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October  2020, 25(10): 3831-3842. doi: 10.3934/dcdsb.2020079

Generalized solutions to models of inviscid fluids

Dominic Breit 1, , Eduard Feireisl 2, and  Martina Hofmanová 3,

1. 

Department of Mathematics, Heriot-Watt University, Riccarton Edinburgh EH14 4AS, UK
2. 

Institute of Mathematics AS CR, Žitná 25,115 67 Praha, Czech Republic and Institute of Mathematics, TU Berlin, Strasse des 17.Juni, Berlin, Germany
3. 

Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany

M.H. gratefully acknowledges the financial support by the German Science Foundation DFG via the Collaborative Research Center SFB1283

Received  July 2019 Revised  December 2019 Published  October 2020 Early access  January 2020

Fund Project: 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

We discuss several approaches to generalized solutions of problems describing the motion of inviscid fluids. We propose a new concept of dissipative solution to the compressible Euler system based on a careful analysis of possible oscillations and/or concentrations in the associated generating sequence. Unlike the conventional measure–valued solutions or rather their expected values, the dissipative solutions comply with a natural compatibility condition – they are classical solutions as long as they enjoy a certain degree of smoothness.

Keywords: Euler system, weak solution, dissipative solution.
Mathematics Subject Classification: 35D30, 35A01, 35Q31.
Citation: Dominic Breit, Eduard Feireisl, Martina Hofmanová. Generalized solutions to models of inviscid fluids. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3831-3842. doi: 10.3934/dcdsb.2020079
References:
[1]

J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures, J. Convex Anal., 4 (1997), 129-147. 

[2]

D. Basarić, Vanishing viscosity limit for the compressible Navier–Stokes system via measure-valued solutions, arXive Preprint Series, arXiv: 1903.05886, 2019.

[3]

D. Breit, E. Feireisl and M. Hofmanová, Dissipative solutions and semiflow selection for the complete Euler system, Commun. Math. Phys. DOI:10.1007/s00220-019-03662-7/ArXive PreprintSeries, arXiv: 1904. 00622, 2019.

[4]

D. Breit, E. Feireisl and M. Hofmanová, Solution semiflow to the isentropic Euler system, Arch. Rational Mech. Anal. DOI: 10.1007/s00205-019-01420-6

[5]

J. E. Cardona and L. Kapitanskii, Semiflow selection and Markov selection theorems, arXive Preprint Series, arXiv: 1707.04778v1, 2017.

[6]

G. Q. Chen and J. Glimm, Kolmogorov-type theory of compressible turbulence and inviscid limit of the Navier–Stokes equations in ${R}^3$, Phys. D, 400 (2019), 132138, 10 pp, arXiv: 1809.09490. doi: 10.1016/j.physd.2019.06.004.

[7]

E. Chiodaroli, A counterexample to well-posedness of entropy solutions to the compressible Euler system, J. Hyperbolic Differ. Equ., 11 (2014), 493-519.  doi: 10.1142/S0219891614500143.

[8]

E. Chiodaroli, C. De Lellis and O. Kreml, Global ill-posedness of the isentropic system of gas dynamics, Comm. Pure Appl. Math., 68 (2015), 1157-1190.  doi: 10.1002/cpa.21537.

[9]

E. Chiodaroli and O. Kreml, On the energy dissipation rate of solutions to the compressible isentropic Euler system, Arch. Ration. Mech. Anal., 214 (2014), 1019-1049.  doi: 10.1007/s00205-014-0771-8.

[10]

E. Chiodaroli, O. Kreml, V. Mácha and S. Schwarzacher, Non niqueness of admissible weak solutions to the compressible Euler equations with smooth initial data, arXive Preprint Series, arXiv: 1812.09917v1, 2019.

[11]

C. De Lellis, L. Székelyhidi and Jr ., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260.  doi: 10.1007/s00205-008-0201-x.

[12]

D. B. Ebin, Viscous fluids in a domain with frictionless boundary, Global Analysis - Analysis on Manifolds, H. Kurke, J. Mecke, H. Triebel, R. Thiele Editors, Teubner-Texte zur Mathematik 57, Teubner, Leipzig, 1983, 93–110.

[13]

E. Feireisl, S. S. Ghoshal and A. Jana, On uniqueness of dissipative solutions to the isentropic Euler system, Comm. Partial Differential Equations, 44 (2019), 1285–1298, arXiv: 1903.11687. doi: 10.1080/03605302.2019.1629958.

[14]

E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Dissipative measure-valued solutions to the compressible Navier–Stokes system, Calc. Var. Partial Differential Equations, 55 (2016), Art. 141, 20 pp. doi: 10.1007/s00526-016-1089-1.

[15]

E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Regularity and energy conservation for the compressible Euler equations, Arch. Ration. Mech. Anal., 223 (2017), 1375-1395.  doi: 10.1007/s00205-016-1060-5.

[16]

E. Feireisl and M. Hofmanová, On convergence of approximate solutions to the compressible Euler system, arXive Preprint Series, arXiv: 1905.02548, 2019.

[17]

E. Feireisl, C. Klingenberg, O. Kreml and S. Markfelder, On oscillatory solutions to the complete Euler system, arXive Preprint Series, arXiv: 1710.10918, 2017.

[18]

E. Feireisl, Š. Matušů-Nečasová, H. Petzeltová and I. Straškraba, On the motion of a viscous compressible flow driven by a time-periodic external flow, Arch. Rational Mech. Anal., 149 (1999), 69-96.  doi: 10.1007/s002050050168.

[19]

P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Weak-strong uniqueness for measure-valued solutions of some compressible fluid models, Nonlinearity, 28 (2015), 3873-3890.  doi: 10.1088/0951-7715/28/11/3873.

[20]

N. V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 691-708. 

[21]

T. Luo, C. Xie and Z. Xin, Non-uniqueness of admissible weak solutions to compressible Euler systems with source terms, Adv. Math., 291 (2016), 542-583.  doi: 10.1016/j.aim.2015.12.027.

[22]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York-Berlin, 1983.

[23]

A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. RIMS Kyoto Univ., 21 (1981), 839-859.  doi: 10.1215/kjm/1250521916.

show all references

References:
[1]

J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures, J. Convex Anal., 4 (1997), 129-147. 

[2]

D. Basarić, Vanishing viscosity limit for the compressible Navier–Stokes system via measure-valued solutions, arXive Preprint Series, arXiv: 1903.05886, 2019.

[3]

D. Breit, E. Feireisl and M. Hofmanová, Dissipative solutions and semiflow selection for the complete Euler system, Commun. Math. Phys. DOI:10.1007/s00220-019-03662-7/ArXive PreprintSeries, arXiv: 1904. 00622, 2019.

[4]

D. Breit, E. Feireisl and M. Hofmanová, Solution semiflow to the isentropic Euler system, Arch. Rational Mech. Anal. DOI: 10.1007/s00205-019-01420-6

[5]

J. E. Cardona and L. Kapitanskii, Semiflow selection and Markov selection theorems, arXive Preprint Series, arXiv: 1707.04778v1, 2017.

[6]

G. Q. Chen and J. Glimm, Kolmogorov-type theory of compressible turbulence and inviscid limit of the Navier–Stokes equations in ${R}^3$, Phys. D, 400 (2019), 132138, 10 pp, arXiv: 1809.09490. doi: 10.1016/j.physd.2019.06.004.

[7]

E. Chiodaroli, A counterexample to well-posedness of entropy solutions to the compressible Euler system, J. Hyperbolic Differ. Equ., 11 (2014), 493-519.  doi: 10.1142/S0219891614500143.

[8]

E. Chiodaroli, C. De Lellis and O. Kreml, Global ill-posedness of the isentropic system of gas dynamics, Comm. Pure Appl. Math., 68 (2015), 1157-1190.  doi: 10.1002/cpa.21537.

[9]

E. Chiodaroli and O. Kreml, On the energy dissipation rate of solutions to the compressible isentropic Euler system, Arch. Ration. Mech. Anal., 214 (2014), 1019-1049.  doi: 10.1007/s00205-014-0771-8.

[10]

E. Chiodaroli, O. Kreml, V. Mácha and S. Schwarzacher, Non niqueness of admissible weak solutions to the compressible Euler equations with smooth initial data, arXive Preprint Series, arXiv: 1812.09917v1, 2019.

[11]

C. De Lellis, L. Székelyhidi and Jr ., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260.  doi: 10.1007/s00205-008-0201-x.

[12]

D. B. Ebin, Viscous fluids in a domain with frictionless boundary, Global Analysis - Analysis on Manifolds, H. Kurke, J. Mecke, H. Triebel, R. Thiele Editors, Teubner-Texte zur Mathematik 57, Teubner, Leipzig, 1983, 93–110.

[13]

E. Feireisl, S. S. Ghoshal and A. Jana, On uniqueness of dissipative solutions to the isentropic Euler system, Comm. Partial Differential Equations, 44 (2019), 1285–1298, arXiv: 1903.11687. doi: 10.1080/03605302.2019.1629958.

[14]

E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Dissipative measure-valued solutions to the compressible Navier–Stokes system, Calc. Var. Partial Differential Equations, 55 (2016), Art. 141, 20 pp. doi: 10.1007/s00526-016-1089-1.

[15]

E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Regularity and energy conservation for the compressible Euler equations, Arch. Ration. Mech. Anal., 223 (2017), 1375-1395.  doi: 10.1007/s00205-016-1060-5.

[16]

E. Feireisl and M. Hofmanová, On convergence of approximate solutions to the compressible Euler system, arXive Preprint Series, arXiv: 1905.02548, 2019.

[17]

E. Feireisl, C. Klingenberg, O. Kreml and S. Markfelder, On oscillatory solutions to the complete Euler system, arXive Preprint Series, arXiv: 1710.10918, 2017.

[18]

E. Feireisl, Š. Matušů-Nečasová, H. Petzeltová and I. Straškraba, On the motion of a viscous compressible flow driven by a time-periodic external flow, Arch. Rational Mech. Anal., 149 (1999), 69-96.  doi: 10.1007/s002050050168.

[19]

P. Gwiazda, A. Świerczewska-Gwiazda and E. Wiedemann, Weak-strong uniqueness for measure-valued solutions of some compressible fluid models, Nonlinearity, 28 (2015), 3873-3890.  doi: 10.1088/0951-7715/28/11/3873.

[20]

N. V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 691-708. 

[21]

T. Luo, C. Xie and Z. Xin, Non-uniqueness of admissible weak solutions to compressible Euler systems with source terms, Adv. Math., 291 (2016), 542-583.  doi: 10.1016/j.aim.2015.12.027.

[22]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York-Berlin, 1983.

[23]

A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. RIMS Kyoto Univ., 21 (1981), 839-859.  doi: 10.1215/kjm/1250521916.

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