Local and global interaction for nongenuinely nonlinear hyperbolic conservation laws (Q2710415)

The author considers a strictly hyperbolic system of conservation laws NEWLINE\[NEWLINEu_t+f(u)_x=0, \quad x\in{\mathbb R},\;t>0,\;u\in{\mathbb R}^n.NEWLINE\]NEWLINE The system is assumed to be cubic nonlinear at \(u=0\) for some characteristic field (thus it is not genuinely nonlinear), that is for some \(i=1,\ldots,n\) NEWLINE\[NEWLINE{\partial\lambda_i\over\partial r_i}(0)=0,\qquad {\partial^2\lambda_i\over\partial r_i^2}(0)\not=0,NEWLINE\]NEWLINE where \(\lambda_i\) is an eigenvalues of the matrix \(f'(u)\) and \(r_i\) is the corresponding right eigenvector. In this case solutions to the Riemann problem have more complicated structure than for genuinely nonlinear systems, they include not only rarefaction and shock curves but also so-called wave curves.NEWLINENEWLINENEWLINEThe author studies all possible interactions between elementary waves and estimates strength of interactions. For the global interaction estimate the author introduces a new norm to measure the wave strength which is equivalent to the total variation norm. It is proved that this norm doesn't increase after interactions. This result allows to establish convergence of the Glimm scheme and it is also usefull for study of long-time behavior of solutions of nongenuinely nonlinear hyperbolic systems.