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 ρ+\px Ψ(ρ, u)=0 \pt u+\px Φ(ρ,u)=0, with $(ρ,u)\in{\cal D}\subset\R^2$, 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 $(ρ_0,u_0)=(0,0)$ and assume ${\cal D}\subset\{ρ\ge0\}$. 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 $(ρ_0,u_0)$ 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 \ptρ+ \px\big(ρu\big)=0 \pt u + \px\big(ρ+ γu^2\big) =0 where the parameter $γ:=\frac12 Φ_{uu}(ρ_0,u_0)$ (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 $γ>1$. [truncated]

69 pages, 3 figures