## Statistical solutions and Onsager’s conjecture.(English)Zbl 1398.35153

Summary: We prove a version of Onsager’s conjecture on the conservation of energy for the incompressible Euler equations in the context of statistical solutions, as introduced recently by the first author et al. [Arch. Ration. Mech. Anal. 226, No. 2, 809–849 (2017; Zbl 1373.35193)]. As a byproduct, we also obtain an alternative proof for the conservative direction of Onsager’s conjecture for weak solutions, under a weaker Besov-type regularity assumption than previously known.

### MSC:

 35Q31 Euler equations 76M35 Stochastic analysis applied to problems in fluid mechanics 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 35D30 Weak solutions to PDEs

Zbl 1373.35193
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### References:

 [1] C. Bardos, E.S. Titi, Onsager’s conjecture for the incompressible Euler equations in bounded domains. Preprint, 2017, arXiv:1707.03115. · Zbl 1390.35241 [2] Onsager, Lars, Statistical hydrodynamics, Nuovo Cimento, 6, Suppl., 279-287, (1949) [3] Eyink, G. L.; Sreenivasan, K. R., Onsager and the theory of hydrodynamic turbulence, Rev. Modern Phys., 78, 87-135, (2006) · Zbl 1205.01032 [4] Constantin, P.; E, W.; Titi, E., Onsager’s conjecture on the energy conservation for solutions of euler’s equation, Comm. Math. Phys., 165, 1, 207-209, (1994) · Zbl 0818.35085 [5] Eyink, G. L., Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer, Physica D, 78, 3, 222-240, (1994) · Zbl 0817.76011 [6] Duchon, J.; Robert, R., Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13, 1, 249-255, (2000) · Zbl 1009.35062 [7] Cheskidov, A.; Constantin, P.; Friedlander, S.; Shvydkoy, R., Energy conservation and onsager’s conjecture for the Euler equations, Nonlinearity, 21, 6, 1233-1252, (2008) · Zbl 1138.76020 [8] J.C. Robinson, J.L. Rodrigo, J.W.D. Skipper, A simple integral condition for energy conservation in the 3D Euler equations. Preprint, 2016. [9] Leslie, T. M.; Shvydkoy, R., The energy balance relation for weak solutions of the density-dependent Navier-Stokes equations, J. Differential Equations, 261, 6, 3719-3733, (2016) · Zbl 1383.76080 [10] Feireisl, Eduard; Gwiazda, Piotr; Świerczewska-Gwiazda, Agnieszka; Wiedemann, Emil, Regularity and energy conservation for the compressible Euler equations, Arch. Ration. Mech. Anal., 223, 3, 1375-1395, (2017) · Zbl 1365.35113 [11] T. Drivas, G. Eyink, An Onsager singularity theorem for turbulent solutions of compressible Euler equations. Preprint, 2017, arXiv:1704.03409. · Zbl 1397.35191 [12] P. Isett, A proof of Onsager’s conjecture. Preprint, 2016, arXiv:1608.08301. · Zbl 1335.58018 [13] T. Buckmaster, C. De Lellis, L. Székelyhidi Jr., V. Vicol, Onsager’s conjecture for admissible weak solutions, Preprint, 2017, arXiv:1701.08678. · Zbl 1480.35317 [14] Frisch, U., Turbulence, (1995), Cambridge University Press [15] Foiaş, C., Statistical study of Navier-Stokes equations I, Rend. Semin. Mat. Univ. Padova, 48, 219-348, (1972) · Zbl 0283.76017 [16] Foiaş, C., Statistical study of Navier-Stokes equations II, Rend. Semin. Mat. Univ. Padova, 49, 9-123, (1973) · Zbl 0283.76018 [17] Fjordholm, U. S.; Lanthaler, S.; Mishra, S., Statistical solutions of hyperbolic conservation laws I: foundations, Arch. Ration. Mech. Anal., 226, 2, 809-849, (2017) · Zbl 1373.35193 [18] DiPerna, R. J., Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal., 88, 223-270, (1985) · Zbl 0616.35055 [19] DiPerna, R. J.; Majda, A. J., Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108, 4, 667-689, (1987) · Zbl 0626.35059 [20] Székelyhidi Jr., L.; Wiedemann, E., Young measures generated by ideal incompressible fluid flows, Arch. Ration. Mech. Anal., 206, 1, 333-366, (2012) · Zbl 1256.35072 [21] Kružkov, S. N., First order quasilinear equations with several independent variables, Mat. Sb. (NS), 81, 123, 228-255, (1970) · Zbl 0202.11203 [22] U.S. Fjordholm, S. Mishra, F. Weber, Statistical solutions of the incompressible Euler equations and Kolmogorov’s theory of turbulence, (in preparation), 2017. [23] Fjordholm, U. S.; Mishra, S.; Tadmor, E., On the computation of measure-valued solutions, Acta Numer., 25, 567-679, (2016) · Zbl 1382.76001
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