Sharp Convergence Rate of the Glimm Scheme for General Nonlinear Hyperbolic Systems

Consider a general strictly hyperbolic, quasilinear system, in one space dimension % $$ u_t+A(u) u_x=0, \eqno (1) $$ % where $u \mapsto A(u)$, $u\in\Omega\subset\real^N$, is a smooth matrix-valued map. Given an initial datum $u(0,\cdot)$ with small total variation, let $u(t,\cdot)$ be the corresponding (unique) vanishing viscosity solution of (1) obtained as limit of solutions to the viscous parabolic approximation $u_t+A(u) u_x=\mu u_{xx}$, as $\mu\to 0$. For every $T\geq 0$, we prove the a-priori bound % $$ \big\|u^\eps(T,\cdot)-u(T,\cdot)\big\|_{\elleuno}=o(1)\cdot\sqrt\eps\,|\log\eps| \eqno (2) $$ % for an approximate solution $u^\eps$ of (1) constructed by the Glimm scheme, with mesh size $\Delta x = \Delta t = \eps$, and with a suitable choice of the sampling sequence. This result provides for general hyperbolic systems the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems of conservation laws $u_t+F(u)_x =0$ satisfying the classical Lax or Liu assumptions on the eigenvalues $\lambda_k(u)$ and on the eigenvectors $r_k(u)$ of the Jacobian matrix $A(u)=DF(u)$. The estimate (2) is obtained introducing a new wave interaction functional with a cubic term that controls the nonlinear coupling of waves of the same family and at the same time decreases at interactions by a quantity that is of the same order of the product of the wave strength times the change in the wave speeds. This is precisely the type of errors arising in a wave tracing analysis of the Glimm scheme, which is crucial to control in order to achieve an accurate estimate of the convergence rate as~(2).