solutions with derivatives Lp, p small, but, which by inclusions of sobolev still remain in C ^ s, s 0 or = 0?

2 / if e (t) is constant (where you can always build non-trivial solutions? In large numbers, eg in terms of non-uniqueness, or everything becomes algebraically trivial with e (t) constant?), Your construction does not give better in x-reg solutions, than when e (t) is variable? = C ^ {1/2} is not accessible (according to you) basically by DS and eg the BF, which are simple and explicit with their algebraic homogeneities etc, allow quickly enough to give / to build quasi-counter-examples / very- strong-indications of optimality at C ^ {1/3}, max, in the DS frame? (=I did not see exactly where it was conclusive in section 7.4 ),

3 / there are 2 extraordinary facts = a / DS and onsager and the resolution of onsager by DS are all 3 optimal at s = 1/3, b / DS was designed first and foremost to break the uniqueness, without being (this said), not at all optimal on this (eg because DS says nothing in non-uniqueness for C ^ s, s in] 1 / 3,1 [, where it remains almost sure, ?even with pressure p_ds = cst.lul^2?) and only then, some years later, without any premeditation, we also have Onsager where DS is, there, optimal and the a/.

4/ and again eg, from my previous public "answers" to my rg-"questions"= "9/8/18/ii= ... IV / the pressure worth in DS, p = cst lul² = p_ds, and in regular theory, p = p_tr = som_ij d_id_j (Lap ^ -1) (u_iu_j) and thus the integral operator singular non-local u -> som_ij d_id_j (Lap ^ -1) (u_iu_j) becomes punctual in DS and is there a link between p_ds and p_tr and how to explain or eclaircize all this in a few words and some remarkable algebraic properties eg in terms of harmonic analysis and functional? and also in a more general way, non-local properties (and linked to singular integrals) that are basic and ubiquitous in TR (regular theory) that become better in DS, eg become punctual or anything else better and what?.."", etc.

=========================================

.22/8/18 = en reponse à la reponse de Camillo De Lellis = +-google translate de mon texte en francais = si je resume (avec quelque remarques et quelques (nouvelles) questions(/voir les "?")) = à nouveau, mes questions ne s'interessent jamais ici à l'identité énergétique violée ou pas violée, ce qui m'interesse c'est la pure existence, par DS (amélioré), de solutions C^s, s dans ]1/3,1[ meme si j'imagine, celles ci ensuite vont conserver l'energie (sauf? ecriture "ultra" faible des equations, ne permettant plus de le montrer), existence tres improbable par le cadre-DS selon vos 2+3/ (c'est bien la courte section 7.4. Beltrami flows. p28 où il ne semble pourtant pas avoir de manipulations et estimations de x-regularités ..? ) 1/ dans Buckmaster et Vicol, a/ on a des solutions avec des derivées Lp, p petit, mais, qui par des inclusions de sobolev restent encore toujours dans C^s, s0 ou = 0 ? 2/ si e(t) est constant (où vous pouvez toujours construire des solutions non-triviales ? en grand nombre, eg en terme de non-unicité, ou alors tout devient algebriquement trivial à e(t) constant ?), votre construction ne donne pas mieux en x-reg des solutions, que quand e(t) est variable ? = C ^ {1/2} n'est pas accessible (selon vous) basiquement par DS et eg les BF, qui sont simples et explicites avec leurs homogeneités algebriques etc, permettent assez vite de donner/construire des quasi-contre-exemples/tres-fortes-indications d'optimalité à C ^ {1/3}, max, dans le cadre DS ? (=je n'ai pas vu exactement où c'etait conclusif dans la section 7.4), 3/ il y a donc 2 faits extraordinaires = a/ DS et onsager et la resolution de onsager par DS sont tous les 3 optimaux à s=1/3, b/ DS a été concu en tout premier lieu pour casser l'unicité, sans etre, cela dit, pas du tout optimal sur ceci (eg car DS ne dit rien en non-unicité pour C^s, s dans ]1/3,1[, où elle reste, par contre, tres vraisemblable) et seulement ensuite, quelques années plus tard, sans aucune premeditation, on a aussi onsager où DS est là optimal et le a/. 4/ deja traduit.

I understand your questions are never interested in energy conservation. But according to experience, if you can implement convex integration, the method itself will give you energy-non-conservative solutions (you do not care, but the method will do it in spite of you :-)). Mind: this is what we saw so far, we don't have

a "proof" that this is always going to be the case.

The reference to my paper was wrong, sorry. I should have said "Section 7.3": there you see the magic number 1/3 coming out of some heuristic. You are right, Section 7.4 is irrevelant, I just misquoted it. Now, although Section 7.3 gives you some explanation of how the 1/3 comes out, this is not that simple: it is simpler than following all the details of the actual proofs, but it is still much more complicated than the algebra behind energy conservation.

The iteration of Buckmaster and Vicol produces a solution which is very far from C^s: Sobolev embedding would not even give you continuity of the solution. Their iteration is different from the ones of the other papers. This happens for a lot of other references: the papers all share some common points, but since often the construction follows different iterations, if in one paper we produce a certain regularity (say in a space X) and in some other paper some other regularity (say in a space Y), it is absolutely not clear whether you can combine the methods and produces something which is in X AND Y

to C. de Lellis= 

1/ in the RESTRICTED spirit of my question, i want to be clear that energy non-conservation is not the matter (and my matter is only pure existing solutions Cs, s>1/3, by DS (augmented) frame, where eg there is always incidentally energy conservation. you (and your descendants, isett etc) solve energy non-conservation and this pb is closed (and is of course very interesting otherwise !),

2/ ((the correct, as CDL said to me:)) section 7.3 ((and not section 7.4 as CDL said to me before)) where there is "θ_0 = 1/3" is still very technical and are you absolutely sure that eg this part with some little more extra new little "scratchings",  "diggings", "hints" and "adaptations" cannot give absolutely at least (the hope of) just one single (more or less more particular) solution, but with Cs, s>1/3 ? (eg with then forced "e(t) cst". independently do you have really nothing more (any criteria) with e(t) cst comparing to e(t) variable ?) = and again the obstacle is "philosophic" or only technic at the moment ?

3/ sorry on what i say on Buckmaster and Vicol = i meant by inclusions of sobolev (ie lflp or lfls < cst lDflq,  lgl_Wp,a < cst lDgl_Wq,b, (p,q) and (s,a,b) appropriate in [1,oo]^2x[0,1]^3) that they will give here always NOT MORE than Cs, s

.5/ about the full technical DS, "not that bad", in the approach of the supposed optimality at s =1/3 = of course ! but/and is there still a little "space" between "not that bad" and "can't be a little better, until one case of Cs, s>1/3" ? and i meant by "some little more extra new little "scratchings", "diggings", "hints" and "adaptations" with some subtil and clever drivings, ideas, (little) adaptations and observations we can reach the "Grall" of at least (the hope of) just one single (more or less more particular) solution, but with Cs, s>1/3= how to believe that the DS technical "monster" (!) reaches a "philosophical" optimal state ?

i precise here that in my memory and in what i say here on this rg-question, e(t) was the energy and NOT, eg, the punctual cst.lu(x,t)l^2 (:cst.lu(x,t)l^2) etc.

(rcts only for the english text) * 30/8/18 = another here completely independent question about, here, the already existing DS solutions (ie x-Cs, s 0, as a regular function of t in [0, T [, but where I suppose, moreover, that lim_t -> (T-) (e (t)) = + oo (weak explosion in T-, eg t-Lp ([0, T [), for a p in [1, + oo] or strong, eg for no p in [1, + oo]), maybe (/can) we say that we have solutions here u (x, t) (distributions, in eg Co,t.Cs,x ([0, T [xlRn), s 1, or C1,x or C1 + s ',x, s'> 0, or rotu L1 or positive and measure etc)? and likewise with e (t) regular outside a discrete set in t in lR*+, but e (t) t-Lp, loc (lR +), for a p? and how to explain then this difference for n = 2 between DS and TR, in practical terms, heuristics, maths, physics, oscillations of the solutions etc? and the TR t-explosion or not in 3d (or nd, n better than 3) is still a very important and open question for decades and min (half of) a century.

=============================

(rcts only for the english text) *30/8/18 = une autre question ici completement independante sur les solutions DS ici deja existantes (ie x-Cs, s0, comme fonction reguliere de t dans [0,T[, mais où je suppose, de plus, que lim_t->(T-) (e(t)) = +oo (explosion faible en T-, eg t-Lp([0,T[), pour un p dans [1,+oo] ou forte, eg pour aucun p dans [1,+oo]), peut-on dire qu'on a ici des solutions u(x,t) (distributions, dans eg Co,t.Cs,x([0,T[xlRn), s < 1/3) de l'equation d'euler incompressible sur lRn, tout n, qui explosent en temps fini T car lul_L2(t) explose en T-, alors qu'en theorie reguliere/TR, avec n=2, elles n'explosent jamais (eg avec uo et donc u dans W1,p,x, p>1, ou C1,x ou C1+s',x, s'>0, ou rotu L1 ou positive et mesure etc) ? et de meme avec e(t) reguliere hors d'un ensemble discret en t dans lR*+, mais e(t) t-Lp,loc(lR+), pour un p ? et comment expliquer alors cette difference pour n=2 entre DS et TR, en termes pratiques, heuristiques, de maths, de physiques, d'oscillations des solutions etc ?

Your question from 30/08: Yes, I think it is possible to construct weak solutions with unbounded energy. But it is questionable what relevance they have. Usually one imposes some energy inequality (e.g. at any time, the energy should not be greater than initially) in order to rule out at least the most obviously unphysical solutions. (With this I am not denying the mathematical interest in, say, weak solutions with compact support in time.)

As for the relation with more classical solutions, let us take for instance the ones of Delort: They emerge as limits of smooth solutions with regularized data, so the energy inequality is inherited in the limit. In fact any weak solution obtained as a limit of some physically meaningful approximation (mollified data in 2D, Leray-Hopf viscosity limit, etc.) will satisfy the energy inequality.

.to the public answer by EW the 31/8/18 = 31/8/18 =it is again my mistake = my matter is here (that we have and we are really able to say that we have (eg because this is a Millennium Prize Problem in TR for euler and nse !!)=) the explosion in finite time T of any type and/or one or some x-norms (taken pointwise in t in [0,T[), and which ones, for solutions on nd incompressible euler eqs, eg via DS augmented frame = is it possible for DS experts (see my tx of 30/8/18 and eg comparisons with TR solutions, 2d, 3d etc) ?, and my matter is not unbounded energy in itself and eg, if i choose for my/this matter this x-norm (taken pointwise in t in [0,T[): lul_2(t), it is only because DS permits extraordinary, for this norm, to choice by advance, its arbitrary profile/behavior/definition in time t, eg an explosive profile in -->T-, and more generally, is this also the case in DS for other norms or quantities/expressions etc (when eg also e(t) is not explosive eg e(t) cst etc) and which ones? = i hope i am more clear here.

.+10+11+13/9/18 = to DS experts on my RG-"question" (see future rcts at this "question" and rg-abstract of my RG-data "rg txs etc")=

"Best (achievable) x-reg for sing sols of nd incompressible euler equations by Székelyhidi jr+de lellis and remarks/links with pressureless (reg) sols?"

= again (and in additions to my previous answers to e. wiedemann and c. de lellis)=

1/ when e(t) is constant, DS sols (De lellis/

Székelyhidi jr) are trivial? and then why ? or some of them are still non trivial ? and in what way ? can we imagine that there are eg in x-C^s, s>1/3 or that they have? any better or other/new properties-(eg regularities) than in the cases e(t) variables(/eg x-C^s'-sols, s'1/3, DS-sols and/or the transition zone going from DS to TR: what DS-"magical" properties can partly survive/mutate in approaching TR) on the regularities of the sols and they perhaps diseappear with liouville type ths with upper levels = (except error of my part=) the pressure is punctual in u: p = cst. lul^2, the IEE itself is, here exceptionally, a punctual/local eq, punctuality/locality which permits convexity methods (punctuality/locality is never the case in TR (:for all this §), where p is a non-local singular integral in u = som_ij didj(lap^-1)(uiuj)), for the coordinates u_i we have ui.uj (or their spatial integrals ?) equal for all i,j (=p/n if i=j), and null if i,j different (and we recover the preceding point), lul^2(x,t) (and then p) can be constant on an arbitrary fixed troncated (circular) vertical cylinder of the (x,t) space with then div(u) =0 on the all (x,t) space and the evolution of the IEE (and then eg what does it gives at least formally, on the x-D of p, u, and the equations IEE ?) etc

----more generally (ie far from DS and its uses on IEE, NSE and other flu mech models) is this the same pbs/cases/maths-lacks for some other/+-all? weak existences of sols obtained by convexity on PDE systems, eg a conservative PDE system with differential order equal to one, D(F(u)) =0, in distributions, where eg u is in Cs, not more (eg with no function derivative), where F is a non linear regular function, and where convexity is used to solve a pure (punctual) functional (non-differential) equation linked to F(u) = in that case and frame, is the distributional writing is not anymore precise enough, ie there are maths lacks and for these reasons "pathological"/etc sols are not (technically) excluded anymore ?

--is the weakness of the distributional writing and the low (x,t)-reg of the sols (with no derivative function asked) permits too easy gluings (on some functional spaces) of 2 (disparate) sols on (t-) and (t+) and to have still then a 3rd weak sol and idem for spatial gluings/superpositions etc ?

--is this a general pb of the convexity methods for existences for PDE systems (or if not, is it, a contrario, only an other exceptional property of DS-convexity frame applied to IEE and other flu mech models) ?

4/ as i say DS permits extraordinary to choose e(t) = integ lul^2(x,t)dx by advance (eg variable and? not bounded in t) and then? to have, with real natural extended proofs and contructions and no efforts (:it seems to me that way= am i wrong ?), exploding sols in t for the x-norm x-L^2 but are there other norms/properties in these cases of t-explosions, even when e(t) is constant, the sols "then" more "physical", but with some efforts, eg for a x-Cs sols, s1 in lR^2 ?

----(the 11=) and at the opposite, i suppose ? that DS can starts 3d IEE globally in time for some dense sets of irregular data AND for the starting a t=0 from the very same sets and very same data, have? other sols at t non null with finite time explosion with no contradiction (because of nonuniquenesses) = is this linked to the lacks said in 3/ ?

.5/ (the 13 =) i mean by "1rsts" (historical) DS-sols, at least those, very particular (in many senses, even inside frames of wide effective non-uniquenesses), with punctual pressures, p = cst.lul^2 (or constructed and (partly and later) derived from these 1rsts ones). Other DS-sols (eg for IEE) were later constructed with other objects as beltrami and mikado flows/fields etc, less? or not? depending on punctual pressures or convexity frames, and later again large similar constuctions were made without beltrami and mikado objects.

-15+16/9/18 = to Wojciech Ożański (and others authors), about your papers on the following (on nse and, iee, visco =0)= hi, do you (or anyone) (nearly or closed to) have clearly explosion in finite time for sols of IEE/3d eg (see also my previous remarks here in this rg-"question" on t-explosions based on that of e(t) and those possible on other terms/norms)= for IEE (IncEulerEqs) in 3d and one T>0, sols in C^0,t([0,T[).C^s,x(lR^3) with a norm of the sols in W^a,p,t([0,T']).W^b,q,x(lR^3), T'T-, for some a,b in lR+, p, q in [1,+oo] (W= sobolev spaces, a, b for indexes of derivative regs, p,q for integrabilities) ? 

--you (and others) seem to have weaker forms of that kind of things for what it is called NS and IE "INEQUALITIES", ie NSE and IEE written as nullities = "d_tu + (u.D)u +Dp - µLap(u) =0" (µ> 0 or null), replaced by the negative scalar produces by the speed = "u.[d_tu + (u.D)u +Dp -µLap(u)] < 0 or =0" (ie, when formally integrated in x, gives again +- the standard (TR+DS) nse energy equality or newly INequality and then also we ask here less than NS equation itself, asked without force).

(Philippe Serfati, PHD/DOCTORATE+ENS+AGREGATION all in MATHS, see RG).

Is it possible to put the question in a simpler English form so that a focused precise answer may be provided?

16/9/18 = to the answer of 16/9/18 by P. K. Karmakar = i believe my english is simple (but perharps not absolutely always good) and perharps you ask for simpler maths instead. i believe that my questions and remarks etc here are +-simple, very natural and often +- heuristical (in terms of maths and for mathematicians and i am only a mathematician) but (became) with many +-independant interests/ways but always on DS sols and frames on IEE and other meca flu models and the answers that i can wait eg from DS experts, could be more precise and technical and "less/not natural", eg beyond my own (heuristical) knowledge on DS theory. but if you want some other new heuristical details and sum-ups from me on what i say here on my "answers" and "sub"-"questions", tell me where and which parts and §s, here or via private messages on RG : i hope i give to you here a 1st good answer to you. and again, i am a TR expert on IEE and the DS frame is a singular theory for IEE (etc) with (very) "surprising" properties (and more) from the point of view of the TR and TR experts (i listed here some of them) and i asked natural questions etc based in this confrontration.

.-17+18/9/18 = 2/to the answer of 16/9/18 by P. K. Karmakar = eg DS (D= De Lellis, S=Székelyhidi jr) we have (+-) surely eg for their 1rsts "historical" singular sols = pressures and IEE and meca flu models are local/punctual in the speed u, sols can be with compact support in x and/or in t, t-energy can be chosen by advance in t, lul(x,t) can be (chosen) cst on an x-open, ui.uj (or? their x-integrals) independant of i if i=j and null otherwise, very easy gluings in x and/or t of disparate sols are possible, sols are holder in x/t not more, 2d-t-explosion and 3-t-global-existence are possible, dense sets or irregular data give existences, non-uniquenesses, non energy conservations, strict decreasings of the energy, every single initial data (from these dense sets) gives? itself another dense sets of t-sols starting from this data (then wild non-uniquenesses), ?=a very large part of this is possible only because almost all manipulations, residues/corrections-terms, (partial) elements of sols, superpositions/additions/constructions of terms/sols, oscillations terms, (conservative) distributional writings, (exact) precisions in the estimations/preserved-properties/constructions etc are (always) strictly local-local (or pointwise-pointwise), diverse (x,t)-regularities asked rigorously at enough low levels and when there are non-local terms they are always more regular residues/corrections-terms (then with no pb in the constructions) etc and all this is absolutely definitely and basically FALSE and disappears (or for the rest, unknown, out of reach at the day, very hard to prove, very very peculiar, to much "surprising" etc) in TR/ regular theory where eg Du or rotu is and remains (with t) a function (or atleast or positive measure) and not only a distribution and eg energy is always conserved. In TR we have permanent fundamental NON-local terms, manipulations, estimations, pbs, considerations etc, eg for IEE the non-local relations: Biot-savart, u +-= (DLap-1)(Du-tDu) or p = (Lap-1)(trDu2) = (divLap-1)(u.Du) = (divdivLap-1)(u.u) etc that do not (basically) exist/appear (in one piece) in DS and are false? for final DS sols (see my "reply" of 23/9/18).

--and one of the other interests is what becomes DS when it gradually and quantitatively approaches TR: in this transition zone in what mutate the surprising properties of the DS sols ?

--"?" = means here = it is a question and/or I am not completely sure and it is then a question (: an ask of corrections/precisions) to and by DS experts.

.23+24+25/9/18 = again = for IEE in TR, we have the non-local relations for divu = 0: Biot-Savart, u +-= (DLap-1)(Du-tDu) (tDu = transposed matrice of Du) or p = (Lap-1)(trDu2) = (divLap-1)(u.Du) = (divdivLap-1)(u.u) = som_ij didjLap-1(ui uj) etc that do not (basically) exist/appear (in one piece) in DS and are false? (?: that is here the/my sub-question) for final DS sols= if the answer is no = eg for some historical DS-sols, we have p = cst. lul^2, ie ? som (i ≠j) didjLap-1(ui uj) = 0 for all x,t (eg a t fixed for all x, static (but very rare) case and much harder, also always during the t-evolution of IEE), or (for some other? DS sols:)  lul(x,t) = 1 on a (x,t) vertical cylinder with x-base an x-open Q, eg bounded, with u=0 outside Q= what does mean this x-compact support of u for all t eg in the linear relation u +-= (DLap-1)(Du-tDu) or its x-Derivative: Du= (D2Lap-1)(Du-tDu), (D2Lap-1) being a standard but major non-local singular integral operator, eg on Qc because in TR, u is never with x-compact support (even if it is supposed a t=0, case where then in TR, its rotu (+-=Du-tDu) has an x-compact support for all t in a [0,T], all T>0 and then we have f =D2Lap-1(g), with both f and g null outside a (fixed) x-compact), ? etc (for the other? DS-sols, when p = cst. lul^2, u is also with x-compact support if it is true at t=0) = all this is made formally, but is it correct ?

--but eg, at the opposite, we can? have also in DS and by nonuniqueness and for all (x,t) and for full strict local punctual properties/frames for the operator and IEE and pressure p = cst.lul^2 (and in spite of them), uo null everywhere then with x-compact support and u(t), t>0, non-null and even with NO x-compact support for all t etc.

.26/9/18 = on the said reactualized answer (of 21/7/18) this day by e. wiedemann (:""The scheme consists of high-frequency oscillations...") as perhaps put again because answering already to my later remarks and sub-questions = if the definitive gap between DS and TR for the speed u, is x-C(1/3) to x-C1 ie a block 2/3-L°°-x-derivative in gap, there is nothing more to say and there is in terms of x-regs eg, no tangency, proximity, zones of transition or "adherence" possibly imaginable between DS and TR. but is this block 2/3-L°°-derivative in definitive gap real ?. all this to be understood modulo C(1/3) or other spaces with the same homogeneities eg via the inclusions of sobolev.

--the latest extensions of DS to models of meca flu and (linear) transport eqs, seem? to use less (or no) convex frames, more, and later less (or no) beltrami and mikado flows/fields, to use (mainly always only?) more basic partial/pieces-of eqs ans sols with eg only stress terms and a lot of partial terms of sols estimate with powers in lambda_q mainly. but DS eg for transport eqs (and? for IEE and nonlin eqs etc ?) reaches already the step of u with a derivative ie Du in an LP, p small, on an interval of times. 

--on the other hand, we can include in TR, one of the worst x-reg known, ie delort's result for IEE in 2d = +- u L2 and rotu measure >0 or eg Du L1 or LP, for one p = is that very far from the best x-regs possible in DS and is the gap DS-TR always here equivalent (eg via the inclusions of sobolev) to an 2/3-L°°-x-derivative ?

--the sub question is then= the gap DS-TR is a definitive block s'-L°°-x-derivative, with s'>0 non null (eg s'=2/3) ? or perhaps the gap is (in the future?) possibly smaller, thinner, or almost null ? and in the 2nd case i reactivate my sub questions and remarks (ie eg, DS approaching TR = transition zones and results and regs, mutations of DS properties and progressive (re-)appearances of forced non-localities, non punctualities, integral (singular) basic relations specific to TR etc).

..rct*2/10/18 = or as the opposite, do we have = eqs of type= D^µ_(x,t) [F_µ(v(x,t))] = 0, with x, µ, Fµ, v, in diverse spaces with different dimensions, eg to mimic the IEE, lµl =1, v= u or (u,p) or (u,p,e).., F separatally homogeneous polynomials of partial degrees = 1 and/or 2, where we have existence by convex frames, ie in a 1st restriction, in mainly solving simple non-differential (:ie with no mention to the "D^µ_(x,t)") algebric (punctual functional) equations linked to the Fµ alone, and the sols are in some spaces W(si),(pj), s_i in (01) and p_j in (1,oo) in a maximal set for/in the ((si),(pj)) (?intersections of half spaces, eventually up to the inverses of s and p..), explicitely simply depending on µ, the partial degrees and dimensions said higher, maximal set very far (ie, with 2 disjoint boundaries ...) to the other maximal set for TR, ie sols obtained with regular theories, eg construted in including the "D^µ_(x,t)" and differential and PDE (additional) methods ?

--.rct = this is perhaps inherent to 0/ convex attack, but when it becomes of minor uses for 1/ the prevalence of beltrami and mikado flows/fields and/or 2/ the further (final?) domination (on cvx f and BM F/F) of estimates in the lambda_q, do we have gains of x-regs between 0/ and 1/, 0/ and 2/, 1/ and 2/ ? and what are the best x-regs hopable for each, 0, 1, 2 ?,

--but again eg for IEE the (worst) TR regs known this day, can be eg in x-2d (see also 3daxi or nd/nilpotent-pressureless sols etc), can be larger, ie u in L°°t(W1,1,x or BVx) and L°°t(L2,x) and is that very far from regs in 0,1,2 ?

--.rct.the 03/10/18 to the reactivation this day of his answer (of 22/7/18) by c. de lellis = yes, in buckmaster and vicol, u in one x-Cs, Dxu is +- in x-L1 (Dxu =rotu?), and eg e(t) can be chosen cst (but is this choice "dangerous" because all becomes trivial in DS ? and if not see below (*) = i would like an answer to this), 

--but eg do we have biot-savart = u = (DLap-1)(Du -tDu) or something explicit and simple near that and idem, do we have the pressure p = som_ij didj(Lap-1)(uiuj) or something explicite and simple near that (and this "nearness" itself) ? and can we hope eg here future non trivial mixes/combinations of partial parts of DS and partial parts of TR, that gives new unknown properties etc of any kind for DS, for TR and more generally on (future) sols of IEE and nse etc (mixes etc by practical adapted +- particuliar (pointwisely?) works on DS and on TR, to bring them closer) ? 

--(*) if not, then, non uniqueness is still open ? for speeds in x-Cs, s in )1/3,1(, eg by DS?, and we can hope in practice that its resolution by DS will be made by construction of sols, eg then incidentally, existence for +- a bit particular but enough general sols in Cs, s in )1/3,1(, then near s=1 and near TR and 

--i am also interested by contacts between DS and TR and eg again, for s near and approaching 1, do we have the 2 integral formula higher, Biot-Savart and p by u or something explicite and simple near that and/or DS (same question for all s in )01( in DS) and 

--eg, is this "something" is linked to one of the surprising properties of some DS sols, ie integ_x (uiuj)dx null if i-j non null, cst in i otherwise, and if yes, how to formalize constructively this with 2 formula for this "something" (and this "nearness" itself, linked to it, said higher) eg for and near BiSa and the integral relation "p by u", that DS will be forced to +- verify eg when s approaches 1 ? or we have to add or take in this (then, same asks on practical explicit constructabilities), another DS surprising properties which exploit the lacks of the distributional derivative writings, not anymore enough exact, trustable, thin, fidele etc in DS frames ?

--rct= or BiSa and p-by-u-integral are always/sometimes? verify in (strict) DS frames (even for the already existing sols ie s

.see on DS, eg the latest short nontechnical survey by D&S themselves=

arXiv:1901.02318  [pdf, ps, other] 

math.AP math.DG

On turbulence and geometry: from Nash to Onsager

Authors: camillo De Lellis, László Székelyhidi Jr

Abstract: This article is a short nontechnical survey of recent progresses in fluid dynamics and differential geometry, relating a conjecture of Lars Onsager to the work of Nash on isometric embeddings.

Submitted 8 January, 2019; originally announced January 2019.

====================================

see also the review article/2019 with (more maths) discussions on a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations =

""Convex integration and phenomenologies in turbulenceTristan Buckmaster, Vlad Vicol (Submitted on 25 Jan 2019) https://arxiv.org/pdf/1901.09023.pdf""