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# Undercompressive Shocks in Systems of Conservation Laws

Conference paper

## Abstract

In this paper, we describe recent progress in our understanding of Riemann problems that involve undercompressive shock waves for 2 × 2 systems of nonstrictly hyperbolic conservation laws. A 2 × 2 system of conservation laws , , where

$$ {U_t} + F{(U)_x} = 0 $$

(1.1)

*U = U(x,t)*∈**R**^{2},*F*:**R**^{2}→**R**^{2}, is*nonstrictly hyperbolic*if the eigenvalues λ_{1}(*U*) ≤ λ_{2}(*U*) of*dF(U)*are real, but not distinct for every*U*. As defined in [4], system (1.1) has an*umbilic point*at*U = U**if*dF(U*)*is a multiple of the identity. Hyperbolic equations with an isolated umbilic point can be classified locally according to properties of the quadratic map*d*^{2}*F(U*)*. Since linear changes of coordinates do not affect the shocks or rarefaction waves for quadratic nonlinearities*F*, the general family of quadratic nonlinearities*F*with a unique umbilic point can be reduced to a two parameter family, which we write as$$ Q(u,v) = d(a{u^3}/3 + b{u^2}v + u{v^2}),a \ne 1 + {b^2} $$

(1.2)

*d*denotes gradient with respect to*U = (u,v)*.### Keywords

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## Copyright information

© Springer-Verlag New York Inc. 1990