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

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## 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. By a viscous profile of this discontinuity 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. We present some results on the stability of the viscous profiles, which means that a small disturbance tends to zero uniformly as*t*→∞. This is done by weighted energy methods, where the weight (depending on*f* and*g*) plays a crucial role.

## Keywords

Sedimentation Initial Data Stationary Solution Solid Particle Step Function## Preview

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## References

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