Stability of planar rarefaction waves under general viscosity perturbation of the isentropic Euler system

Feireisl, Eduard; Novotný, Antonín

doi:10.1016/j.anihpc.2021.01.001

Feireisl, Eduard ^1,² ; Novotný, Antonín ³

¹ a Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67 Praha 1, Czech Republic
² b Institute of Mathematics, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
³ c Université de Toulon, IMATH, EA 2134, BP 20132, 83957 La Garde, France

Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1725-1737.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

We consider the vanishing viscosity limit for a model of a general non-Newtonian compressible fluid in $R^{d}$ , $d = 2, 3$ . We suppose that the initial data approach a profile determined by the Riemann data generating a planar rarefaction wave for the isentropic Euler system. Under these circumstances the associated sequence of dissipative solutions approaches the corresponding rarefaction wave strongly in the energy norm in the vanishing viscosity limit. The result covers the particular case of a linearly viscous fluid governed by the Navier–Stokes system.

Reçu le : 2020-02-11
Révisé le : 2020-12-01
Accepté le : 2021-01-07

MR Zbl

DOI : 10.1016/j.anihpc.2021.01.001

Mots-clés : Vanishing viscosity limit, Non–linear viscous fluid, Rarefaction wave, Isentropic Euler system

Affiliations des auteurs :

Feireisl, Eduard ^{1, 2} ; Novotný, Antonín ³

@article{AIHPC_2021__38_6_1725_0,
     author = {Feireisl, Eduard and Novotn\'y, Anton{\'\i}n},
     title = {Stability of planar rarefaction waves under general viscosity perturbation of the isentropic {Euler} system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1725--1737},
     publisher = {Elsevier},
     volume = {38},
     number = {6},
     year = {2021},
     doi = {10.1016/j.anihpc.2021.01.001},
     mrnumber = {4327895},
     zbl = {1494.35132},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2021.01.001/}
}

TY  - JOUR
AU  - Feireisl, Eduard
AU  - Novotný, Antonín
TI  - Stability of planar rarefaction waves under general viscosity perturbation of the isentropic Euler system
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2021
SP  - 1725
EP  - 1737
VL  - 38
IS  - 6
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2021.01.001/
DO  - 10.1016/j.anihpc.2021.01.001
LA  - en
ID  - AIHPC_2021__38_6_1725_0
ER  -

%0 Journal Article
%A Feireisl, Eduard
%A Novotný, Antonín
%T Stability of planar rarefaction waves under general viscosity perturbation of the isentropic Euler system
%J Annales de l'I.H.P. Analyse non linéaire
%D 2021
%P 1725-1737
%V 38
%N 6
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2021.01.001/
%R 10.1016/j.anihpc.2021.01.001
%G en
%F AIHPC_2021__38_6_1725_0

Feireisl, Eduard; Novotný, Antonín. Stability of planar rarefaction waves under general viscosity perturbation of the isentropic Euler system. Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1725-1737. doi : 10.1016/j.anihpc.2021.01.001. https://www.numdam.org/articles/10.1016/j.anihpc.2021.01.001/

Bibliographie
Cité par

[1] Abbatiello, A.; Feireisl, E.; Novotný, A. Generalized solutions to models of compressible viscous fluids, Discrete Contin. Dyn. Syst., Ser. A, Volume 41 (2021) no. 1, pp. 1-28 | DOI | MR | Zbl

[2] Dafermos, C.M. The second law of thermodynamics and stability, Arch. Ration. Mech. Anal., Volume 70 (1979), pp. 167-179 | DOI | MR | Zbl

[3] Chen, G.-Q.; Chen, J. Stability of rarefaction waves and vacuum states for the multidimensional Euler equations, J. Hyperbolic Differ. Equ., Volume 4 (2007) no. 1, pp. 105-122 | DOI | MR | Zbl

[4] Chen, G.-Q.; Perepelitsa, M. Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow, Commun. Pure Appl. Math., Volume 63 (2010) no. 11, pp. 1469-1504 | DOI | MR | Zbl

[5] Chiodaroli, E.; De Lellis, C.; Kreml, O. Global ill-posedness of the isentropic system of gas dynamics, Commun. Pure Appl. Math., Volume 68 (2015) no. 7, pp. 1157-1190 | DOI | MR | Zbl

[6] Chiodaroli, E.; Kreml, O. On the energy dissipation rate of solutions to the compressible isentropic Euler system, Arch. Ration. Mech. Anal., Volume 214 (2014) no. 3, pp. 1019-1049 | DOI | MR | Zbl

[7] Chiodaroli, E.; Kreml, O.; Mácha, V.; Schwarzacher, S. Non–uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data, 2019 (Archive Preprint Series) | arXiv | MR | Zbl

[8] Feireisl, E.; Ghoshal, S.S.; Jana, A. On uniqueness of dissipative solutions to the isentropic Euler system, Commun. Partial Differ. Equ., Volume 44 (2019) no. 12, pp. 1285-1298 | DOI | MR | Zbl

[9] Feireisl, E.; Kreml, O. Uniqueness of rarefaction waves in multidimensional compressible Euler system, J. Hyperbolic Differ. Equ., Volume 12 (2015) no. 3, pp. 489-499 | DOI | MR | Zbl

[10] Feireisl, E.; Kreml, O.; Vasseur, A. Stability of the isentropic Riemann solutions of the full multidimensional Euler system, SIAM J. Math. Anal., Volume 47 (2015) no. 3, pp. 2416-2425 | DOI | MR | Zbl

[11] Feireisl, E.; Kwon, Y.-S.; Novotný, A. On the long–time behavior of dissipative solutions to models of non-Newtonian compressible fluids, 2020 (Archive Preprint Series) | arXiv | MR | Zbl

[12] Goodman, J.; Xin, Z.P. Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Ration. Mech. Anal., Volume 121 (1992) no. 3, pp. 235-265 | DOI | MR | Zbl

[13] Hoff, D.; Liu, T.-P. The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data, Indiana Univ. Math. J., Volume 38 (1989) no. 4, pp. 861-915 | DOI | MR | Zbl

[14] Kwon, Y.S.; Novotný, A. Dissipative solutions to compressible Navier-Stokes equations with general inflow-outflow data: existence, stability and weak-strong uniqueness, J. Math. Fluid Mech. (2021) (in press) | MR | Zbl

[15] Li, L.-A.; Wang, D.; Wang, Y. Vanishing viscosity limit to the planar rarefaction wave for the two-dimensional compressible Navier–Stokes equations, Commun. Math. Phys., Volume 376 (2020), pp. 353-384 | DOI | MR | Zbl

[16] Lions, P.-L. Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998 | MR | Zbl

[17] Markfelder, S.; Klingenberg, C. The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock, Arch. Ration. Mech. Anal., Volume 227 (2018) no. 3, pp. 967-994 | DOI | MR | Zbl

[18] Matsumura, A.; Nishihara, K. Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Jpn. J. Appl. Math., Volume 3 (1986) no. 1, pp. 1-13 | DOI | MR | Zbl

[19] Perepelitsa, M. Asymptotics toward rarefaction waves and vacuum for 1-D compressible Navier-Stokes equations, SIAM J. Math. Anal., Volume 42 (2010) no. 3, pp. 1404-1412 | DOI | MR | Zbl

[20] Xin, Z.P. Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Commun. Pure Appl. Math., Volume 46 (1993) no. 5, pp. 621-665 | DOI | MR | Zbl

Cité par Sources :