Multi-dimensional centered rarefaction waves of gas dynamics with vacuum states

We consider the two-dimensional isentropic compressible Euler equations for a polytropic gas, $p(\rho )=k{\rho }^{\gamma }$. The $R$–vacuum state–$R$ configuration—two rarefaction waves bounding a vacuum—occurs when the two constant states move apart with sufficiently large relative velocity and is essentially the only vacuum-forming Riemann pattern in the absence of external forces. Local-in-time rarefaction fronts have been constructed only away from vacuum, where the sound speed is bounded away from zero; their construction through the vacuum edge, and hence the structural stability of the $R$–vacuum state–$R$ configuration, has remained open. We establish this stability for $\gamma \in (1,1.3]\cup \{\frac{5}{3},\frac{7}{5}\}$: for sufficiently small perturbations, the Riemann problem admits a continuous entropy solution on a fixed time interval, with the domain partitioned by four characteristic surfaces and the solution in the three central regions belonging to $W^{2,\mathrm{\infty }}$ in the self-similar coordinates $(t,x_1\mathrm{/}t,x_2)$, even though it may contain logarithmic singularities of the form $c^{\alpha }(\ln c{)}^{\beta }$ at the vacuum edge. The proof develops a geometric formulation directly on the first-order system, together with a direct analysis of the initial slice at the vacuum edge, where the formal high-order expansion fails.