Global strong solutions of compressible Navier–Stokes equations with small scaling invariant

This paper is concerned with the Cauchy problem for the three-dimensional isentropic compressible Navier–Stokes equations, considering both far-field vacuum and non-vacuum cases. Our main novelties are twofold: First, we establish a new and refined scaling-invariant initial condition, which is simpler and less restrictive than those used in recent works {e.g., Wen [Adv. Math. 482, 110628 (2025)]}. Specifically, global existence and uniqueness of strong solutions are proven under the smallness of the quantity: ρ̄3E0∇u0L22+ρ̄γE01+ρ̄1+γ/3E02/3. This framework dispenses with the previously required ‖ρ0γ‖L22 term, thereby relaxing the initial data assumptions. A key challenge addressed is controlling the supercritical nonlinear convective terms in the momentum equation without relying on the L2 norm of the pressure. This is achieved via a combination of energy estimates, interpolation trick, and a refined analysis of the density’s upper bound, showing that the L2 pressure estimate is unnecessary. Second, the result is extended to the physically relevant case of non-zero far-field density (ρ∞ > 0), a scenario with additional complexity due to the non-trivial equilibrium state. This generalization is achieved by adapting our a priori estimates to a modified energy functional, further demonstrating the robustness and generality of our new framework.