Abstract
Recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces. The motivation comes from Onsager’s conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field.
In a recent paper the third author has improved upon the methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent — albeit weaker than the one conjectured by Onsager. In this paper we give a shorter proof of this final result, adhering more to the original scheme of the second and fourth authors and introducing some new devices. More precisely we show that for any positive $\varepsilon$, there exist periodic solutions of the 3D incompressible Euler equations that dissipate the total kinetic energy and belong to the Hölder class $C^{1/5\varepsilon}$.

[Buckmaster] T. Buckmaster, "Onsager’s conjecture almost everywhere in time," Comm. Math. Phys., vol. 333, p. 24, 2015.
@article{Buckmaster,
author = {Buckmaster, T.},
title = {Onsager's conjecture almost everywhere in time},
journal={Comm. Math. Phys.},
VOLUME={333},
YEAR={2015},
NOTE={published online 7 January 2015},
PAGES={24pp.},
DOI={10.1007/s002200142262z},
} 
[CCFS2007] A. Cheskidov, P. Constantin, S. Friedlander, and R. Shvydkoy, "Energy conservation and Onsager’s conjecture for the Euler equations," Nonlinearity, vol. 21, iss. 6, pp. 12331252, 2008.
@article{CCFS2007, mrkey = {2422377},
author = {Cheskidov, A. and Constantin, P. and Friedlander, S. and Shvydkoy, R.},
title = {Energy conservation and {O}nsager's conjecture for the {E}uler equations},
journal = {Nonlinearity},
fjournal = {Nonlinearity},
volume = {21},
year = {2008},
number = {6},
pages = {12331252},
issn = {09517715},
coden = {NONLE5},
mrclass = {76B03 (76F02)},
mrnumber = {2422377},
mrreviewer = {Hee Chul Pak},
doi = {10.1088/09517715/21/6/005},
zblnumber = {1138.76020},
} 
[ConstantinETiti] P. Constantin, W. E, and E. S. Titi, "Onsager’s conjecture on the energy conservation for solutions of Euler’s equation," Comm. Math. Phys., vol. 165, iss. 1, pp. 207209, 1994.
@article{ConstantinETiti, mrkey = {1298949},
author = {Constantin, Peter and E, Weinan and Titi, Edriss S.},
title = {Onsager's conjecture on the energy conservation for solutions of {E}uler's equation},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {165},
year = {1994},
number = {1},
pages = {207209},
issn = {00103616},
coden = {CMPHAY},
mrclass = {76C99 (35Q30 76F99)},
mrnumber = {1298949},
zblnumber = {0818.35085},
doi = {10.1007/BF02099744},
} 
[CDSz] S. Conti, C. De Lellis, and L. relax Székelyhidi Jr., "$h$principle and rigidity for $C^{1,\alpha}$ isometric embeddings," in Nonlinear Partial Differential Equations, New York: SpringerVerlag, 2012, vol. 7, pp. 83116.
@incollection{CDSz,
author = {Conti, S. and De Lellis, C. and {\relax Sz{é}kelyhidi, Jr.},
L.},
title = {$h$principle and rigidity for {$C^{1,\alpha}$} isometric embeddings},
booktitle = {Nonlinear Partial Differential Equations},
volume = {7},
series = {Abel Symposia},
year = {2012},
pages = {83116},
publisher = {SpringerVerlag},
address = {New York},
zblnumber = {1255.53038},
} 
[MR0213764] R. Courant, K. Friedrichs, and H. Lewy, "On the partial difference equations of mathematical physics," IBM J. Res. Develop., vol. 11, pp. 215234, 1967.
@article{MR0213764, mrkey = {0213764},
author = {Courant, R. and Friedrichs, K. and Lewy, H.},
title = {On the partial difference equations of mathematical physics},
journal = {IBM J. Res. Develop.},
fjournal = {International Business Machines Corporation. Journal of Research and Development},
volume = {11},
year = {1967},
pages = {215234},
issn = {00188646},
mrclass = {39.20 (69.00)},
mrnumber = {0213764},
doi = {10.1147/rd.112.0215},
zblnumber = {0145.40402},
} 
[Daneri] S. Daneri, "Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations," Comm. Math. Phys., vol. 329, iss. 2, p. 745, 2014.
@article{Daneri,
author = {Daneri, S.},
title = {Cauchy problem for dissipative {H}ölder solutions to the incompressible {E}uler equations},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {329},
number = {2},
year = {2014},
pages = {745—786},
zblnumber = {1298.35140},
doi = {10.1007/s0022001419735},
} 
[DS1] C. De Lellis and L. Székelyhidi Jr., "The Euler equations as a differential inclusion," Ann. of Math., vol. 170, iss. 3, pp. 14171436, 2009.
@article{DS1, mrkey = {2600877},
author = {De Lellis, Camillo and Sz{é}kelyhidi, Jr., L{á}szl{ó}},
title = {The {E}uler equations as a differential inclusion},
journal = {Ann. of Math.},
fjournal = {Annals of Mathematics. Second Series},
volume = {170},
year = {2009},
number = {3},
pages = {14171436},
issn = {0003486X},
coden = {ANMAAH},
mrclass = {35Q31 (34A60 35D30 76B03)},
mrnumber = {2600877},
mrreviewer = {Fr{é}d{é}ric Charve},
doi = {10.4007/annals.2009.170.1417},
zblnumber = {05710190},
} 
[DS2] C. De Lellis and L. Székelyhidi Jr., "On admissibility criteria for weak solutions of the Euler equations," Arch. Ration. Mech. Anal., vol. 195, iss. 1, pp. 225260, 2010.
@article{DS2, mrkey = {2564474},
author = {De Lellis, Camillo and Sz{é}kelyhidi, Jr., L{á}szl{ó}},
title = {On admissibility criteria for weak solutions of the {E}uler equations},
journal = {Arch. Ration. Mech. Anal.},
fjournal = {Archive for Rational Mechanics and Analysis},
volume = {195},
year = {2010},
number = {1},
pages = {225260},
issn = {00039527},
mrclass = {35Q31 (35A02 35L65 76N15)},
mrnumber = {2564474},
mrreviewer = {Stefano Bianchini},
doi = {10.1007/s002050080201x},
zblnumber = {1192.35138},
} 
@misc{DS4,
author = {De Lellis, Camillo and Sz{é}kelyhidi, Jr., L{á}szl{ó}},
title = {Dissipative {E}uler flows and {O}nsager's conjecture},
fjournal = {Journal of the European Mathematical Society (JEMS)},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {16},
number = {7},
pages = {14671505},
mrnumber = {3254331},
zblnumber = {06353607},
doi = {10.4171/JEMS/466},
} 
[DSSurvey] C. De Lellis and L. Székelyhidi Jr., "The $h$principle and the equations of fluid dynamics," Bull. Amer. Math. Soc., vol. 49, iss. 3, pp. 347375, 2012.
@article{DSSurvey, mrkey = {2917063},
author = {De Lellis, Camillo and Sz{é}kelyhidi, Jr., L{á}szl{ó}},
title = {The {$h$}principle and the equations of fluid dynamics},
journal = {Bull. Amer. Math. Soc.},
fjournal = {American Mathematical Society. Bulletin. New Series},
volume = {49},
year = {2012},
number = {3},
pages = {347375},
issn = {02730979},
coden = {BAMOAD},
mrclass = {76F02 (35D30 35Q31 35Q35)},
mrnumber = {2917063},
mrreviewer = {Isabelle Gruais},
doi = {10.1090/S027309792012013769},
zblnumber = {1254.35180},
} 
[DS3] C. De Lellis and L. Székelyhidi Jr., "Dissipative continuous Euler flows," Invent. Math., vol. 193, iss. 2, pp. 377407, 2013.
@article{DS3, mrkey = {3090182},
author = {De Lellis, Camillo and Sz{é}kelyhidi, Jr., L{á}szl{ó}},
title = {Dissipative continuous {E}uler flows},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {193},
year = {2013},
number = {2},
pages = {377407},
issn = {00209910},
mrclass = {35Q31 (35A01 35B10 35B65 35D30 76B03)},
mrnumber = {3090182},
mrreviewer = {Francesco Fanelli},
doi = {10.1007/s0022201204299},
zblnumber = {1280.35103},
} 
[RobertDuchon] J. Duchon and R. Robert, "Inertial energy dissipation for weak solutions of incompressible Euler and NavierStokes equations," Nonlinearity, vol. 13, iss. 1, pp. 249255, 2000.
@article{RobertDuchon, mrkey = {1734632},
author = {Duchon, Jean and Robert, Raoul},
title = {Inertial energy dissipation for weak solutions of incompressible {E}uler and {N}avier{S}tokes equations},
journal = {Nonlinearity},
fjournal = {Nonlinearity},
volume = {13},
year = {2000},
number = {1},
pages = {249255},
issn = {09517715},
coden = {NONLE5},
mrclass = {76D05 (35Q30 76B99)},
mrnumber = {1734632},
mrreviewer = {Emmanuel Grenier},
doi = {10.1088/09517715/13/1/312},
zblnumber = {1009.35062},
} 
[Eyink] G. L. Eyink, "Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer," Phys. D, vol. 78, iss. 34, pp. 222240, 1994.
@article{Eyink, mrkey = {1302409},
author = {Eyink, Gregory L.},
title = {Energy dissipation without viscosity in ideal hydrodynamics. {I}. {F}ourier analysis and local energy transfer},
journal = {Phys. D},
fjournal = {Physica D. Nonlinear Phenomena},
volume = {78},
year = {1994},
number = {34},
pages = {222240},
issn = {01672789},
coden = {PDNPDT},
mrclass = {76D05 (35Q35 76F99)},
mrnumber = {1302409},
mrreviewer = {P. L. Sulem},
doi = {10.1016/01672789(94)901171},
zblnumber = {0817.76011},
} 
[EyinkSreenivasan] G. L. Eyink and K. R. Sreenivasan, "Onsager and the theory of hydrodynamic turbulence," Rev. Modern Phys., vol. 78, iss. 1, pp. 87135, 2006.
@article{EyinkSreenivasan, mrkey = {2214822},
author = {Eyink, Gregory L. and Sreenivasan, Katepalli R.},
title = {Onsager and the theory of hydrodynamic turbulence},
journal = {Rev. Modern Phys.},
fjournal = {Reviews of Modern Physics},
volume = {78},
year = {2006},
number = {1},
pages = {87135},
issn = {00346861},
coden = {RMPHAT},
mrclass = {76F02 (01A60 7603 76F55 8203)},
mrnumber = {2214822},
mrreviewer = {Oleg V. Zaboronsky},
doi = {10.1103/RevModPhys.78.87},
zblnumber = {1205.01032},
} 
[FrischBook] U. Frisch, Turbulence, Cambridge: Cambridge Univ. Press, 1995.
@book{FrischBook, mrkey = {1428905},
author = {Frisch, Uriel},
title = {Turbulence},
titlenote = {The legacy of {A. N. K}olmogorov},
publisher = {Cambridge Univ. Press},
address = {Cambridge},
year = {1995},
pages = {xiv+296},
isbn = {0521451035},
mrclass = {7602 (35Q30 76D05 76Fxx 76M35)},
mrnumber = {1428905},
mrreviewer = {Philip J. Holmes},
zblnumber = {0832.76001},
} 
[GT] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: SpringerVerlag, 2001.
@book{GT, mrkey = {1814364},
author = {Gilbarg, David and Trudinger, Neil S.},
title = {Elliptic Partial Differential Equations of Second Order},
series = {Classics in Math.},
note = {reprint of the 1998 edition},
publisher = {SpringerVerlag},
year = {2001},
pages = {xiv+517},
isbn = {3540411607},
mrclass = {3502 (35Jxx)},
mrnumber = {1814364},
address = {New York},
zblnumber = {1042.35002},
} 
@book{Gromov, mrkey = {0864505},
author = {Gromov, Mikhael},
title = {Partial Differential Relations},
series = {Ergeb. Math. Grenzgeb.},
volume = {9},
publisher = {SpringerVerlag},
year = {1986},
pages = {x+363},
isbn = {3540121773},
mrclass = {58G99 (35A99 35B99 53C42 5802)},
mrnumber = {0864505},
mrreviewer = {HungHsi Wu},
doi = {10.1007/9783662022672},
address = {New York},
zblnumber = {0651.53001},
} 
[Isett] P. Isett, Hölder continuous Euler flows in three dimensions with compact support in time, 2012.
@misc{Isett,
author = {Isett, P.},
title = {H{ö}lder continuous {E}uler flows in three dimensions with compact support in time},
note = {preprint},
year = {2012},
} 
[Kolmogorov] A. N. Kolmogorov, "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers," Proc. Roy. Soc. London Ser. A, vol. 434, iss. 1890, pp. 913, 1991.
@article{Kolmogorov, mrkey = {1124922},
author = {Kolmogorov, A. N.},
title = {The local structure of turbulence in incompressible viscous fluid for very large {R}eynolds numbers},
note = {translated from the Russian by V. Levin, Turbulence and stochastic processes: Kolmogorov's ideas 50 years on},
journal = {Proc. Roy. Soc. London Ser. A},
fjournal = {Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences},
volume = {434},
year = {1991},
number = {1890},
pages = {913},
issn = {09628444},
coden = {PRLAAZ},
mrclass = {76F05},
mrnumber = {1124922},
doi = {10.1098/rspa.1991.0075},
zblnumber = {1142.76389},
} 
@article{Nash54, mrkey = {0065993},
author = {Nash, John},
title = {{$C\sp 1$} isometric imbeddings},
journal = {Ann. of Math.},
fjournal = {Annals of Mathematics. Second Series},
volume = {60},
year = {1954},
pages = {383396},
issn = {0003486X},
mrclass = {53.0X},
mrnumber = {0065993},
mrreviewer = {S. Chern},
doi = {10.2307/1969840},
zblnumber = {0058.37703},
} 
[Onsager] L. Onsager, "Statistical hydrodynamics," Nuovo Cimento, vol. 6, iss. Supplemento, 2(Convegno Internazionale di Meccanica Statistica), pp. 279287, 1949.
@article{Onsager, mrkey = {0036116},
author = {Onsager, L.},
title = {Statistical hydrodynamics},
journal = {Nuovo Cimento},
volume = {6},
year = {1949},
number = {Supplemento, 2(Convegno Internazionale di Meccanica Statistica)},
pages = {279287},
mrclass = {76.1X},
mrnumber = {0036116},
mrreviewer = {J. V. Wehausen},
} 
[Robert] R. Robert, "Statistical hydrodynamics (Onsager revisited)," in Handbook of Mathematical Fluid Dynamics, Vol. II, Amsterdam: NorthHolland, 2003, pp. 154.
@incollection{Robert, mrkey = {1983588},
author = {Robert, Raoul},
title = {Statistical hydrodynamics ({O}nsager revisited)},
booktitle = {Handbook of Mathematical Fluid Dynamics, {V}ol. {II}},
pages = {154},
publisher = {NorthHolland},
address = {Amsterdam},
year = {2003},
mrclass = {76A02 (35Q35 76B03 76D06 76F55 76M35 82D15)},
mrnumber = {1983588},
mrreviewer = {Benedetta Ferrario},
doi = {10.1016/S18745792(03)800034},
} 
[Scheffer93] V. Scheffer, "An inviscid flow with compact support in spacetime," J. Geom. Anal., vol. 3, iss. 4, pp. 343401, 1993.
@article{Scheffer93, mrkey = {1231007},
author = {Scheffer, Vladimir},
title = {An inviscid flow with compact support in spacetime},
journal = {J. Geom. Anal.},
fjournal = {The Journal of Geometric Analysis},
volume = {3},
year = {1993},
number = {4},
pages = {343401},
issn = {10506926},
mrclass = {35Q35 (28A80 76C99)},
mrnumber = {1231007},
mrreviewer = {Helena J. Nussenzveig Lopes},
doi = {10.1007/BF02921318},
zblnumber = {0836.76017},
} 
[Shnirelmandecrease] A. Shnirelman, "Weak solutions with decreasing energy of incompressible Euler equations," Comm. Math. Phys., vol. 210, iss. 3, pp. 541603, 2000.
@article{Shnirelmandecrease, mrkey = {1777341},
author = {Shnirelman, A.},
title = {Weak solutions with decreasing energy of incompressible {E}uler equations},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {210},
year = {2000},
number = {3},
pages = {541603},
issn = {00103616},
coden = {CMPHAY},
mrclass = {76B03 (35D05 35Q30 76F99)},
mrnumber = {1777341},
mrreviewer = {Emmanuel Grenier},
doi = {10.1007/s002200050791},
zblnumber = {1011.35107},
}