Abstract
We study the time-asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one-dimensional viscous media, with strictly hyperbolic flux functions. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, “Burgers” rarefaction wave, for initial perturbations w 0 with small mass and localized as w 0(x)=\(\mathcal{O}(|x|^{ - 1} )\)
The proof proceeds by iteration of a pointwise ansatz for the error, using integral representations of its various components, based on Green's functions. We estimate the Green's functions by careful use of the Hopf-Cole transformation, combined with a refined parametrix method. As a consequence of our method, we also obtain rates of decay and detailed pointwise estimates for the error.
This pointwise method has been used successfully in studying stability of shock and constant-state solutions. New features in the rarefaction case are time-varying coefficients in the linearized equations and error waves of unbounded mass \(\mathcal{O}\) (log (t)). These “diffusion waves” have amplitude \(\mathcal{O}\)(t -1/2logt) in linear degenerate transversal fields and \(\mathcal{O}\)(t -1/2logt) in genuinely nonlinear transversal fields, a distinction which is critical in the stability proof.
Similar content being viewed by others
References
Friedman, A., Partial Differential Equation of Parabolic Type, Prentice-Hall, 1964.
Goodman, J., Szepessy, A. & Zumbrun, K., A remark on the stability of viscous shocks, preprint (1992) TRITA-NA-9211, Royal Inst. of Technology, S-100 44 Stockholm, to appear in SIAM J. Math. Anal
Goodman, J., Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986) 325–344.
Harebetian, E., Rarefactions and large-time behavior for parabolic equations and monotone schemes, Comm. Math. Phys. 114 (1986) 527–536.
Hoff, D. & Smoller, J., Solutions in the large for certain nonlinear parabolic systems, Analyse non Lin. 2 (1985) 213–235.
Hopf, E., The partial differential equation u t +uu x =μ xx , Comm. Pure Appl. Math. 3 (1950) 201–230.
Il'in, A. M. & Oleinik, O. A., Behavior of the solution of the Cauchy problem for certain quasi linear equations for unbounded increase of time, Amer. Math. Soc. Translations, Ser. 2, 42 (1964) 19–23.
Kawashima, S., Matsumura, A. & Nishihara, K., Asymptotic behavior of the solutions for the equations of a viscous head-conductive gas. Proc. Japan Acad. 62 (1986) 249–252.
Levi, E. E., Sulle equazioni lineari totalmente ellittiche alle derivate parziali, Rend. Circ. Mat. Palermo, 24 (1907) 275–317.
Liu, T.-P., Linear and nonlinear large-time behavior of solutions to general systems of hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977) 767–797.
Liu, T.-P., Interaction of nonlinear hyperbolic waves, in Nonlinear Analysis, Eds. F.-C. Liu & T.-P. Liu, World Scientific, 1991, 171–184.
Liu, T.-P., Nonlinear Stability of Shock Waves for Viscous Conservation Laws, Memoirs of Amer. Math. Soc., 328 (1986).
Liu, T.-P. Pointwise convergence to shock waves for the system of viscous conservation laws, to appear.
Liu, T.-P. & Zeng, Y., Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, preprint (1994).
Liu, T.-P. & Zumbrun, K., Nonlinear stability of an undercompressive shock of complex Burgers equation, to appear, Comm. Math. Phys.
Liu, T.-P. & Zumbrun, K., On stability of general undercompressive viscous shock waves, preprint (1994).
Matsumura, A. & Nishihara K., Asymptotics towards the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan. J. Appl. Math. 3 (1986) 1–13.
Matsumura, A. & Nishihara, K. Global stability of the rarefaction wave of a one-dimensional model system for compressible gas, Comm. Math. Phys. 144 (1992), 335–335.
Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.
Szepessy, A. & Xin, Z., Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal. 122 (1993) 53–103.
Xin, Z., Asymptotic stability of rarefaction waves for 2×2 viscous hyperbolic conservation laws, J. Diff. Eqs. 73 (1988) 45–77.
Zingano, P., Thesis, New York University, 1990.
Author information
Authors and Affiliations
Additional information
Communicated by {deT.-P. Liu}
Rights and permissions
About this article
Cite this article
Szepessy, A., Zumbrun, K. Stability of rarefaction waves in viscous media. Arch. Rational Mech. Anal. 133, 249–298 (1996). https://doi.org/10.1007/BF00380894
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00380894