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

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 $\mu(\rho)=\nu\rho^{\alpha}$ and $\lambda(\rho)=2\nu(\alpha-1)\rho^{\alpha}$, and that the Korteweg stress tensor complies with a generalized Bohm identity of the form $\kappa(\rho)=\varepsilon^2\alpha^2\rho^{2\alpha-3}$, 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 $\nu$.

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.