[Analyse de Lyapunov et stabilisation vers l'état d'équilibre pour les solutions des equations de Navier–Stokes compressibles unidimensionnelles]
Dans cette Note, nous établissons de nouvelles estimées pour le comportement asymptotique en temps des solutions des équations unidimensionnelles de Navier–Stokes pour un fluide compressible barotropique, associées à des conditions aux limites homogènes de Dirichlet, pour de larges conditions initiales, sous l'influence de larges forces externes telles que la densité stationnaire peut s'annuler : un problème fortement singulier. Comme conséquence nous apportons une réponse nouvelle à la question du taux de convergence.
In this Note, we establish new estimates for the long time behavior of the solutions to the Navier–Stokes Equations for a compressible barotropic fluid in 1D, with homogeneous Dirichlet boundary conditions, with large initial data, and under the influence of a large mass force in the case when the stationary density admits vacua: a highly singular problem. As a consequence we bring new answers to the question of the stabilizing rate of convergence.
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@article{CRMATH_2007__345_2_67_0,
author = {Patrick Penel and Ivan Stra\v{s}kraba},
title = {Lyapunov analysis and stabilization to the rest state for solutions to the {1D-barotropic} compressible {Navier{\textendash}Stokes} equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {67--72},
publisher = {Elsevier},
volume = {345},
number = {2},
year = {2007},
doi = {10.1016/j.crma.2007.04.018},
language = {en},
}
TY - JOUR AU - Patrick Penel AU - Ivan Straškraba TI - Lyapunov analysis and stabilization to the rest state for solutions to the 1D-barotropic compressible Navier–Stokes equations JO - Comptes Rendus. Mathématique PY - 2007 SP - 67 EP - 72 VL - 345 IS - 2 PB - Elsevier DO - 10.1016/j.crma.2007.04.018 LA - en ID - CRMATH_2007__345_2_67_0 ER -
%0 Journal Article %A Patrick Penel %A Ivan Straškraba %T Lyapunov analysis and stabilization to the rest state for solutions to the 1D-barotropic compressible Navier–Stokes equations %J Comptes Rendus. Mathématique %D 2007 %P 67-72 %V 345 %N 2 %I Elsevier %R 10.1016/j.crma.2007.04.018 %G en %F CRMATH_2007__345_2_67_0
Patrick Penel; Ivan Straškraba. Lyapunov analysis and stabilization to the rest state for solutions to the 1D-barotropic compressible Navier–Stokes equations. Comptes Rendus. Mathématique, Volume 345 (2007) no. 2, pp. 67-72. doi : 10.1016/j.crma.2007.04.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.04.018/
[1] An -theory for the n-dimensional, stationary, compressible Navier–Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions, Commun. Math. Phys., Volume 109 (1987), pp. 229-248
[2] On the static–limit solutions to the Navier–Stokes equations of compressible flow, J. Math Fluid Mech., Volume 3 (2001), pp. 393-408
[3] On the zero-velocity-limit of solutions to the Navier–Stokes equations of compressible flow, Manuscr. Math., Volume 97 (1998), pp. 109-116
[4] Stabilization of solutions to compressible Navier–Stokes equations, J. Math. Kyoto Univ., Volume 40 (2000) no. 2, pp. 217-245
[5] Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2003
[6] P. Penel and I. Straškraba, Construction of a Lyapunov functional for 1D-viscous compressible barotropic fluid equations admitting vacua, Preprint of the Mathematical Institute of the Czech Academy of Sciences, Prague, 2006, 14 pp
[7] Asymptotic development of vacuums for 1-d Navier–Stokes equations of compressible flow, Nonlinear World, Volume 3 (1996), pp. 519-535
[8] On a decay rate for 1D-viscous compressible barotropic fluid equations, J. Evolution Equations, Volume 2 (2002), pp. 69-96
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