Some Remarks on Structured Keyfitz-Kranzer Systems

Several applications for systems of conservation laws of the form $U_t + (Φ(U) U)_x =0$, $U: R_t\times R_x\rightarrow R^n$ , $n\geq 2$, with $Φ(U) = ϕ(r, Θ): R^n\rightarrow R$, $r = |U|$, and $Θ= U/|U|\in S^n-1$, are obtained by imposing structural conditions to provide a classification framework for solutions dependent on the form of $ϕ(U)$. By prescribing the evolution of a particular eigenvalue, we can categorize classes of such functions, $ϕ$, for which this evolution is met, into depending on either a scalar field $z=rK(Θ)$, where $K: S^n-1\rightarrow R$, or on the vector field $Θ$, 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.