Rarefaction Wave Interaction and Existence of a Global Smooth Solution in the Blood Flow Model With Time-Dependent Body Force

This paper presents the collision between two rarefaction waves for the one-dimensional blood flow model with non-constant body force. We establish the Riemann solutions using phase plane analysis, which are no longer self-similar. By employing Riemann invariants, we transform the system into a non-reducible diagonal system in Riemann invariant coordinates. This interaction problem then becomes a Goursat boundary value problem (GBVP) within the interaction region. We demonstrate that no vacuum forms within the interaction domain if the boundary data on characteristics is vacuum-free, and a vacuum only appears if the two rarefaction waves fail to penetrate each other within a finite time. Furthermore, we prove the existence and uniqueness of the $C^1$ solution to the GBVP throughout the entire interaction region using $\text{a priori}{C}^1$ bounds. Finally, we present the results of the interaction, showing that either the rarefaction waves completely penetrate each other or form a vacuum in the solution at a sufficiently large time during the process of penetration.