We study the convergence to equilibrium for the full compressible Navier-Stokes equations on the torus . Under the conditions that both the density ρ and the temperature θ possess uniform in time positive lower and upper bounds, it is shown that global regular solutions converge to equilibrium with exponential rate. We improve the previous result obtained by Villani in (2009) [28] on two levels: weaker conditions on solutions and faster decay rates.
Mots-clés : Convergence to equilibrium, Global smooth solutions, Full compressible Navier-Stokes equations
@article{AIHPC_2020__37_2_457_0, author = {Zhang, Zhifei and Zi, Ruizhao}, title = {Convergence to equilibrium for the solution of the full compressible {Navier-Stokes} equations}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {457--488}, publisher = {Elsevier}, volume = {37}, number = {2}, year = {2020}, doi = {10.1016/j.anihpc.2019.09.001}, mrnumber = {4072802}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2019.09.001/} }
TY - JOUR AU - Zhang, Zhifei AU - Zi, Ruizhao TI - Convergence to equilibrium for the solution of the full compressible Navier-Stokes equations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2020 SP - 457 EP - 488 VL - 37 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2019.09.001/ DO - 10.1016/j.anihpc.2019.09.001 LA - en ID - AIHPC_2020__37_2_457_0 ER -
%0 Journal Article %A Zhang, Zhifei %A Zi, Ruizhao %T Convergence to equilibrium for the solution of the full compressible Navier-Stokes equations %J Annales de l'I.H.P. Analyse non linéaire %D 2020 %P 457-488 %V 37 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2019.09.001/ %R 10.1016/j.anihpc.2019.09.001 %G en %F AIHPC_2020__37_2_457_0
Zhang, Zhifei; Zi, Ruizhao. Convergence to equilibrium for the solution of the full compressible Navier-Stokes equations. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 2, pp. 457-488. doi : 10.1016/j.anihpc.2019.09.001. http://www.numdam.org/articles/10.1016/j.anihpc.2019.09.001/
[1] Quelques modèles diffusifs capillaires de type Korteweg, C. R. Mecanique, Volume 332 (2004), pp. 881–886 | DOI | Zbl
[2] On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl., Volume 86 (2006), pp. 362–368 | DOI | MR | Zbl
[3] On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl., Volume 87 (2007), pp. 57–90 | DOI | MR | Zbl
[4] Solution of some vector analysis problems connected with operators div and grad, Tr. Semin. S.L. Soboleva, Volume 80 (1980), pp. 5–40 (in Russian) | MR | Zbl
[5] Existence results for viscous polytropic fluids with vacuum, J. Differ. Equ., Volume 228 (2006), pp. 377–411 | MR | Zbl
[6] On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math., Volume 159 (2005), pp. 245–316 | DOI | MR | Zbl
[7] Optimal convergence rates for the compressible Navier-Stokes equations with potential forces, Math. Models Methods Appl. Sci., Volume 17 (2007), pp. 737–758 | DOI | MR | Zbl
[8] A blow-up criterion for two dimensional compressible viscous heat-conductive flows, Nonlinear Anal., Volume 75 (2012), pp. 3130–3141 | DOI | MR | Zbl
[9] Decay estimates for isentropic compressible Navier-Stokes equations in bounded domain, J. Math. Anal. Appl., Volume 386 (2012), pp. 939–947 | DOI | MR | Zbl
[10] On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., Volume 3 (2001), pp. 358–392 | DOI | MR | Zbl
[11] Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004 | Zbl
[12] An Introduction to the Mathematical Theory of the Navier-Stokes Equations, i, Springer-Verlag, New York, 1994 | Zbl
[13] Global stability of large solutions to the 3D compressible Navier-Stokes equations | arXiv
[14] Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differ. Equ., Volume 120 (1995), pp. 215–254 | DOI | MR | Zbl
[15] Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids, Arch. Ration. Mech. Anal., Volume 139 (1997), pp. 303–354 | DOI | MR | Zbl
[16] Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations, Arch. Ration. Mech. Anal., Volume 227 (2018), pp. 995–1059 | DOI | MR
[17] The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier-Stokes equations, Bull. Soc. Math. Fr., Volume 127 (1999), pp. 473–517 | Numdam | MR | Zbl
[18] Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pure Appl. Math., Volume 34 (1981), pp. 481–524 | DOI | MR | Zbl
[19] Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in , Commun. Math. Phys., Volume 200 (1999), pp. 621–659 | DOI | MR | Zbl
[20] Mathematical Topics in Fluid Mechanics, Vols. 2, Compressible Models, Oxford Science Publication, Oxford, 1998 | MR | Zbl
[21] The initial value problem for the equation of compressible viscous and heat-conductive fluids, Proc. Jpn. Acad. Ser., Volume 55 (1979), pp. 337–342 | MR | Zbl
[22] The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., Volume 20 (1980), pp. 67–104 | MR | Zbl
[23] Computing Methods in Applied Sciences and Engineering, vol. V, Initial-boundary value problems for the equations of motion of general fluids, North-Holland, Amsterdam (1982), pp. 389–406 (Versailles, 1981) | MR | Zbl
[24] Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. Fr., Volume 90 (1962), pp. 487–497 | Numdam | MR | Zbl
[25] Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., Volume 9 (1985), pp. 339–418 | DOI | MR | Zbl
[26] On the uniqueness of compressible fluid motion, Arch. Ration. Mech. Anal., Volume 3 (1959), pp. 271–288 | DOI | MR | Zbl
[27] A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Ration. Mech. Anal., Volume 201 (2011), pp. 727–742 | MR | Zbl
[28] Hypocoercivity, Mem. Am. Math. Soc., Volume 202 (2009) | MR | Zbl
[29] Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Adv. Math., Volume 248 (2013), pp. 534–572 | MR | Zbl
[30] Global solutions to the three-dimensional full compressible Navier-Stokes equations with vacuum at infinity in some classes of large data, SIAM J. Math. Anal., Volume 49 (2017), pp. 162–221 | MR
Cité par Sources :