Perturbation of singular equilibria of hyperbolic two-component systems: a universal hydrodynamic limit

Abstract

We consider one-dimensional, locally finite interacting particle systems with two conservation laws which under Eulerian hydrodynamic limit lead to two-by-two systems of conservation laws: \pt \rho +\px \Psi(\rho, u)=0 \pt u+\px \Phi(\rho,u)=0, with $(\rho,u)\in{\cal D}\subset\R^2$, where ${\cal D}$ is a convex compact polygon in $\R^2$. The system is typically strictly hyperbolic in the interior of ${\cal D}$ with possible non-hyperbolic degeneracies on the boundary $\partial {\cal D}$. We consider the case of isolated singular (i.e. non hyperbolic) point on the interior of one of the edges of ${\cal D}$, call it $(\rho_0,u_0)=(0,0)$ and assume ${\cal D}\subset\{\rho\ge0\}$. This can be achieved by a linear transformation of the conserved quantities. We investigate the propagation of small nonequilibrium perturbations of the steady state of the microscopic interacting particle system, corresponding to the densities $(\rho_0,u_0)$ of the conserved quantities. We prove that for a very rich class of systems, under proper hydrodynamic limit the propagation of these small perturbations are \emph{universally} driven by the two-by-two system \pt\rho + \px\big(\rho u\big)=0 \pt u + \px\big(\rho + \gamma u^2\big) =0 where the parameter $\gamma:=\frac12 \Phi_{uu}(\rho_0,u_0)$ (with a proper choice of space and time scale) is the only trace of the microscopic structure. The proof is valid for the cases with $\gamma>1$. [truncated]