A proof of Onsager's conjecture

Annals of Mathematics

Volumn 188, Issue 3, 2018, Pages 871-963

  Isett, Philip ^a,b  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Anomalous dissipation; Conservation of energy; Convex integration; Dissipation; Energy conservation; Energy dissipation; Euler equations; Fluid dynamics; Incompressible; Nonuniqueness; Onsager conjecture; Onsager's conjecture; Regularity; Turbulence; Weak solutions

Indexed keywords

EID: 85055915274     PISSN: 0003486X     EISSN: 19398980     Source Type: Journal    
DOI: 10.4007/annals.2018.188.3.4     Document Type: Article

References (42)

1. C. Bardos and U. Frisch, Finite-time regularity for boundedand unbounded ideal incompressible fluids using Hölder estimates, Lecture Notes inMath. 565 (1976), 1-13,Proc.Conf.,Univ.Paris-Sud,Orsay, 1975. MR 0467034. Zbl 0355.76016
2. C. W. Bardos and E. S. Titi, Mathematics and turbulence:where do westand?, J. Turbul. 14 no. 3(2013),42-76. MR 3174319. https: //doi.org/10.1080/14685248.2013.771838
3. T. Buckmaster, Onsager's conjecture almost everywhere in time, Comm. Math. Phys. 333 no. 3(2015),1175-1198. MR 3302631. Zbl 1308.35184. https://doi.org/10.1007/s00220-014-2262-z
4. T. Buckmaster, C. DeLellis, and L. Székelyhidi, Jr., Trans-porting microstructures and dissipative Euler flows,2013,preprint. arXiv 1302.2815
5. T. Buckmaster, C. DeLellis, P. Isett, and L. Székelyhidi, Jr., Anomalous dissipation for 1/5-Hölder Euler flows, Ann. of Math. (2) 182 no. 1(2015),127-172. MR 3374958. Zbl 1330.35303. https://doi. org/10.4007/annals.2015.182.1.3
6. T. Buckmaster, C. DeLellis, and L. Székelyhidi, Jr., Dissipative Euler flows with Onsager-critical spatial regularity, Comm. PureAppl. Math. 69 no. 9(2016),1613-1670. MR 3530360. Zbl 1351.35109. https: //doi.org/10.1002/cpa.21586
7. A. Cheskidov, P. Constantin, S. Friedlander, and R. Shvydkoy, Energy conservation and Onsager's conjecture for the Eulere quations, Nonlinearity 21 no. 6(2008),1233-1252. MR 2422377. Zbl 1138.76020. https://doi.org/10.1088/0951-7715/21/6/005
8. A. Cheskidov, M. LopesFilho, H. NussenzveigLopes, and R. Shvydkoy, On the Onsager conjecture in two dimensions,2015. arXiv 1509.03213
9. A. Cheskidov, R. Shvydkoy, and S. Friedlander, A continuous model for turbulent energy cascade,in Mathematical Aspect sofFluid Mechanics, LondonMath. Soc. LectureNoteSer. 402, CambridgeUniv. Press, Cambridge,2012,pp.52-69. MR 3050291. Zbl 1296.76059. https://doi.org/10.1017/CBO9781139235792.004
10. A. Choffrut, h-principles for the incompressible Eulere quations, Arch. Ration. Mech. Anal. 210 no. 1(2013),133-163. MR 3073150. Zbl 1291.35200. https://doi.org/10.1007/s00205-013-0639-3
11. A. Choffrut, C. DeLellis, and L. Székelyhidi, Jr., Dissipative continuous Euler flows in two and three dimensions,2012, preprint. arXiv 1205.1226
12. A. Choffrut and L. Székelyhidi, Jr., Weak solutions to thesta-tionary incompressible Eulere quations, SIAM J. Math. Anal. 46 no. 6 (2014), 4060-4074. MR 3505175. Zbl 1311.35194. https:// doi. org/10. 1137/140957354
13. P. Constantin, W. E, and E. S.Titi, Onsager's conjecture on the energy conservation for solutions of Euler'sequation, Comm. Math. Phys. 165 no. 1(1994),207-209. MR 1298949. Zbl 0818.35085. https: // doi.org/10.1007/BF02099744
14. 1,α isometric embeddings,in Nonlinear Partial Differential Equations, AbelSymp. 7, Springer, Heidelberg, 2012, pp. 83-116. MR 3289360. Zbl 1255.53038. https:// doi. org/10.1007/ 978-3-642-25361-4 5
15. D. Cordoba, D. Faraco, and F. Gancedo, Lack of uniqueness for weak solutions of the incompressible porous media equation, Arch. Ra-tion. Mech. Anal. 200 no. 3(2011),725-746. MR 2796131. Zbl 1241. 35156. https://doi.org/10.1007/s00205-010-0365-z
16. S. Daneri and L. Székelyhidi, Jr., Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Eulere quations,2016. arXiv 1603.09714
17. C. DeLellis and L. Székelyhidi, Jr., Continuous dissipative Euler flows and a conjecture of Onsager,in European Congress of Mathematics, Eur. Math. Soc., Zürich,2013,pp.13-29. MR 3469113. Zbl 1364.35250
18. C. DeLellis and L. Székelyhidi, Jr., Dissipative continuous Euler ows, Invent. Math. 193 no. 2(2013),377-407. MR 3090182. Zbl 1280. 35103. https://doi.org/10.1007/s00222-012-0429-9
19. C. DeLellis and L. Székelyhidi, Jr., Dissipative Euler ows and Onsager's conjecture, J. Eur. Math. Soc. (JEMS) 16 no. 7(2014),1467-1505. MR 3254331. Zbl 1307.35205. https://doi.org/10.4171/JEMS/ 466
20. C. DeLellis and L. Székelyhidi, Jr., On h-principle and On-sager's conjecture, Eur. Math. Soc. Newsl. no.95(2015),19-24. MR 3330471. Zbl 1335.35189. Availableat http://www.ems-ph.org/ journals/newsletter/pdf/2015-03-95.pdf
21. J. Duchon and R. Robert, Inertial energy dissipation for weak so-lutions of incompressible Eulerand Navier-Stoke sequations, Nonlinearity 13 no. 1(2000),249-255. MR 1734632. Zbl 1009.35062. https: //doi.org/10.1088/0951-7715/13/1/312
22. G. L.Eyink, Energy dissipation without viscosity in ideal hydrody-namics. I.Fourier analysis and local energy transfer, Phys. D 78 no. 3-4 (1994),222-240. MR 1302409. Zbl 0817.76011. https://doi.org/10. 1016/0167-2789(94)90117-1
23. G. L.Eyink and K. R.Sreenivasan, Onsager and the theory of hydro dynamic turbulence, Rev. ModernPhys. 78 no. 1(2006),87-135. MR 2214822. Zbl 1205.01032. https://doi.org/10.1103/RevModPhys. 78.87
24. P. Isett, Hölder Continuous Euler Flows with Compact Supportin Time, Pro QuestLLC, AnnArbor, MI, 2013, Thesis (Ph.D.)-Princeton University. MR 3153420. Available at http:// gateway. proquest. com/ openurl?url ver=Z39.88-2004&rft val fmt=info:ofi/fmt:kev:mtx: dissertation&res dat=xri:pqm&rft dat=xri:pqdiss:3562201
25. P. Isett, Regularity in time along the coarse scale flow for the Euler equations, 2013, preprint. arXiv 1307.0565
26. P. Isett, Hölder continuous Euler flows in three dimension swith com-pact support in time, Annals of Mathematics Studies 196, Princeton University Press, Princeton, NJ, 2017. MR 3616258. Zbl 1367.35001. https://doi.org/10.1515/9781400885428
27. P. Isett and S.-J. Oh, Aheat flow approach to Onsager's conjecture for the Eulere quations on manifolds, Trans. Amer. Math. Soc. 368 no. 9 (2016), 6519-6537. MR 3461041. Zbl 1335.58018. https://doi.org/10. 1090/tran/6549
28. P. Isett and S.-J. Oh, Aheat flow approach to Onsager's conjecture for the Eulere quations on manifolds, Trans. Amer. Math. Soc. 368 no. 9 (2016), 6519-6537. MR 3461041. Zbl 1335.58018. https://doi.org/10. 1090/tran/6549
29. P. Isett and V. Vicol, Hölder continuous solution so factives calar equations, Ann. PDE 1 no. 1(2015),Art.2,77. MR 3479065. Zbl 06919574. https://doi.org/10.1007/s40818-015-0002-0
30. X. Luo and R. Shvydkoy, 2D homogeneous solutions to the Euler equation, Comm. Partial Differential Equations 40 no. 9(2015),1666-1687. MR 3359160. Zbl 1329.35237. https://doi.org/10.1080/03605302. 2015.1045073
31. J. Nash, C1 isometric imbeddings, Ann. of Math. (2) 60 (1954), 383-396. MR 0065993. Zbl 0058.37703. https://doi.org/10.2307/1969840
32. L. Onsager, Statistical hydro dynamics, Nuovo Cimento (9) 6 no. Sup-plemento, 2(Convegno Internazionale di MeccanicaStatistica) (1949), 279-287. MR 0036116
33. J. C.Robinson, J. L.Rodrigo, and J. W.Skipper, A simple inte-gral condition for energy conservation in the 3d Euler equations, 2016. arXiv 1611.00181
34. V. Scheffer, An inviscid flow with compact supportin space-time, J. Geom. Anal. 3 no. 4(1993),343-401. MR 1231007. Zbl 0836.76017. https://doi.org/10.1007/BF02921318
35. A. Shnirelman, On the non uniqueness of weak solution of the Eu-ler equation, Comm. PureAppl. Math. 50 no. 12(1997),1261-1286. MR 1476315. Zbl 0909.35109. https://doi.org/fm39m4
36. A. Shnirelman, Weak solutions with decreas in genergy of incompress-ible Eulere quations, Comm. Math. Phys. 210 no. 3(2000),541-603. MR 1777341. Zbl 1011.35107. https://doi.org/10.1007/s002200050791
37. R. Shvydkoy, On the energy of inviscid singular flows, J. Math. Anal. Appl. 349 no. 2(2009),583-595. MR 2456214. Zbl 1184.35256. https: //doi.org/10.1016/j.jmaa.2008.09.007
38. R. Shvydkoy, Lectures on the Onsager conjecture, Discrete Contin. Dyn. Syst. Ser. S 3 no. 3(2010),473-496. MR 2660721. Zbl 1210. 76086. https://doi.org/10.3934/dcdss.2010.3.473
39. R. Shvydkoy, Convex integration for a class of actives calare quations, J. Amer. Math. Soc. 24 no. 4(2011),1159-1174. MR 2813340. Zbl 1231. 35177. https://doi.org/10.1090/S0894-0347-2011-00705-4
40. R. Shvydkoy, Homogeneous solutions to the 3D Euler system, Trans. Amer. Math. Soc. 370 no. 4(2018),2517-2535. MR 3748575. Zbl 1386. 35333. https://doi.org/10.1090/tran/7022
41. T. Tao and L. Zhang, Hölder continuous solutions of Boussinesq equa-tion with compact support, J. Funct. Anal. 272 no. 10(2017),4334-4402. MR 3626042. Zbl 1375.35351. https://doi.org/10.1016/j.jfa.2017. 01.013.
42. T. Tao and L. Zhang, On the continuous periodic weak solutions of Boussinesq equations, SIAM J. Math. Anal. 50 no. 1(2018),1120-1162. MR 3763091. Zbl 06840324. https://doi.org/10.1137/17M1115526