Some remarks on structured Keyfitz-Kranzer systems

Abstract

Several applications for systems of conservation laws of the form $U_t + (\Phi (U) U)_x =0$, $U: R_t\times R_x\rightarrow R^n$ , $n\geq 2$, with $\Phi (U) = \phi (r, \Theta): R^n\rightarrow R$, $r = |U|$, and $\Theta = U/|U|\in S^{n-1}$, are obtained by imposing structural conditions to provide a classification framework for solutions dependent on the form of $\phi(U)$. By prescribing the evolution of a particular eigenvalue, we can categorize classes of such functions, $\phi$, for which this evolution is met, into depending on either a scalar field $z=rK(\Theta)$, where $K: S^{n-1}\rightarrow R$, or on the vector field $\Theta$, and find for which the amplitude of solutions may blow up in finite time. As a consequence, solutions to the corresponding Riemann problems can be divided into those with are classical and those which involve delta-shocks and/or vacuum states.