Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A heat flow approach to Onsager's conjecture for the Euler equations on manifolds

Authors: Philip Isett and Sung-Jin Oh
Journal: Trans. Amer. Math. Soc. 368 (2016), 6519-6537
MSC (2010): Primary 58J35, 35Q31
Published electronically: November 17, 2015
MathSciNet review: 3461041
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a simple proof of Onsager's conjecture concerning energy conservation for weak solutions to the Euler equations on any compact Riemannian manifold, extending the results of Constantin-E-Titi and Cheskidov-Constantin-Friedlander-Shvydkoy in the flat case. When restricted to $ \mathbb{T}^{d}$ or $ \mathbb{R}^{d}$, our approach yields an alternative proof of the sharp result of the latter authors.

Our method builds on a systematic use of a smoothing operator defined via a geometric heat flow, which was considered by Milgram-Rosenbloom as a means to establish the Hodge theorem. In particular, we present a simple and geometric way to prove the key nonlinear commutator estimate, whose proof previously relied on a delicate use of convolutions.

References [Enhancements On Off] (What's this?)

  • [1] Thierry Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1636569 (99i:58001)
  • [2] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275 (58 #2349)
  • [3] T. Buckmaster, Onsager's conjecture almost everywhere in time, (2013).
  • [4] T. Buckmaster, C. De Lellis, and L. Székelyhidi, Jr., Transporting microstructure and dissipative Euler flows, (2013).
  • [5] A. Cheskidov, P. Constantin, S. Friedlander, and R. Shvydkoy, Energy conservation and Onsager's conjecture for the Euler equations, Nonlinearity 21 (2008), no. 6, 1233-1252. MR 2422377 (2009g:76008),
  • [6] Peter Constantin, Weinan E, and Edriss S. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Comm. Math. Phys. 165 (1994), no. 1, 207-209. MR 1298949 (96e:76025)
  • [7] Antoine Choffrut, $ h$-principles for the incompressible Euler equations, Arch. Ration. Mech. Anal. 210 (2013), no. 1, 133-163. MR 3073150,
  • [8] A. Choffrut, C. De Lellis, and L. Székelyhidi, Jr.,
    Dissipative continuous Euler flows in two and three dimensions, Preprint.
  • [9] C. De Lellis and L. Székelyhidi, Jr., The Euler equations as a differential inclusion, Annals of Mathematics (2009).
  • [10] C. De Lellis and L. Székelyhidi, Jr., Dissipative continuous Euler flows, (2012).
  • [11] C. De Lellis and L. Székelyhidi, Jr., Dissipative Euler Flows and Onsager's Conjecture, (2012).
  • [12] Jean Duchon and Raoul Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity 13 (2000), no. 1, 249-255. MR 1734632 (2001c:76032),
  • [13] Gregory L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer, Phys. D 78 (1994), no. 3-4, 222-240. MR 1302409 (95m:76020),
  • [14] E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University Courant Institute of Mathematical Sciences, New York, 1999. MR 1688256 (2000e:58011)
  • [15] P. Isett, Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time, (2012).
  • [16] S. Klainerman and I. Rodnianski, A geometric approach to the Littlewood-Paley theory, Geom. Funct. Anal. 16 (2006), no. 1, 126-163. MR 2221254 (2007e:58046),
  • [17] L. Onsager, Statistical hydrodynamics, Nuovo Cimento (9) 6 (1949), no. Supplemento, 2(Convegno Internazionale di Meccanica Statistica), 279-287. MR 0036116 (12,60f)
  • [18] A. N. Milgram and P. C. Rosenbloom, Harmonic forms and heat conduction. I. Closed Riemannian manifolds, Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 180-184. MR 0042769 (13,160a)
  • [19] Roman Shvydkoy, Lectures on the Onsager conjecture, Discrete Contin. Dyn. Syst. Ser. S 3 (2010), no. 3, 473-496. MR 2660721 (2011h:76051),
  • [20] Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory., Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961 (40 #6176)
  • [21] Robert S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), no. 1, 48-79. MR 705991 (84m:58138),
  • [22] Hans Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992. MR 1163193 (93f:46029)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 58J35, 35Q31

Retrieve articles in all journals with MSC (2010): 58J35, 35Q31

Additional Information

Philip Isett
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Sung-Jin Oh
Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720

Received by editor(s): May 4, 2014
Received by editor(s) in revised form: August 25, 2015
Published electronically: November 17, 2015
Additional Notes: The second author is a Miller research fellow, and would like to thank the Miller Institute for support
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society