Abstract
We show that the Reynolds defect measure for a dissipative weak solution of the compressible Euler system vanishes for large time. This may be seen as a piece of evidence that the dissipative solutions are asymptotically close to weak solutions in the turbulent regime, whence suitable for describing compressible fluid flows in the long run.
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D. Breit, E. Feireisl, and M. Hofmanová. Dissipative solutions and semiflow selection for the complete Euler system. Comm. Math. Phys., 376(2):1471–1497, 2020.
D. Breit, E. Feireisl, and M. Hofmanová. Solution semiflow to the isentropic Euler system. Arch. Ration. Mech. Anal., 235(1):167–194, 2020.
E. Chiodaroli. A counterexample to well-posedness of entropy solutions to the compressible Euler system. J. Hyperbolic Differ. Equ., 11(3):493–519, 2014.
E. Chiodaroli, C. De Lellis, and O. Kreml. Global ill-posedness of the isentropic system of gas dynamics. Comm. Pure Appl. Math., 68(7):1157–1190, 2015.
E. Chiodaroli and O. Kreml. On the energy dissipation rate of solutions to the compressible isentropic Euler system. Arch. Ration. Mech. Anal., 214(3):1019–1049, 2014.
E. Chiodaroli, O. Kreml, V. Mácha, and S. Schwarzacher. Non–uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data. Trans. Amer. Math. Soc., 374(4): 2269–2295, 2021.
C. M. Dafermos. The entropy rate admissibility criterion for solutions of hyperbolic conservation laws. J. Differential Equations, 14:202–212, 1973.
C. M. Dafermos. Maximal dissipation in equations of evolution. J. Differential Equations, 252(1):567–587, 2012.
C.M. Dafermos. The second law of thermodynamics and stability. Arch. Rational Mech. Anal., 70:167–179, 1979.
E. Feireisl and M. Hofmanová. On convergence of approximate solutions to the compressible Euler system. Ann. PDE, 6(2):24, 2020. (Paper No. 11)
E. Feireisl, M. Lukáčová-Medvidová, and H. Mizerová. Convergence of finite volume schemes for the Euler equations via dissipative measure–valued solutions. Found. Comput. Math., 20(4):923–966, 2020.
E. Feireisl, M. Lukáčová-Medvidová, H. Mizerová, B. She, and Wang. Computing oscillatory solutions to the Euler system via K-convergence. 2019. arxiv preprint No.arXiv:1910.03161.
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Dedicated to Matthias Hieber on the occasion of his 60-th birthday.
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The research of E.F. leading to these results has received funding from the Czech Sciences Foundation (GAČR), Grant Agreement 18-05974S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.
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Feireisl, E. A note on the long-time behavior of dissipative solutions to the Euler system. J. Evol. Equ. 21, 2807–2814 (2021). https://doi.org/10.1007/s00028-021-00696-0
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DOI: https://doi.org/10.1007/s00028-021-00696-0