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Higher-order time-stepping schemes for fluid-structure interaction problems
Generalized solutions to models of inviscid fluids
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 |
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.
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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