Advertisement

Communications in Mathematical Physics

, Volume 176, Issue 1, pp 45–71 | Cite as

Scalar conservation laws with discontinuous flux function: II. On the stability of the viscous profiles

  • S. Diehl
  • N. -O. Wallin
Article

Abstract

The equation\(\frac{{\partial u}}{{\partial t}} + \frac{\partial }{{\partial x}}\left( {H(x)f(u) + \left( {1 - H(x)} \right)g(u)} \right) = 0\), whereH is Heaviside's step function, appears for example in continuous sedimentation of solid particles in a liquid, in two-phase flow, in traffic-flow analysis and in ion etching. The discontinuity of the flux function atx=0 causes a discontinuity of a solution, which is not uniquely determined by the initial data. By a viscous profile of this discontinuity we mean a stationary solution ofu t +(Fδ) x u xx , whereFδ is a smooth approximation of the discontinuous flux, i.e.,H is smoothed. We present some results on the stability of the viscous profiles, which means that a small disturbance tends to zero uniformly ast→∞. This is done by weighted energy methods, where the weight (depending onf andg) plays a crucial role.

Keywords

Sedimentation Initial Data Stationary Solution Solid Particle Step Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chancelier, J.-Ph., Cohen de Lara, M., and Pacard, F.: Analysis of a conservation pde with discontinuous flux: A model of settler. SIAM J. Appl. Math. 54(4), 954–995 (1994)Google Scholar
  2. 2.
    Diehl, S., and Wallin, N.-O.: Scalar conservation laws with discontinuous flux function: I. The viscous profile condition. To appear in Commun. Math. Phys.Google Scholar
  3. 3.
    Diehl, S.: On scalar conservation laws with point source and discontinuous flux function. To appear in SIAM J. Math. Anal., 1995Google Scholar
  4. 4.
    Diehl, S.: A conservation law with point source and discontinuous flux function modelling continuous sedimentation. To appear in SIAM J. Appl. Math., 1996Google Scholar
  5. 5.
    Gimse, T., and Risebro, N. H.: Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal. 23(3), 635–648 (1992)Google Scholar
  6. 6.
    Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95, 325–344 (1986)Google Scholar
  7. 7.
    Hörmander, H.: Non-linear hyperbolic differential equations Technical report, Department of Mathematics, Lund University, 1988. ISSN 0327-8475Google Scholar
  8. 8.
    Il'in, A. M., and Olinik, O. A.: Behaviour of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of the time. Amer. Math. Soc. Transl. Ser. 2, 42, 19–23 (1964)Google Scholar
  9. 9.
    Jones, C. K. R., Gardner, R., and Kapitula, T.: Stability of travelling waves for non-convex scalar viscous conservation laws. Comm. Pure Appl. Math. 46, 505–526 (1993)Google Scholar
  10. 10.
    Kawashima, S., and Matsumura, A.: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101, 97–127 (1985)Google Scholar
  11. 11.
    Liu, T.-P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Am. Math. Soc. 56(328), 1–108 (1985)Google Scholar
  12. 12.
    Liu, T-P., and Xin, Z.: Stability of viscous shock waves associated with a system of nonstrictly hyperbolic conservation laws. Comm. Pure Appl. Math. 45, 361–388 (1992)Google Scholar
  13. 13.
    Matsumura, A., and Nishihara, K.: On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2, 17–25 (1985)Google Scholar
  14. 14.
    Matsumura, A., and Nishihara, K.: Asymptotic stability of travelling waves for scalar conservation laws with non-convex nonlinearity. Commun. Math. Phys. 165, 83–96 (1994)Google Scholar
  15. 15.
    Mochon, S.: An analysis of the traffic on highways with changing surface conditions. Math. Model. 9(1), 1–11 (1987)Google Scholar
  16. 16.
    Osher, S., and Ralston, J.:L 1 stability of travelling waves with applications to convective porous media flow. Comm. Pure Appl. Math. 35, 737–749 (1982)Google Scholar
  17. 17.
    Ross, D. S.: Two new moving boundary problems for scalar conservation laws. Comm. Pure Appl. Math. 41, 725–737 (1988)Google Scholar
  18. 18.
    Sattinger, D. H.: On the stability of waves of nonlinear parabolic systems. Adv. Math. 22, 312–355 (1976)Google Scholar
  19. 19.
    Szepessy, A., and Xin, Z.: Nonlinear stability of viscous shock waves. Arch. Rational Mech. Anal. 122, 53–103 (1993)Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • S. Diehl
    • 1
  • N. -O. Wallin
    • 1
  1. 1.Department of MathematicsLund Institute of TechnologyLundSweden

Personalised recommendations