Abstract
This note addresses the issue of energy conservation for the 2D Euler system with an L p-control on vorticity. We provide a direct argument, based on a mollification in physical space, to show that the energy of a weak solution is conserved if \({\omega = \nabla \times u \in L^{\frac{3}{2}}}\). An example of a 2D field in the class \({\omega \in L^{\frac{3}{2} - \epsilon}}\) for any \(\epsilon > 0\), and \({u \in B^{1/3}_{3,\infty}}\) (Onsager critical space, see Shvydkoy in Discr Contin Dyn Syst Ser S 3(3):473–496, 2010) is constructed with non-vanishing energy flux. This demonstrates sharpness of the kinematic argument, which does not differentiate between 2D and 3D, and requires Onsager’s regularity control on the solution. Next, we show that for physically realizable solutions there is a mechanism preventing the anomalous dissipation in 2D that does not require such a control. Namely, we prove that any solution to the Euler equations produced via a vanishing viscosity limit from the Navier–Stokes equations, with \({\omega \in L^p}\), for p > 1, conserves energy.
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References
Bardos C., Titi E.S.: Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations. Discr. Contin. Dyn. Syst. Ser. S 3(2), 185–197 (2010)
Buckmaster, T., De Lellis, C., Székelyhidi, L.: Dissipative Euler flows with Onsager-critical spatial regularity. To appear in Commun. Tur. Appl. Math. http://arxiv.org/abs/1404.6915
Buckmaster T., De Lellis C., Isett P., Székelyhidi L. Jr: Anomalous dissipation for 1/5-Hölder Euler flows. Ann. Math. 182, 127–172 (2015)
Cheskidov A., Constantin P., Friedlander S., Shvydkoy R.: Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity 21(6), 1233–1252 (2008)
Cheskidov A., Shvydkoy R.: Ill-posedness of the basic equations of fluid dynamics in Besov spaces. Proc. Am. Math. Soc. 138(3), 1059–1067 (2010)
Choffrut A.: h-Principles for the incompressible Euler equations. Arch. Ration. Mech. Anal. 210(1), 133–163 (2013)
Constantin P., Weinan E., Titi Edriss S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165(1), 207–209 (1994)
Crippa G., Spirito S.: Renormalized solutions of the 2D Euler equations. Commun. Math. Phys. 339(1), 191–198 (2015)
Lellis C., Székelyhidi L. Jr: The Euler equations as a differential inclusion. Ann. Math. (2) 170(3), 1417–1436 (2009)
DiPerna R.J., Majda A.J.: Concentrations in regularizations for 2-D incompressible flow. Commun. Pure Appl. Math. 40(3), 301–345 (1987)
Duchon J., Robert R.: Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13(1), 249–255 (2000)
Eyink G.L.: Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Phys. D 78(3–4), 222–240 (1994)
Eyink G.L., Sreenivasan K.R.: Onsager and the theory of hydrodynamic turbulence. Rev. Modern Phys. 78(1), 87–135 (2006)
Isett, P.: Holder continuous Euler flows with compact support in time. ProQuest LLC, Ann Arbor, MI. Thesis Ph.D., Princeton University (2013)
Lopes Filho M.C., Nussenzveig Lopes H.J., Tadmor E.: Approximate solutions of the incompressible Euler equations with no concentrations. Ann. Inst. H. Poincaré Anal. Non Linéaire 17(3), 371–412 (2000)
Luo X., Shvydkoy R.: 2D homogeneous solutions to the Euler equation. Commun. Partial Differ. Equ. 40(9), 1666–1687 (2015)
Onsager L.: Statistical hydrodynamics. Nuovo Cimento (9) 6(Supplemento, 2 (Convegno Internazionale di Meccanica Statistica)), 279–287 (1949)
Scheffer V.: An inviscid flow with compact support in space-time. J. Geom. Anal. 3(4), 343–401 (1993)
Shnirelman A.: On the nonuniqueness of weak solution of the Euler equation. Commun. Pure Appl. Math. 50(12), 1261–1286 (1997)
Shvydkoy R.: On the energy of inviscid singular flows. J. Math. Anal. Appl. 349(2), 583–595 (2009)
Shvydkoy R.: Lectures on the Onsager conjecture. Discr. Contin. Dyn. Syst. Ser. S 3(3), 473–496 (2010)
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Cheskidov, A., Filho, M.C.L., Lopes, H.J.N. et al. Energy Conservation in Two-dimensional Incompressible Ideal Fluids. Commun. Math. Phys. 348, 129–143 (2016). https://doi.org/10.1007/s00220-016-2730-8
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DOI: https://doi.org/10.1007/s00220-016-2730-8