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We show the convergence of the zero relaxation limit in systems of $2 \times2$ hyperbolic conservation laws with stochastic initial data. Precisely,solutions converge to a solution of the local equilibrium approximation as therelaxation time tends to zero. The initial data are assumed to depend onfinitely many random variables, and the convergence is then proved via theappropriate analogues of the compensated compactness methods used in treatingthe deterministic case. We also demonstrate the validity of this limit in thecase of the semi-linear $p$-system; the well-posedness of both the system andits equilibrium approximation are proved, and the convergence is shown with noa priori conditions on solutions. This model serves as a prototype forunderstanding how asymptotic approximations can be used as control variates forhyperbolic balance laws with uncertainty.

Analysis of the Zero Relaxation Limit of Systems of Hyperbolic Conservation Laws with Random Initial Data