no. 3
On the kinetic energy profile of Hölder continuous Euler flows
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 3, pp. 711-730.

In [8], the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space Lt∞B3,∞1/3 due to low regularity of the energy profile.

The present paper is the second in a series of two papers whose results may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5. The main result of this paper shows that any non-negative function with compact support and Hölder regularity 1/2 can be prescribed as the energy profile of an Euler flow in the class Ct,x1/5−ϵ. The exponent 1/2 is sharp in view of a regularity result of Isett [8]. The proof employs an improved greedy algorithm scheme that builds upon that in Buckmaster–De Lellis–Székelyhidi [1].

DOI : 10.1016/j.anihpc.2016.05.002
Mots clés : Convex integration, Incompressible Euler equations, Weak solutions, Conservation of energy 
@article{AIHPC_2017__34_3_711_0,
     author = {Isett, Philip and Oh, Sung-Jin},
     title = {On the kinetic energy profile of {H\"older} continuous {Euler} flows},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {711--730},
     publisher = {Elsevier},
     volume = {34},
     number = {3},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.05.002},
     zbl = {1383.35147},
     mrnumber = {3633742},
     language = {en},
     url = {http://www.numdam.org./articles/10.1016/j.anihpc.2016.05.002/}
}
TY  - JOUR
AU  - Isett, Philip
AU  - Oh, Sung-Jin
TI  - On the kinetic energy profile of Hölder continuous Euler flows
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2017
SP  - 711
EP  - 730
VL  - 34
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org./articles/10.1016/j.anihpc.2016.05.002/
DO  - 10.1016/j.anihpc.2016.05.002
LA  - en
ID  - AIHPC_2017__34_3_711_0
ER  - 
%0 Journal Article
%A Isett, Philip
%A Oh, Sung-Jin
%T On the kinetic energy profile of Hölder continuous Euler flows
%J Annales de l'I.H.P. Analyse non linéaire
%D 2017
%P 711-730
%V 34
%N 3
%I Elsevier
%U http://www.numdam.org./articles/10.1016/j.anihpc.2016.05.002/
%R 10.1016/j.anihpc.2016.05.002
%G en
%F AIHPC_2017__34_3_711_0
Isett, Philip; Oh, Sung-Jin. On the kinetic energy profile of Hölder continuous Euler flows. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 3, pp. 711-730. doi : 10.1016/j.anihpc.2016.05.002. http://www.numdam.org./articles/10.1016/j.anihpc.2016.05.002/

[1] T. Buckmaster, C. De Lellis, L. Székelyhidi Jr., Transporting microstructures and dissipative Euler flows, preprint, 2013.

[2] Cheskidov, A.; Constantin, P.; Friedlander, S.; Shvydkoy, R. Energy conservation and Onsager's conjecture for the Euler equations, Nonlinearity, Volume 21 (2008) no. 6, pp. 1233–1252 | DOI | MR | Zbl

[3] Constantin, Peter; E, Weinan; Titi, Edriss S. Onsager's conjecture on the energy conservation for solutions of Euler's equation, Commun. Math. Phys., Volume 165 (1994) no. 1, pp. 207–209 | MR | Zbl

[4] De Lellis, C.; Székelyhidi, L. Jr. Dissipative Euler flows and Onsager's conjecture, J. Eur. Math. Soc., Volume 16 (2014) no. 7, pp. 1467–1505 | DOI | MR | Zbl

[5] De Lellis, Camillo; Székelyhidi, László Dissipative continuous Euler flows, Invent. Math., Volume 193 (2013) no. 2, pp. 377–407 | MR | Zbl

[6] Isett, P.; Oh, S.-J. On nonperiodic Euler flows with Hölder regularity, Arch. Ration. Mech. Anal., Volume 221 (2016) no. 2, pp. 725–804 | DOI | MR | Zbl

[7] P. Isett, Hölder continuous Euler flows in three dimensions with compact support in time, preprint, 2012.

[8] P. Isett, Regularity in time along the coarse scale flow for the incompressible Euler equations, preprint, 2013. | MR

Cité par Sources :