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arXiv · 2610.18177

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

Jin Jia·Tao Luo
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Abstract

We consider the two-dimensional isentropic compressible Euler equations for a polytropic gas, $p(ρ)=kρ^γ$. 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 $γ\in(1,1.3]\cup\{\frac{5}{3},\frac{7}{5}\}$: 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 $W^{2,\infty}$ in the self-similar coordinates $(t,x_{1}/t,x_{2})$, even though it may contain logarithmic singularities of the form $c^α(\ln c)^β$ 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.

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Jin Jia, Tao Luo. 2026-10-13. Multi-dimensional centered rarefaction waves of gas dynamics with vacuum states. https://arxiv.org/abs/2610.18177

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