[Stabilisation pour un milieu continu compressible avec pression non monotone]
Nous étudions l'évolution 1d d'un milieu continu compressible conducteur de la chaleur. La pression est donnée par p(η,θ)=p0(η)+p1(η)θ, où p0 et p1 sont des fonctions non monotones assez générales pour permettre de traiter à la fois des modèles de fluides nucléaires et des solides thermo-visco-élastiques. Pour un problème aux limites d'évolution associé, avec grandes données, nous prouvons la stabilisation pour t→∞ au sens suivant : convergence ponctuelle et dans Lq pour le volume spécifique η, dans Lq pour la vitesse v, pour tout q∈[2,∞), et dans L2 pour la temperature θ.
We consider the system of quasilinear equations for 1d-motion of viscous compressible heat-conducting media. The state function has the form p(η,θ)=p0(η)+p1(η)θ, with general nonmonotone p0 and p1, which allows us to treat both nuclear fluids and thermoviscoelastic solids (for fluids, p, η, and θ are the pressure, specific volume, and temperature). For an initial boundary value problem with large data, we establish stabilization as t→∞: pointwise and in Lq for η, in Lq for v (the velocity), for any q∈[2,∞), and in L2 for θ.
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@article{CRMATH_2002__334_2_119_0,
author = {Ducomet, Bernard and Zlotnik, Alexander},
title = {Stabilization for viscous compressible heat-conducting media equations with nonmonotone state functions},
journal = {Comptes Rendus. Math\'ematique},
pages = {119--124},
publisher = {Elsevier},
volume = {334},
number = {2},
year = {2002},
doi = {10.1016/S1631-073X(02)02227-6},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02227-6/}
}
TY - JOUR AU - Ducomet, Bernard AU - Zlotnik, Alexander TI - Stabilization for viscous compressible heat-conducting media equations with nonmonotone state functions JO - Comptes Rendus. Mathématique PY - 2002 SP - 119 EP - 124 VL - 334 IS - 2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02227-6/ DO - 10.1016/S1631-073X(02)02227-6 LA - en ID - CRMATH_2002__334_2_119_0 ER -
%0 Journal Article %A Ducomet, Bernard %A Zlotnik, Alexander %T Stabilization for viscous compressible heat-conducting media equations with nonmonotone state functions %J Comptes Rendus. Mathématique %D 2002 %P 119-124 %V 334 %N 2 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02227-6/ %R 10.1016/S1631-073X(02)02227-6 %G en %F CRMATH_2002__334_2_119_0
Ducomet, Bernard; Zlotnik, Alexander. Stabilization for viscous compressible heat-conducting media equations with nonmonotone state functions. Comptes Rendus. Mathématique, Tome 334 (2002) no. 2, pp. 119-124. doi : 10.1016/S1631-073X(02)02227-6. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02227-6/
[1] Global generalized solutions of the equations of the one-dimensional motion of a viscous heat-conducting gas, Soviet Math. Dokl., Volume 38 (1989), pp. 1-5
[2] Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, 1990
[3] Simplified models of quantum fluids in nuclear physics Mathematica Bohemica, 126 (2001), pp. 323-336
[4] Ducomet B., Zlotnik A.A., Stabilization for equations of one-dimensional viscous heat-conducting media with nonmonotone equation of state (in preparation)
[5] Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, 1997
[6] Large-time behaviour of solutions to the equations of one-dimensional nonlinear thermoviscoelasticity, Quart. Appl. Math., Volume 56 (1998), pp. 201-219
[7] On the asymptotic behavior of the motion of a viscous heat-conducting one-dimensional real gas, Math. Z., Volume 216 (1994), pp. 317-336
[8] On the outer pressure problem of the one-dimensional polytropic ideal gas, Japan J. Appl. Math., Volume 15 (1988), pp. 53-85
[9] Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability, Arch. Rational Mech. Anal., Volume 97 (1987), pp. 353-394
[10] Global existence and large-time behaviour of the solution to the system in one-dimensional nonlinear thermoviscoelasticity, Quart. Appl. Math., Volume 59 (2001), pp. 113-142
[11] Global existence and asymptotic behaviour in nonlinear thermoviscoelasticity, J. Differential Equations, Volume 134 (1997), pp. 46-67
[12] Global existence and asymptotic behaviour of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions, Quart. Appl. Math., Volume 57 (1999), pp. 93-116
[13] On equations for one-dimensional motion of a viscous barotropic gas in the presence of a body force, Siberian Math. J., Volume 33 (1992), pp. 798-815
[14] Uniform estimates and the stabilization of symmetric solutions to one system of quasilinear equations, Differential Equations, Volume 36 (2000), pp. 701-716
[15] Properties and asymptotic behaviour of solutions of some problems of one-dimensional motion of a viscous barotropic gas, Math. Notes, Volume 55 (1994), pp. 471-482
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