Global-in-time strong solutions for the 2D and 3D generalized compressible Navier-Stokes-Korteweg system with arbitrarily large initial data

Abstract

In 1901, Korteweg formulated a constitutive equation for the Cauchy stress tensor to provide a continuum mechanical model for capillarity within fluids. Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133,1985] in 1985 further modified the system of compressible fluids based on the Korteweg theory of capillarity. Since then, for the 2D and 3D compressible Navier-Stokes-Korteweg system, the global existence of strong solutions with arbitrarily large initial data have remained a challenging open problem. In this paper, we provide an affirmative answer to this longstanding open problem. Specifically, under the assumption that the viscosity coefficients satisfy a BD-type algebraic relation of the form $μ(ρ)=νρ^α$ and $λ(ρ)=2ν(α-1)ρ^α$, and that the Korteweg stress tensor complies with a generalized Bohm identity of the form $κ(ρ)=\varepsilon^2α^2ρ^{2α-3}$, we establish the global existence of strong solutions for the 2D and 3D systems in torus with arbitrarily large regular initial data. The analysis is carried out in the intermediary non-dispersive regime, characterized by the condition that the capillarity coefficient constant $\varepsilon$ does not exceed the viscosity constant $ν$. This result provides the first proof of the global-in-time existence of strong solutions for the 3D general Navier-Stokes-Korteweg system with arbitrarily large initial data in the non-dispersive regime.