Convergence rate of approximate solutions to weakly coupled nonlinear systems

We study the convergence rate of approximate solutions to nonlinear hyperbolic systems which are weakly coupled through linear source terms. Such weakly coupled $2 \times 2$ systems appear, for example, in the context of resonant waves in gas dynamics equations.

This work is an extension of our previous scalar analysis. This analysis asserts that a One Sided Lipschitz Condition (OSLC, or $\mathrm{Lip}^+$-stability) together with $W^{-1,1}$-consistency imply convergence to the unique entropy solution. Moreover, it provides sharp convergence rate estimates, both global (quantified in terms of the $W^{s,p}$-norms) and local.

We focus our attention on the $\mathrm{Lip}^+$-stability of the viscosity regularization associated with such weakly coupled systems. We derive sufficient conditions, interesting for their own sake, under which the viscosity (and hence the entropy) solutions are $\mathrm{Lip}^+$-stable in an appropriate sense. Equipped with this, we may apply the abovementioned convergence rate analysis to approximate solutions that share this type of $\mathrm{Lip}^+$-stability.