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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.
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.
Penel, Patrick 1 ; Straškraba, Ivan 2
@article{CRMATH_2007__345_2_67_0,
author = {Penel, Patrick and Stra\v{s}kraba, Ivan},
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},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2007.04.018/}
}
TY - JOUR AU - Penel, Patrick AU - Straškraba, Ivan 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 UR - http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2007.04.018/ DO - 10.1016/j.crma.2007.04.018 LA - en ID - CRMATH_2007__345_2_67_0 ER -
%0 Journal Article %A Penel, Patrick %A Straškraba, Ivan %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 %U http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2007.04.018/ %R 10.1016/j.crma.2007.04.018 %G en %F CRMATH_2007__345_2_67_0
Penel, Patrick; Straškraba, Ivan. Lyapunov analysis and stabilization to the rest state for solutions to the 1D-barotropic compressible Navier–Stokes equations. Comptes Rendus. Mathématique, Tome 345 (2007) no. 2, pp. 67-72. doi : 10.1016/j.crma.2007.04.018. http://geodesic.mathdoc.fr/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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