Abstract:This paper is devoted to the study of strong solutions for the compressible Navier-Stokes/Allen-Cahn system in bounded domain $\Omega\subset\mathbb R^3$, allowing for the presence of initial vacuum. A characteristic of this system is the strong coupling between density and the Allen-Cahn equation, which leads to strong degeneracy in vacuum regions. Under a compatibility condition on the initial phase-field variable, we establish the local existence and uniqueness of strong solutions for $0\le\rho_0\in W^{1,q}$ with $q\in(3,6)$, $u_0\in H_0^1$ and $\chi_0\in H^2$. Owing to time-weighted estimates, no compatibility condition is required for the velocity, but these estimates introduce a singularity in proving uniqueness. We then establish a criterion for the possible breakdown of such a local strong solution at finite time in terms of blow-up of the quantities $\|\nabla u\|_{L_t^{1} L_x^{\infty}}$, $\|u\|_{L_t^{2} L_x^{\infty}}$ and $\|\nabla \chi\|_{L_t^{2} L_x^{\infty}}$.
| Subjects: | Analysis of PDEs (math.AP) |
| Cite as: | arXiv:2605.24568 [math.AP] |
| (or arXiv:2605.24568v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24568 arXiv-issued DOI via DataCite (pending registration) |
From: Yinghua Li [view email]
[v1]
Sat, 23 May 2026 13:02:40 UTC (42 KB)
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