Some remarks on structured Keyfitz-Kranzer systems

Abstract

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

36 pages