Convergence of first-order quasilinear hyperbolic systems to hyperbolic-parabolic systems

We provide a framework to study the zero relaxation time limit of Cauchy problem for first-order quasilinear hyperbolic systems with relaxation in several space dimensions. For this purpose, we construct an approximate solution by a formal asymptotic expansion with initial layer corrections. The system of the leading term in the expansion is governed by a hyperbolic-parabolic system. Under appropriate structural and partial dissipation conditions, we justify rigorously the validity of the asymptotic expansion on a time interval independent of the relaxation time, provided that the system of the leading term admits a local-in-time smooth solution. The main theorem of the present paper includes the results obtained in previous works and applies to additional examples of models arising in fluid mechanics.