# Scalar conservation laws with discontinuous flux function: I. The viscous profile condition

- 149 Downloads
- 23 Citations

## Abstract

The equation\(\frac{{\partial u}}{{\partial t}} + \frac{\partial }{{\partial x}}\left( {H(x)f(u) + \left( {1 - H(x)} \right)g(u)} \right) = 0\), where*H* 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 at*x*=0 causes a discontinuity of a solution, which is not uniquely determined by the initial data. The equation can be written as a triangular 2×2 non-strictly hyperbolic system. This augmentation is non-unique and a natural definition is given by means of viscous profiles. By a viscous profile we mean a stationary solution of*u*_{ t }+(*F*^{δ})_{ x }=ε*u*_{ xx }, where*F*^{δ} is a smooth approximation of the discontinuous flux, i.e.,*H* is smoothed. In terms of the 2×2 system, the discontinuity at*x*=0 is either a regular Lax, an under-or overcompressive, a marginal under- or overcompressive or a degenerate shock wave. In some cases, depending on*f* and*g*, there is a unique viscous profile (e.g. undercompressive and regular Lax waves) and in some cases there are infinitely many (e.g. overcompressive waves). The main purpose of the paper is to show the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution at*x*=0 and the viscous profile condition.

## Keywords

Shock Wave Sedimentation Solid Particle Uniqueness Condition Hyperbolic System## Preview

Unable to display preview. Download preview PDF.

## References

- 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.Diehl, S.: On scalar conservation laws with point source and discontinuous flux function. To appear in SIAM J. Math. Anal., 1995Google Scholar
- 3.Diehl, S.: A conservation law with point source and discontinuous flux function modelling continuous sedimentation. To appear in SIAM J. Appl. Math., 1996Google Scholar
- 4.Diehl, S., and Wallin, N.-O.: Scalar conservation laws with discontinuous flux function: II. On the stability of the viscous, profiles. To appear in Commun. Math. Phys.Google Scholar
- 5.Gimse, T., and Risebro, N. H.: Riemann problems with a discontinuous flux function. In: Engquist, B., and Gustavsson, B., editors, Third International Conference on Hyperbolic Problems, Theory, Numerical Methods and Applications, volume I, 1990, pp. 488–502Google Scholar
- 6.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
- 7.Godunov, S. K.: A finite difference method for the numerical computations of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 271–306 (1959)Google Scholar
- 8.Holden, L., and Høegh-Krohn, R.: A class of
*N*nonlinear hyperbolic conservation laws. J. Differential Equations 84, 73–99 (1990)Google Scholar - 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.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
- 11.Mochon, S.: An analysis of the traffic on highways with changing surface conditions. Math. Model. 9(1), 1–11 (1987)Google Scholar
- 12.Oleinik, O. A.: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation Uspekhi Mat. Nauk 14, 165–170 (1959) Amer. Math. Soc. Transl. Ser. 2 33, 285–290 (1964)Google Scholar
- 13.Ross, D. S.: Two new moving boundary problems for scalar conservation laws. Comm. Pure Appl. Math. 41, 725–737 (1988)Google Scholar