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Large Time Asymptotics for Partially Dissipative Hyperbolic Systems

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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.

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References

  • Aregba-Driollet, D., Natalini, R.: Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37, 1973–2004 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Chern, I.-L.: Long time effect of relaxation for hyperbolic conservation laws. Comm. Math. Phys. 172(1), 39–55 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Dafermos, C.M.: Hyperbolic conservation laws in continuum physics. Springer, Berlin, 2000

    MATH  Google Scholar 

  • Dafermos, C.M.: Hyperbolic systems of balance laws with weak dissipation. J. Hyperbolic Differ. Equ. 3, 505–527 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Haraux, A.: Semi-linear hyperbolic problems in bounded domains. Math. Rep. 3(1), 1–281 (1987)

    MathSciNet  Google Scholar 

  • Haraux, A., Zuazua, E.: Decay estimates for some semilinear damped hyperbolic problems. Arch. Ration. Mech. Anal. 100(2), 191–206 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    MATH  Google Scholar 

  • Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variable. Springer, New York, 1984

    Book  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    MATH  MathSciNet  Google Scholar 

  • 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)

    MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Zeng, Y.: Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Ration. Mech. Anal. 150(3), 225–279 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev., 47(2), 197–243 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. CMLA, ENS Cachan, CNRS, Universud, 61 avenue du Président Wilson, 94230, Cachan, France

    Karine Beauchard

  2. BCAM – Basque Center for Applied Mathematics, Bizkaia Technology Park, Building 500 E-48160 Derio, Basque Country, Spain

    Enrique Zuazua

Authors
  1. Karine Beauchard
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  2. Enrique Zuazua
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Correspondence to Enrique Zuazua.

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Communicated by C. Le Bris

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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

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  • DOI: https://doi.org/10.1007/s00205-010-0321-y

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