Five Open Problems — Compressible Navier–Stokes near vacuum

Definitions and mathematical statement

Denis Serre's Open Problem 3 concerns compressible Navier--Stokes flow with density-independent constant viscosities near vacuum. In a standard barotropic model the unknowns are density $\rho\geq0$ and velocity $u\in{\mathbb{R}}^d$, satisfying \[ \partial_t\rho+\operatorname{div}(\rho u)=0, \qquad \partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)+\nabla p(\rho) =\operatorname{div}\mathbb S(u), \] with constant-coefficient viscous stress \[ \mathbb S(u)=\mu(\nabla u+\nabla u^{\mathsf T}) +\lambda(\operatorname{div}u)I. \] This notation illustrates the constant-viscosity issue; the source does not specify whether the intended system is barotropic or includes energy/temperature, nor does it supply a pressure law, viscosity inequalities, dimension-dependent parameters, domain, boundary data, initial data, or solution class.

The question is whether the same unphysical one-dimensional consequence identified by Hoff and Serre has a counterpart in dimensions $d\geq2$. A rigorous analogue must identify that consequence and the one-dimensional hypotheses, then formulate the corresponding higher-dimensional property to prove or disprove. Neither the consequence nor its precise theorem is included in the supplied statement, so no particular vacuum-interface pathology, nonexistence result, or regularity failure is substituted for it. "Near vacuum" also requires a specified regime, such as vanishing density or a limiting low-density family.

The supplied research summary reports the one-dimensional constant-viscosity/vacuum incompatibility as established, but the multidimensional analogue unresolved. It describes the one-dimensional mechanism as forcing solutions into "only recursively weaker classes," without defining those classes or reproducing a result. It mentions later vacuum-interface work by Hoff--Serre and followers but supplies no precise citation settling or disproving the requested analogue. These are supplied assessments, not independently verified progress or a new failure example.

Short English statement

Serre asks whether a known one-dimensional pathology of constant-viscosity compressible flow near vacuum also occurs in higher dimensions. The record omits the exact Hoff--Serre consequence and its hypotheses, so the counterpart cannot yet be stated as a unique theorem or counterexample problem.

Sources

• Local UnsolvedMath registry, permanent ID 2700003, exact source code catalogue entry, "Five Open Problems --- Compressible Navier--Stokes near vacuum"; proposer Denis Serre, 2012; status partially solved, classification PARTIAL-PROGRESS. The provenance's default difficulty is L3, while the supplied later suggestion and current metadata give L4.
• Serre, problems (2012), Open Problem 3, https://perso.ens-lyon.fr/denis.serre/DPF/Ouverts.pdf. The provenance records PDF-text extraction, accessed July 29, 2026, and says the corrected author PDF and an open journal version present this among five open problems. No separate journal citation is supplied.
• Hoff and Serre, credited for the one-dimensional phenomenon, and unspecified followers studying multidimensional vacuum interfaces. No exact paper titles, theorem statements, dates, or authoritative citations are included. The supplied summary says no explicit multidimensional settlement was verified.