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Springer
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: 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 +(F δ) x =εu xx , whereF 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 ast→∞. This is done by weighted energy methods, where the weight (depending onf andg) plays a crucial role.
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