Initial-data-independent smallness condition for bounded-domain global well-posedness

Establish global well-posedness for the three-dimensional barotropic compressible Navier–Stokes equations with vacuum in a bounded domain under a smallness condition analogous to those available for the Cauchy problem, with a threshold independent of any initial-data quantities.

Background

The paper proves global well-posedness in a smooth bounded three-dimensional domain with Navier-slip boundary conditions when the initial velocity gradient and initial pressure deviation are sufficiently small in the specified norms. The resulting threshold depends on the known physical and geometric parameters and prescribed density bounds, but it remains dependent on the initial velocity through the smallness quantity itself.

The authors explicitly identify the stronger goal of obtaining a smallness condition analogous to the scaling-invariant conditions used for the Cauchy problem, whose threshold would be independent of any initial data. They leave the bounded-domain version of this question unresolved for future investigation.

References

Certainly, one may wonder whether we can establish the global well-posedness of a1--a4 under a smallness condition analogous to the Cauchy problem in whose threshold is independent of any initial dada. We leave this problem for future investigation.

— Global well-posedness for the compressible Navier--Stokes equations in 3D bounded domains under a new smallness condition  (2610.23091 - Xu et al., 16 Oct 2026) in Remark immediately following Theorem 1.1, Section 1.1 (Background and main result)