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

*(English)*Zbl 0845.35067The 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.

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 |

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