Global Solutions for a Simplified Shallow Elastic Fluids Model

Abstract

The Cauchy problem for a simplified shallow elastic fluids model, one $3\times 3$ system of Temple’s type, is studied and a global weak solution is obtained by using the compensated compactness theorem coupled with the total variation estimates on the first and third Riemann invariants, where the second Riemann invariant is singular near the zero layer depth $\left(\rho =0\right)$. This work extends in some sense the previous works, (Serre, 1987) and (Leveque and Temple, 1985), which provided the global existence of weak solutions for $2\times 2$ strictly hyperbolic system and (Heibig, 1994) for $n\times n$ strictly hyperbolic system with smooth Riemann invariants.