Global strong solutions with large initial data for the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Korteweg system

In this paper, we establish global strong solutions for arbitrarily large initial data to the 2D and 3D compressible Navier-Stokes-Korteweg system, also referred to as the quantum Navier-Stokes equations, originally derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985]. Specifically, we prove the existence of global strong solutions for arbitrarily large initial data in the case $N=2$ when $γ\ge 1$, and $N=3$ with $1 \le γ< 8/3$ for the associated Cauchy problem. By employing techniques from Littlewood-Paley theory, range truncation analysis, refined Nash-Moser and De Giorgi iteration methods, we derive positive upper and lower bounds for the density. As a consequence, we are able to treat the whole-space case with strictly positive far-field density. To the best of our knowledge, this is the first result that establishes global strong solutions for physically relevant compressible Navier-Stokes equations in the whole space, without imposing any symmetry or special geometric assumptions on the initial data.

💡 Research Summary

The paper addresses the long‑standing open problem of global strong solutions for the multi‑dimensional compressible Navier‑Stokes‑Korteweg (NSK) system—also known as the quantum Navier‑Stokes equations—on the whole space ℝⁿ without any symmetry or special geometric constraints on the initial data. The authors consider the physically relevant case of shallow‑water type viscosity μ(ρ)=ρ, λ(ρ)=0 and a specific capillarity coefficient κ(ρ)=1/ρ, which leads to the system

∂ₜρ + ∇·(ρu)=0,
∂ₜ(ρu)+∇·(ρu⊗u)+∇P(ρ)=∇·(2ρD(u)) + ∇·(ρ∇∇log ρ).

The Cauchy problem is supplemented with far‑field conditions ρ(x,t)→\barρ>0, u(x,t)→0 as |x|→∞. The main result (Theorem 1.1) proves that for N=2 with γ≥1 and for N=3 with 1≤γ<8/3, any initial data satisfying 0<\underlineρ≤ρ₀≤\barρ, (ρ₀−\barρ)∈H³(ℝⁿ) and u₀∈H²(ℝⁿ) (no smallness assumption) generate a unique global strong solution (ρ,u). Moreover, the solution enjoys uniform positive bounds C(T)⁻¹≤ρ≤C(T) for any finite time interval, and possesses the regularity

(ρ−\barρ)∈C(