Asymptotic stability of scalar multi-D inviscid shock waves
Serre, Denis1
1 U.M.P.A., UMR 5669, CNRS–ENSL ENS de Lyon, 46 allée d’Italie 69364 Lyon cedex 07 (France)
Annales de l'Institut Fourier, Online first, 20 p.

In several space dimensions, scalar shock waves between two constant states are not necessarily planar. We describe them in detail. Then we prove their asymptotic stability in L 1 -distance, assuming that they are uniformly non-characteristic. Our result is conditional for a general flux, while unconditional for the multi-D Burgers equation.

En plusieurs variables d’espace, les chocs scalaires entre deux constantes u ± ne sont pas nécessairement des chocs plans. Nous les décrivons en détail. Puis nous prouvons leur stabilité asymptotique en distance L 1 , sous l’hypothèse qu’ils ne sont pas caractéristiques. Pour un flux général, notre résultat suppose que la donée initiale est à valeurs dans l’intervalle [u + ,u - ]. Pour l’équation de Burgers multi-D, il est valable pour des perturbations arbitrairement grandes.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3569
Classification: 35L65, 35B40
Keywords: Scalar conservation laws, Burgers equation, shock waves, contraction semi-group, asymptotic stability.
Keywords: Lois de conservation scalaires, équation de Burgers, ondes de choc, semi-groupe de contractions, stabilité asymptotique.
Serre, Denis 1

1 U.M.P.A., UMR 5669, CNRS–ENSL ENS de Lyon, 46 allée d’Italie 69364 Lyon cedex 07 (France) 
@unpublished{AIF_0__0_0_A19_0,
     author = {Serre, Denis},
     title = {Asymptotic stability of scalar {multi-D} inviscid shock waves},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2023},
     doi = {10.5802/aif.3569},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Serre, Denis
TI  - Asymptotic stability of scalar multi-D inviscid shock waves
JO  - Annales de l'Institut Fourier
PY  - 2023
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3569
LA  - en
ID  - AIF_0__0_0_A19_0
ER  - 
%0 Unpublished Work
%A Serre, Denis
%T Asymptotic stability of scalar multi-D inviscid shock waves
%J Annales de l'Institut Fourier
%D 2023
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3569
%G en
%F AIF_0__0_0_A19_0
Serre, Denis. Asymptotic stability of scalar multi-D inviscid shock waves. Annales de l'Institut Fourier, Online first, 20 p.

[1] Duchêne, V.; Rodrigues, L. M. Large-time asymptotic stability of Riemann shocks of scalar balance laws, SIAM J. Math. Anal., Volume 52 (2020) no. 1, pp. 792-820 | DOI | MR | Zbl

[2] Freistühler, Heinrich; Serre, Denis ℒ 1 stability of shock waves in scalar viscous conservation laws, Comm. Pure Appl. Math., Volume 51 (1998) no. 3, pp. 291-301 | DOI | MR | Zbl

[3] Goodman, Jonathan; Miller, Judith R. Long-time behavior of scalar viscous shock fronts in two dimensions, J. Dynam. Differential Equations, Volume 11 (1999) no. 2, pp. 255-277 | DOI | MR | Zbl

[4] Hoff, David; Zumbrun, Kevin Asymptotic behavior of multidimensional scalar viscous shock fronts, Indiana Univ. Math. J., Volume 49 (2000) no. 2, pp. 427-474 | DOI | MR | Zbl

[5] Kružkov, S. N. First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), Volume 81 (123) (1970), pp. 228-255 | MR | Zbl

[6] Lions, P.-L.; Perthame, B.; Tadmor, E. A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc., Volume 7 (1994) no. 1, pp. 169-191 | DOI | MR | Zbl

[7] Majda, Andrew The existence of multidimensional shock fronts, Mem. Amer. Math. Soc., 43, American Mathematical Society, Providence, RI, 1983 no. 281, v+93 pages | DOI | MR | Zbl

[8] Majda, Andrew The stability of multidimensional shock fronts, Mem. Amer. Math. Soc., 41, American Mathematical Society, Providence, RI, 1983 no. 275, iv+95 pages | DOI | MR | Zbl

[9] Osher, Stanley; Ralston, James L 1 stability of travelling waves with applications to convective porous media flow, Comm. Pure Appl. Math., Volume 35 (1982) no. 6, pp. 737-749 | DOI | MR | Zbl

[10] Serre, Denis L 1 -stability of nonlinear waves in scalar conservation laws, Evolutionary equations. Vol. I (Handb. Differ. Equ.), North-Holland, Amsterdam, 2004, pp. 473-553 | MR | Zbl

[11] Serre, Denis; Silvestre, Luis Multi-dimensional Burgers equation with unbounded initial data: well-posedness and dispersive estimates, Arch. Ration. Mech. Anal., Volume 234 (2019) no. 3, pp. 1391-1411 | DOI | MR | Zbl

Cited by Sources: