On vanishing pressure limit of continuous solutions to the isentropic Euler equations

The vanishing pressure limit of continuous solutions isentropic Euler equations is analyzed, which is formulated as small parameter [Formula: see text] goes to [Formula: see text]. Due to the characteristics being degenerated in the limiting process, the resonance may cause the mass concentration. It is shown that in the pressure vanishing process, for the isentropic Euler equations, the continuous solutions with compressive initial data converge to the mass concentration solution of pressureless Euler equations, and with rarefaction initial data converge to the continuous solutions globally. It is worth to point out: [Formula: see text] converges in [Formula: see text], while [Formula: see text] converges in [Formula: see text], due to the structure of pressureless Euler equations. To handle the blow-up of density [Formula: see text] and spatial derivatives of velocity [Formula: see text], a new level set argument is introduced. Furthermore, we consider the convergence rate with respect to [Formula: see text], both [Formula: see text] and the area of characteristic triangle are [Formula: see text] order, while the rates of [Formula: see text] and [Formula: see text] depend on the further regularity of the initial data of [Formula: see text].