Global well-posedness of isentropic compressible Navier--Stokes equations with smallness on scaling-invariant quantity in a half-space

Abstract

We investigate the initial-boundary value problem for the three-dimensional isentropic compressible Navier--Stokes equations in the upper half-space with the slip boundary conditions. We prove the global existence and uniqueness of strong solutions in the presence of vacuum and large oscillations. Although scaling frameworks for compressible flows in domains with boundaries have been developed in several settings, the global well-posedness result in the half-space remains far from complete. The system with far-field vacuum admits a natural scaling structure that preserves both the half-space geometry and the slip boundary conditions. Motivated by this scaling, we identify the following \textit{scaling-invariant initial quantity}: $$ \left[ \|\rho_{0}\|_{L^{\infty}}^3 \left( \frac12\|\sqrt{\rho_0} u_0\|_{L^{2}}^{2} +\frac{1}{\gamma-1}\|P(\rho_{0})\|_{L^{1}} \right) +\|\rho_{0}\|_{L^{\infty}}^{\frac{3-\gamma}{2}} \right] \left( \|\nabla u_0\|_{L^2}^2 +\|P(\rho_{0})\|_{L^2}^2 \right). $$ Under the assumption that this quantity is sufficiently small, we establish the global well-posedness of strong solutions. This result provides a half-space counterpart of the scaling-invariant global theory for the Cauchy problem established by Wen (Adv. Math. 482 (2025), Paper No. 110628) and shows that the slip boundary condition is compatible with the system.