Abstract
We study the equations describing the motion of a thermal non-equilibrium gas in three space dimensions. It is a hyperbolic system of six equations with a relaxation term. The dissipation mechanism induced by the relaxation is weak in the sense that the Shizuta-Kawashima criterion is violated. This implies that a perturbation of a constant equilibrium state consists of two parts: one decays in time while the other stays. In fact, the entropy wave grows weakly along the particle path as the process is irreversible. We study thermal properties related to the well-posedness of the nonlinear system. We also obtain a detailed pointwise estimate on the Green’s function for the Cauchy problem when the system is linearized around an equilibrium constant state. The Green’s function provides a complete picture of the wave pattern, with an exact and explicit leading term. Comparing with existing results for one dimensional flows, our results reveal a new feature of three dimensional flows: not only does the entropy wave not decay, but the velocity also contains a non-decaying part, strongly coupled with its decaying one. The new feature is supported by the second order approximation via the Chapman-Enskog expansions, which are the Navier-Stokes equations with vanished shear viscosity and heat conductivity.
Similar content being viewed by others
References
Bianchini S., Hanouzet B., Natalini R.: Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Commun. Pure Appl. Math. 60(11), 1559–1622 (2007)
Butler B.J. Jr.: Perturbation series for eigenvalues of analytic non-symmetric operators. Arch. Math. 10, 21–27 (1959)
Chen G.Q., Levermore C.D., Liu T.-P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure. Appl. Math. 47(6), 787–830 (1994)
Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin, 2000
Friedrichs K.O.: Symmetric hyperbolic linear differential equations. Commun. Pure. Appl. Math. 7, 345–392 (1954)
Hanouzet B., Natalini R.: Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Ration. Mech. Anal. 169(2), 89–117 (2003)
Hsiao L., Liu T.-P.: Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Phys. 143(3), 599–605 (1992)
Kato T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181–205 (1975)
Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, New York, 1976
Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Doctoral Thesis, Kyoto University, 1983
Kawashima S., Yong W.-A.: Decay estimates for hyperbolic balance laws. Z. Anal. Anwend. 28(1), 1–33 (2009)
Li D.L.: The Green’s function of the Navier-Stokes equations for gas dynamics in \({\mathbb{R}^3}\) . Commun. Math. Phys. 257, 579–619 (2005)
Liu T.-P.: Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108(1), 153–175 (1987)
Liu, T.-P., Noh, S.E.: Wave propagation for the compressible Navier-Stokes equations. J. Hyperbolic Differ. Equ. (accepted)
Liu T.-P., Wang W.: The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions. Commun. Math. Phys. 196(1), 145–173 (1998)
Liu T.-P., Yu S.-H.: Green’s function for Boltzmann equation, 3-D waves. Bullet. Inst. Math. Academia Sinica 1(1), 1–78 (2006)
Liu T.-P., Yu S.-H.: Solving Boltzmann equation, part I: Green’s function. Bullet. Inst. Math. Academia Sinica 6(2), 115–243 (2011)
Liu, T.-P., Zeng, Y.: Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Mem. Am. Math. Soc. 125(599), viii+120 (1997)
Nishida, T.: Nonlinear hyperbolic equations and related topics in fluid dynamics. Publications Mathématiques D’Orsay, Département de Mathématique, Université de Paris-Sud, Orsay, pp. 78–82, 1978
Shizuta Y., Kawashima S.: Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14, 249–275 (1985)
Shu C.-W., Zeng Y.: High-order essentially non-oscillatory scheme for viscoelasticity with fading memory. Quart. Appl. Math. 55(3), 459–484 (1997)
Vincenti, W., Kruger, C. Jr: Introduction to Physical Gas Dynamics. Krieger, Malabar, 1986
Yong W.-A.: Singular perturbations of first-order hyperbolic systems with stiff source terms. J. Differ. Equ. 155, 89–132 (1999)
Yong W.-A.: Entropy and global existence for hyperbolic balance laws. Arch. Ration. Mech. Anal. 172(2), 247–266 (2004)
Zeng, Y.: L 1 asymptotic behavior of compressible, isentropic, viscous 1-D flow. Commun. Pure Appl. Math. 47, 1053–1082 (1994)
Zeng Y.: Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Ration. Mech. Anal. 150(3), 225–279 (1999)
Zeng Y.: Gas flows with several thermal nonequilibrium modes. Arch. Ration. Mech. Anal. 196, 191–225 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Bressan
Rights and permissions
About this article
Cite this article
Zeng, Y. Thermal Non-Equilibrium Flows in Three Space Dimensions. Arch Rational Mech Anal 219, 27–87 (2016). https://doi.org/10.1007/s00205-015-0892-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-015-0892-8