Perturbation of Singular Equilibria of Hyperbolic Two-Component Systems: A Universal Hydrodynamic Limit

Balint Toth and Benedek Valko

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 $(\rho,u)\in{\cal D}\subset{\mathbb R}^2$, where ${\cal D}$ is a convex compact polygon in ${\mathbb R^2$. The system is \emph{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 \emph{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 $\gamma>1$.

The proof relies on the relative entropy method and thus, it is valid only in the regime of smooth solutions of the pde. But there are essentially new elements: in order to control the fluctuations of the terms with Poissonian (rather than Gaussian) decay coming from the low density approximations we have to apply refined pde estimates. In particular Lax entropies of these pde systems play a \emph{not merely technical} key role in the main part of the proof.

Paper:
Available as PDF (728 Kbytes).
Author(s):
Balint Toth, <balint@math.bme.hu>
Benedek Valko, <valko@math.bme.hu>
Publishing information:
Journal of Statistical Physics, vol. 112 (2003) pp. 497-521