Approximating viscosity solutions of the Euler system
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- by Eduard Feireisl, Mária Lukáčová-Medvid’ová, Simon Schneider and Bangwei She;
- Math. Comp. 91 (2022), 2129-2164
- DOI: https://doi.org/10.1090/mcom/3738
- Published electronically: June 1, 2022
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Abstract:
Applying the concept of S-convergence, based on averaging in the spirit of Strong Law of Large Numbers, the vanishing viscosity solutions of the Euler system are studied. We show how to efficiently compute a viscosity solution of the Euler system as the S-limit of numerical solutions obtained by the viscosity finite volume method. Theoretical results are illustrated by numerical simulations of the Kelvin–Helmholtz instability problem.References
- Erik J. Balder, Lectures on Young measure theory and its applications in economics, Rend. Istit. Mat. Univ. Trieste 31 (2000), no. suppl. 1, 1–69. Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1997). MR 1798830
- H. Brenner, Kinematics of volume transport, Phys. A 349 (2005), 11–59.
- Howard Brenner, Navier-Stokes revisited, Phys. A 349 (2005), no. 1-2, 60–132. MR 2120925, DOI 10.1016/j.physa.2004.10.034
- H. Brenner, Fluid mechanics revisited, Phys. A 349 (2006) 190–224.
- Alberto Bressan and Ryan Murray, On self-similar solutions to the incompressible Euler equations, J. Differential Equations 269 (2020), no. 6, 5142–5203. MR 4104468, DOI 10.1016/j.jde.2020.04.005
- Tristan Buckmaster, Camillo de Lellis, László Székelyhidi Jr., and Vlad Vicol, Onsager’s conjecture for admissible weak solutions, Comm. Pure Appl. Math. 72 (2019), no. 2, 229–274. MR 3896021, DOI 10.1002/cpa.21781
- Tristan Buckmaster and Vlad Vicol, Convex integration and phenomenologies in turbulence, EMS Surv. Math. Sci. 6 (2019), no. 1-2, 173–263. MR 4073888, DOI 10.4171/emss/34
- Gui-Qiang Chen and Mikhail Perepelitsa, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow, Comm. Pure Appl. Math. 63 (2010), no. 11, 1469–1504. MR 2683391, DOI 10.1002/cpa.20332
- Gui-Qiang G. Chen and James Glimm, Kolmogorov-type theory of compressible turbulence and inviscid limit of the Navier-Stokes equations in $\Bbb R^3$, Phys. D 400 (2019), 132138, 10. MR 4008031, DOI 10.1016/j.physd.2019.06.004
- Elisabetta Chiodaroli, Camillo De Lellis, and Ondřej Kreml, Global ill-posedness of the isentropic system of gas dynamics, Comm. Pure Appl. Math. 68 (2015), no. 7, 1157–1190. MR 3352460, DOI 10.1002/cpa.21537
- Elisabetta Chiodaroli, Ondřej Kreml, Václav Mácha, and Sebastian Schwarzacher, Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data, Trans. Amer. Math. Soc. 374 (2021), no. 4, 2269–2295. MR 4223016, DOI 10.1090/tran/8129
- Camillo De Lellis and László Székelyhidi Jr., Dissipative continuous Euler flows, Invent. Math. 193 (2013), no. 2, 377–407. MR 3090182, DOI 10.1007/s00222-012-0429-9
- Ronald J. DiPerna and Andrew J. Majda, Concentrations in regularizations for $2$-D incompressible flow, Comm. Pure Appl. Math. 40 (1987), no. 3, 301–345. MR 882068, DOI 10.1002/cpa.3160400304
- Ronald J. DiPerna and Andrew J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), no. 4, 667–689. MR 877643, DOI 10.1007/BF01214424
- Ronald J. DiPerna and Andrew Majda, Reduced Hausdorff dimension and concentration-cancellation for two-dimensional incompressible flow, J. Amer. Math. Soc. 1 (1988), no. 1, 59–95. MR 924702, DOI 10.1090/S0894-0347-1988-0924702-6
- Weinan E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 2, 207–218. MR 1778702, DOI 10.1007/s101140000034
- David G. Ebin, Viscous fluids in a domain with frictionless boundary, Global analysis–analysis on manifolds, Teubner-Texte Math., vol. 57, Teubner, Leipzig, 1983, pp. 93–110. MR 730604
- Tarek M. Elgindi and In-Jee Jeong, Finite-time singularity formation for strong solutions to the axi-symmetric 3D Euler equations, Ann. PDE 5 (2019), no. 2, Paper No. 16, 51. MR 4029562, DOI 10.1007/s40818-019-0071-6
- Volker Elling, Nonuniqueness of entropy solutions and the carbuncle phenomenon, Hyperbolic problems: theory, numerics and applications. I, Yokohama Publ., Yokohama, 2006, pp. 375–382. MR 2667260
- Volker Elling, A possible counterexample to well posedness of entropy solutions and to Godunov scheme convergence, Math. Comp. 75 (2006), no. 256, 1721–1733. MR 2240632, DOI 10.1090/S0025-5718-06-01863-1
- Volker Elling, The carbuncle phenomenon is incurable, Acta Math. Sci. Ser. B (Engl. Ed.) 29 (2009), no. 6, 1647–1656. MR 2589096, DOI 10.1016/S0252-9602(10)60007-0
- Eduard Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 26, Oxford University Press, Oxford, 2004. MR 2040667
- Eduard Feireisl, (S)-convergence and approximation of oscillatory solutions in fluid dynamics, Nonlinearity 34 (2021), no. 4, 2327–2349. MR 4246460, DOI 10.1088/1361-6544/abbd84
- Eduard Feireisl and Martina Hofmanová, On convergence of approximate solutions to the compressible Euler system, Ann. PDE 6 (2020), no. 2, Paper No. 11, 24. MR 4135633, DOI 10.1007/s40818-020-00086-8
- Eduard Feireisl, Mária Lukáčová-Medviďová, and Hana Mizerová, $\mathcal K$-convergence as a new tool in numerical analysis, IMA J. Numer. Anal. 40 (2020), no. 4, 2227–2255. MR 4165465, DOI 10.1093/imanum/drz045
- Eduard Feireisl, Mária Lukáčová-Medvid’ová, Hana Mizerová, and Bangwei She, Numerical analysis of compressible fluid flows, MS&A. Modeling, Simulation and Applications, vol. 20, Springer, Cham, [2021] ©2021. MR 4390192, DOI 10.1007/978-3-030-73788-7
- Eduard Feireisl, Mária Lukáčová-Medvid’ová, Hana Mizerová, and Bangwei She, Convergence of a finite volume scheme for the compressible Navier-Stokes system, ESAIM Math. Model. Numer. Anal. 53 (2019), no. 6, 1957–1979. MR 4031688, DOI 10.1051/m2an/2019043
- Eduard Feireisl, Mária Lukáčová-Medvid’ová, Bangwei She, and Yue Wang, Computing oscillatory solutions of the Euler system via $\mathcal K$-convergence, Math. Models Methods Appl. Sci. 31 (2021), no. 3, 537–576. MR 4260207, DOI 10.1142/S0218202521500123
- Eduard Feireisl, Mária Lukáčová-Medvid’ová, and Hana Mizerová, A finite volume scheme for the Euler system inspired by the two velocities approach, Numer. Math. 144 (2020), no. 1, 89–132. MR 4050088, DOI 10.1007/s00211-019-01078-y
- H. J. S. Fernando, Turbulent mixing in stratified fluids, Ann. Rev. Fluid Mech. 23 (1991), 455–493.
- Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, and Eitan Tadmor, Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws, Found. Comput. Math. 17 (2017), no. 3, 763–827. MR 3648106, DOI 10.1007/s10208-015-9299-z
- Ulrik S. Fjordholm, Siddhartha Mishra, and Eitan Tadmor, On the computation of measure-valued solutions, Acta Numer. 25 (2016), 567–679. MR 3509212, DOI 10.1017/S0962492916000088
- Jean-Luc Guermond and Bojan Popov, Viscous regularization of the Euler equations and entropy principles, SIAM J. Appl. Math. 74 (2014), no. 2, 284–305. MR 3176331, DOI 10.1137/120903312
- H. von Helmhotz, On the discontinuous movements of fluids, Monatsberichte der Königlichen Preussische Akademie der Wissenschaften zu Berlin 23 (1868), 215–278.
- Philip Isett, A proof of Onsager’s conjecture, Ann. of Math. (2) 188 (2018), no. 3, 871–963. MR 3866888, DOI 10.4007/annals.2018.188.3.4
- W.T. Kelvin, Hydrokinetic solutions and observations, Philosophical Magazine 42 (1871), 362–377.
- Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press, Oxford University Press, New York, 1998. Compressible models; Oxford Science Publications. MR 1637634
- Akitaka Matsumura and Takaaki Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys. 89 (1983), no. 4, 445–464. MR 713680, DOI 10.1007/BF01214738
- P. I. Plotnikov and W. Weigant, Isothermal Navier-Stokes equations and Radon transform, SIAM J. Math. Anal. 47 (2015), no. 1, 626–653. MR 3305369, DOI 10.1137/140960542
- H. P. Rosenthal, Weakly independent sequences and the Banach–Saks property. In Proceedings of the Durham Symposium on the relations between infinite dimensional and finite dimensional convexity, p. 26. Durham, 1975.
- Yongzhong Sun, Chao Wang, and Zhifei Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Ration. Mech. Anal. 201 (2011), no. 2, 727–742. MR 2820362, DOI 10.1007/s00205-011-0407-1
- Alberto Valli and Wojciech M. Zajączkowski, Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys. 103 (1986), no. 2, 259–296. MR 826865, DOI 10.1007/BF01206939
- Cédric Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454, DOI 10.1007/978-3-540-71050-9
Bibliographic Information
- Eduard Feireisl
- Affiliation: Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, CZ-115 67 Praha 1, Czech Republic; and Institute of Mathematics, TU Berlin, Strasse des 17. Juni, Berlin, Germany
- MR Author ID: 65780
- Email: feireisl@math.cas.cz
- Mária Lukáčová-Medvid’ová
- Affiliation: Institute of Mathematics, Johannes Gutenberg-University Mainz, Staudingerweg 9, 55 128 Mainz, Germany
- Email: lukacova@uni-mainz.de
- Simon Schneider
- Affiliation: Institute of Mathematics, Johannes Gutenberg-University Mainz, Staudingerweg 9, 55 128 Mainz, Germany
- Email: sschne15@uni-mainz.de
- Bangwei She
- Affiliation: Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, CZ-115 67 Praha 1, Czech Republic
- Address at time of publication: Academy for Multidisciplinary studies, Capital Normal University, West 3rd Ring North Road 105, 100048 Beijing, People’s Republic of China
- MR Author ID: 1165111
- ORCID: 0000-0002-5025-0070
- Email: she@math.cas.cz
- Received by editor(s): May 9, 2021
- Received by editor(s) in revised form: October 31, 2021
- Published electronically: June 1, 2022
- Additional Notes: The research of the first and fourth authors leading to these results received funding from the Czech Sciences Foundation (GAČR), Grant Agreement 21-02411S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic was supported by RVO:67985840. The second author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number 233630050 – TRR 146 as well as by TRR 165 Waves to Weather. She is grateful to the Gutenberg Research College for supporting her research. The research of the third author was funded by Mainz Institute of Multiscale Modelling.
The fourth author is the corresponding author. - © Copyright 2022 American Mathematical Society
- Journal: Math. Comp. 91 (2022), 2129-2164
- MSC (2020): Primary 76N06, 35Q31
- DOI: https://doi.org/10.1090/mcom/3738
- MathSciNet review: 4451458