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Scalar conservation laws with discontinuous flux function. I: The viscous profile condition. (English) Zbl 0845.35067
The equation $${\partial u\over \partial t} {\partial\over \partial x} (H(x) f(u)(1- H(x)) g(u))= 0$$, where $$H$$ is Heviside’s step function, arises on continuous sedimentation of solid particles in a liquid, in two-phase flow in porous media etc. The discontinuity of the flux function causes a discontinuity of a solution, which is not uniquely determined by the initial data. The equation can be written as a triangular $$2\times 2$$ nonstrictly hyperbolic system. This augmentation is nonunique and a natural definition is given by means a viscous profiles.
A viscous profile is a stationary solution of $$u_t(F^\delta)_x= \varepsilon u_{xx}$$, where $$F^\delta$$ is a smooth approximation of the discontinuous flux, i.e. $$H$$ is smoothed. In terms of the $$2\times 2$$, the discontinuity at $$x= 0$$ is either a regular Lax, an under- or over-compressive or a degenerate shock wave. In some cases, depending on $$f$$ and $$g$$, there is unique viscous profile (e.g. undercompressive and regular Lax waves) and in some cases there are infinitely many (e.g. overcompressive waves). It is proved the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution at $$x= 0$$ and the viscous profile condition.
Reviewer: L.G.Vulkov (Russe)

##### MSC:
 35L67 Shocks and singularities for hyperbolic equations 35L60 First-order nonlinear hyperbolic equations 76S05 Flows in porous media; filtration; seepage 76T99 Multiphase and multicomponent flows
##### Keywords:
discontinuous flux; viscous profiles
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##### References:
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