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

ABSTRACT 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 solution to the GBVP throughout the entire interaction region using 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.