Abstract
We give sufficient conditions on the regularity of solutions to the inhomogeneous incompressible Euler and the compressible isentropic Euler systems in order for the energy to be conserved. Our strategy relies on commutator estimates similar to those employed by Constantin et al. for the homogeneous incompressible Euler equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Buckmaster T., De Lellis C., Isett P., Székelyhidi L. Jr.: Anomalous dissipation for 1/5-Hölder Euler flows. Ann. Math. (2) 182(1), 127–172 (2015)
Buckmaster, T., De Lellis, C., Székelyhidi, Jr., L.: Dissipative Euler flows with Onsager-critical spatial regularity. Comm. Pure Appl. Math. (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)
Chiodaroli E.: A counterexample to well-posedness of entropy solutions to the compressible Euler system. J. Hyperbolic Differ. Equ. 11(3), 493–519 (2014)
Chiodaroli E., De Lellis C., Kreml O.: Global ill-posedness of the isentropic system of gas dynamics. Comm. Pure Appl. Math. 68(7), 1157–1190 (2015)
Constantin, P.,W. E, Titi, E. S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm. Math. Phys., 165(1), 207–209 (1994)
Danchin R.: On the well-posedness of the incompressible density-dependent Euler equations in the L p framework. J. Differ. Equ. 248(8), 2130–2170 (2010)
Danchin R., Fanelli F.: The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces. J. Math. Pures Appl. (9) 96(3), 253–278 (2011)
De Lellis C., Székelyhidi L. Jr.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260 (2010)
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)
Feireisl E.: Maximal dissipation and well-posedness for the compressible Euler system. J. Math. Fluid Mech. 16(3), 447–461 (2014)
Feireisl E., Liao X., Málek J.: Global weak solutions to a class of non-Newtonian compressible fluids. Math. Methods Appl. Sci., 38(16), 3482–3494 (2015)
Frehse, J., Málek, J., Ružicka, M.: Large data existence result for unsteady flows of inhomogeneous shear-thickening heat-conducting incompressible fluids. Comm. Partial Differ. Equ., 35(10), 1891–1919 (2010)
Isett, P.: A proof of Onsager’s conjecture. Preprint, 2016
Leslie T. M., Shvydkoy R.: The energy balance relation for weak solutions of the density-dependent Navier-Stokes equations. J. Differ. Equ. 261(6), 3719–3733 (2016)
Marsden J. E.: Well-posedness of the equations of a non-homogeneous perfect fluid. Comm. Partial Differ. Equ., 1(3), 215–230 (1976)
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.: Weak solutions with decreasing energy of incompressible Euler equations. Comm. Math. Phys., 210(3), 541–603 (2000)
Wróblewska-Kamińska A.: Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces. Discrete Contin. Dyn. Syst. 33(6), 2565–2592 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Constantin
Rights and permissions
About this article
Cite this article
Feireisl, E., Gwiazda, P., Świerczewska-Gwiazda, A. et al. Regularity and Energy Conservation for the Compressible Euler Equations. Arch Rational Mech Anal 223, 1375–1395 (2017). https://doi.org/10.1007/s00205-016-1060-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-016-1060-5