Abstract
This work is concerned with (n-component) hyperbolic systems of balance laws in m space dimensions. First, we consider linear systems with constant coefficients and analyze the possible behavior of solutions as t → ∞. Using the Fourier transform, we examine the role that control theoretical tools, such as the classical Kalman rank condition, play. We build Lyapunov functionals allowing us to establish explicit decay rates depending on the frequency variable. In this way we extend the previous analysis by Shizuta and Kawashima under the so-called algebraic condition (SK). In particular, we show the existence of systems exhibiting more complex behavior than the one that the (SK) condition allows. We also discuss links between this analysis and previous literature in the context of damped wave equations, hypoellipticity and hypocoercivity. To conclude, we analyze the existence of global solutions around constant equilibria for nonlinear systems of balance laws. Our analysis of the linear case allows proving existence results in situations that the previously existing theory does not cover.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aregba-Driollet, D., Natalini, R.: Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37, 1973–2004 (2000)
Bianchini, S., Hanouzet, B., Natalini, R.: Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Comm. Pure Appl. Math. 60(11), 1559–1622 (2007)
Bouchut, F.: Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Stat. Phys. 95, 113–170 (1999)
Carbou, G., Hanouzet, B.: Relaxation approximation of Kerr model for the three dimensional initial boundary value problem. JHDE (2010, in press)
Carbou, G., Hanouzet, B., Natalini, R.: Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation. J. Differ. Equ. 246(1), 291–319 (2009)
Chern, I.-L.: Long time effect of relaxation for hyperbolic conservation laws. Comm. Math. Phys. 172(1), 39–55 (1995)
Coron, J.-M.: Control and nonlinearity. Mathematical Surveys and Monographs, Vol. 136. American Mathematical Society, New York, 2007
Coulombel, J.-F., Goudon, T.: The strong relaxation limit of the multidimensional isothermal Euler equations. Trans. Am. Math. Soc. 359(2), 637–648 (2007)
Dafermos, C.M.: A system of hyperbolic conservation laws with frictional damping. Theoretical, experimental and numerical contributions to the mecanics of fluids and solids. Z. Angew. Math. Phys. 46, S294–S307 (1995)
Dafermos, C.M.: Hyperbolic conservation laws in continuum physics. Springer, Berlin, 2000
Dafermos, C.M.: Hyperbolic systems of balance laws with weak dissipation. J. Hyperbolic Differ. Equ. 3, 505–527 (2006)
Hanouzet, B., Huynh, P.: Approximation par relaxation d un systFme de Maxwell non linTaire. C.R. Acad. Sci. Paris STr. I Math. 330, 193–198 (2000)
Hanouzet, B., Natalini, R.: Global existence of smooth solutions for partially dissipative hyperbolic systems with convex entropy. Arch. Ration. Mech. Anal. 169, 89–117 (2003)
Haraux, A.: Semi-linear hyperbolic problems in bounded domains. Math. Rep. 3(1), 1–281 (1987)
Haraux, A., Zuazua, E.: Decay estimates for some semilinear damped hyperbolic problems. Arch. Ration. Mech. Anal. 100(2), 191–206 (1988)
Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)
Hosono, T., Shuichi, K.: Decay property of regularity-loss type and application to some nonlinear hyperbolic–elliptic systems. Math. Models Methods Appl. Sci. 16(11), 1839–1859 (2006)
Hsiao, L., Liu, T.-P.: Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Comm. Math. Physics 143, 599–605 (1992)
Hsiao, L., Serre, D.: Global existence of solutions for the system of compressible adiabatic flow through porous media. SIAM J. Math. Anal. 27, 70–77 (1996)
Ide, K., Haramoto, K., Kawashima, S.: Decay property of regularity-loss type for dissipative Timoshenko system. Math. Models Methods Appl. Sci. 18(5), 647–667 (2008)
Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Springer, Berlin, 1995 (Reprint of the 1980 edition)
Li, T.-T.: Global Classical Solutions for Quasilinear Hyperbolic Systems. Masson, Paris, 1994
Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variable. Springer, New York, 1984
Mascia, C., Natalini, R.: On relaxation hyperbolic systems violating the Shizuta– Kawashima condition. ARMA (2010, in press)
Natalini, R.: Recent results on hyperbolic relaxation problems. Analysis of systems of conservation laws (Aachen, 1997). Chapman and Hall, Boca Raton, 128–198, 1999
Nishida, T.: Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics. Département de mathématiques, Université Paris Sud, Orsay, Pré publications mathématiques d Orsay, 78-02, 1978
Orive, R., Zuazua, E.: Long-time behavior of solutions to a nonlinear hyperbolic relaxation system. J. Differ. Equ. 228(1), 17–38 (2006)
Ruggeri, T., Serre, D.: Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. Q. Appl. Math. 62, 163–179 (2004)
Serre, D.: Systè mes de lois de conservation, tome 1. Diderot editeur, Arts et Sciences, Paris, New York, Amsterdam, 1996
Shizuta, Y., Kawashima, S.: Systems of equations of hyperbolic–parabolic type with application to the discrete boltzmann equation. Hokkaido Math. J. 14, 249–275 (1985)
Sideris, T.C., Thomases, B., Wang, D.: Long time behaviour of solutions to the 3D compressible Euler equations with damping. Comm. PDE 28, 795–816 (2003)
Trelat, E.: Contrôle optimal: théorie et applications. Vuilbert, collection Mathématiques Concrè tes (2005)
Tzavaras, A.: Materials with internal variables and relaxation of conservation laws. Arch. Rational Mech. Anal. 146, 129–155 (1999)
Villani, C.: Hypocoercive diffusion operators. International Congress of Mathematicians, Vol. III. European Mathematical Society, Zürich, 473–498, 2006
Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 182 (2010, to appear). http://www.umpa.ens-lyon.fr/cvillani/Cedrif/B09.Hypoco.pdf
Yong, W.A.: Entropy and global existence for hyperbolic balance laws. Arch. Ration. Mech. Anal. 172, 247–266 (2004)
Zeng, Y.: Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Ration. Mech. Anal. 150(3), 225–279 (1999)
Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev., 47(2), 197–243 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Le Bris
Rights and permissions
About this article
Cite this article
Beauchard, K., Zuazua, E. Large Time Asymptotics for Partially Dissipative Hyperbolic Systems. Arch Rational Mech Anal 199, 177–227 (2011). https://doi.org/10.1007/s00205-010-0321-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-010-0321-y