(abstract 2017) Stratified regularity and 3D Euler equation at large time Abstract — We solve the Euler equation 1/ in all dimension at small time, for data with stratified regularity, D2Δ-1 conserving this type of regularity (: only an announcement of a part of our Ch4B, thesis, 92, univ Paris 6-PMC, France, see also below) and 2/ we give an almost (ie except only on a simple estimate on a regularization of the initial axi-data and classical maths on euler eqs) self contained proof, with no paradifferential-calculus, at large time in the 3D axiymmetrical case (and newly without condition of sign linked to the curl), firstly for general under-lipchitz speeds secondly in the vortex patches 3d frame (VP3 :ie local C1+s(R3) deformations of L°°(R,C1+s(R2))), precising fully and newly in this last case, the matricial discontinuous part of the gradient of the speed. our here well existing under-lipchitz axi3d solutions are, by our other "Pertes de régularité pour le laplacien et l'équation d'Euler sur Rn", lagrangian C°° in time. Ch4B is refined, reworked, extended and included in results of the paper = ""Vortex patches dans Rn et régularité stratifiée pour le laplacien"" (but ch4B has still a remaining interest on small differences in some approaches). ---- our axi part here, in particular, on the novelty of the absence of condition of sign linked to the curl is +- "similar, simultaneous and independent" (at least for me), to Saint Raymond, X. (1994). Remarks on axisymmetric solutions of the incompressible Euler system. Communications in partial differential equations, 19(1-2), 321-334, this one extending and completing the other paper on its axi-part with the condition of sign linked to the curl = Gamblin, P., & Saint Raymond, X. (1995). On three-dimensional vortex patches. Bulletin de la Société Mathématique de France, 123(3), 375-424. ------- 15 to 25 citations (with google (scholar) and me). GS14, ap17.
https://www.researchgate.net/publication/266432609_Regularite_stratifiee_et_equation_d%27Euler_3d_a_temps_grand?_iepl%5BviewId%5D=MxK1034nfk3sbQgOOKv1kvlL&_iepl%5BprofilePublicationItemVariant%5D=default&_iepl%5Bcontexts%5D%5B0%5D=prfpi&_iepl%5BinteractionType%5D=publicationTitle
Article Régularité stratifiée et équation d'Euler 3d à temps grand
Veuillez consulter ces informations sur le Web aussi ..
http://www.numdam.org/article/M2AN_2000__34_2_201_0.pdf
my question was a mistake= i wanted only to advise axiflow project people of my axi result, but tell me why your paper is linked to this. thanks
+(rct1/10/18) (intend to be) said and asked to (certain and/or) all authors (on their papers) on euler (and nse) on 3Daxi sols (without swirl) eg when x.st-Raymond1994 (on euler) is (/should be) cited, eg on extension to nse (eg major papers on nse on 3Daxi sols (without swirl))=
/under-quasi-log lip vp3d axi glob in t no positive curl condition 1st result serfati ph.1.93-94/= from SERFATI Philippe, PhD+ Ec.Norm.Sup.-PS+AGREGATION in MATHS. 240 citations by 120 international university authors. --about your paper(s) on the Axisymmetric 3D Euler (+ns) Equations Without Swirl, see my "Serfati, P. (1994). Régularité stratifiée et équation d'Euler 3D à temps grand. CR Acad. Sci. Paris Ser. I Math, 318, 925-928."/submitted1-93 (see full text on my profile on researchgate or on the site gallica and the/its 14 citations, sp18, according to my google citations), and its theorem 3, where i give a short self contained proof (except the last line at the end of p926 on a basic regularization of the initial data) with no paradifferential calculus and no positivity condition on the initial curl, for global existence of 3d euler eqs axisymmetrical case with no swirl and for under-lipschitz speeds and (L1 and L°°) discontinuous bounded (initial) curls with then NO positivity condition (: curl_o.(-x2,x1,0) >=0, condition asked before with also here over-lipschitz speeds (a/ isotrope + b/ aniso/VP), by eg a/ majda/cpam1986 (see my ref [4]) and a very short time before me, by b/ gamblin&x.st-raymond, my ref [3]/preprint1-93), simultaneously and independently (at least for me) of the classical/major reference on these precise non-signed points: "Saint Raymond, X. (1994). Remarks on axisymmetric solutions of the incompressible Euler system. Communications in partial differential equations, 19(1-2), 321-334."/5-93 (see eg :https://www.tandfonline.com/doi/abs/10.1080/03605309408821018 which says the receiving date "5-93" = then 4 months after my own submitted date, and(/but?) i remember that i went to a talk (because i was just aware of it) around these subjects made by x. st R. to whom i gave my finalized paper, him and gamblin, in their previous (:for x.st-R) paper together, didn't ((be able to) (think to)) prove the/my 2 results (iso quasilip +VPaniso overlip) without the positivity condition)= --my th3 (and th4) are then the 1st historical results with these natural conditions (ie under/quasi/log- lip axi 3d global in t for euler (with no positivity condition on the curl_o), and also in the 2 over-lip sub-cases, iso/C(1+s)+ aniso/VP, and eg this last including the 1st, this 1st also trivially extendable from quasilip to iso-overlip), then i believe that you have to cite my paper. --the/my other ths (fully proved or announced) here are of independant interests on striated x-regs (eg the b/ said before but always with no positivity on the curl_o, fully proved, this case proved again also in "Saint Raymond, X. (1994). Remarks..". --there exist a "third" proof, posterior (!), in the no-sign case, with u C(1+s), over-lip iso sub-case (included in our VP case, to StR and me) = Majda, A. J., & Bertozzi, A. L. (((eg:)) 2002). Vorticity and incompressible flow (Vol. 27). Cambridge University Press., "".. ((p)) 166 4 Particle-Trajectory Method for Solutions to the Euler Equation .. Notes for Chapter 4.. . A simple proof of Proposition 4.4 ((see below)) for axisymmetric flows without swirl was given by Majda (1986 ((ie MajdaCpam1986)) ). .. he proved global existence by assuming fixed sign of the vorticity... ((Proposition 4.4, being then the same result, WITHOUT assuming fixed sign of the vorticity, majda&B didn't cite for this no-sign case, with no apparent explanation, the 2 earlier St-Raymond1994 and earlier, me cras1994reg.strat.lap.euler))"". This book has other earlier versions: 2001, 1997?, etc, but apparently always posterior to 1994 and us 2, StR and me. https://cdn.preterhuman.net/texts/science_and_technology/physics/theoretical_Physics/Vorticity%20and%20incompressible%20flow%20-%20Majda,%20Bertozzi..pdf --tell me what you will do.
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--nb =nobody, majda86, G/StR, StR, and me cite " Ukhovski, M., and Iudovich, V. J. Appl. Math. Mech., 32, 52-62 (1968). " at the time and i didn't know this reference. to us 4, our method is +- to estimate lX(x,t)^-1l^-1, that is different from that of UI68, based fundamentally and non-trivially on the conservation of the energy, that is not our case to us 4.
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