Asymptotical Behavior of Global Solutions of the Navier-Stokes-Korteweg Equations with Respect to Capillarity Number at Infinity

TL;DR

This paper studies the behavior of solutions to the 3D Navier-Stokes-Korteweg equations as the capillarity number becomes very large, showing convergence to 2D Navier-Stokes-like equations under certain conditions.

Contribution

It is the first to analyze the asymptotic behavior of global solutions of NSK equations with large capillarity number in a 3D setting, revealing convergence to 2D NS-like equations.

Findings

Solutions converge to 2D NS-like equations as capillarity number increases.

Global strong solutions exist for small initial perturbations.

Convergence holds under well-prepared initial data.

Abstract

Vanishing capillarity in the Navier-Stokes-Korteweg (NSK) equations has been widely investigated, in particular, it is well-known that the NSK equations converge to the Navier-Stokes (NS) equations by vanishing capillarity number. To our best knowledge, this paper first investigates the behavior of large capillary number, denoted by $\kappa^2$, for the global(-in-time) strong solutions with small initial perturbations of the three-dimensional (3D) NSK equations in a slab domain with Navier(-slip) boundary condition. Under the well-prepared initial data, we can construct a family of global strong solutions of the 3D incompressible NSK equations with respect to $\kappa>0$, where the solutions converge to a unique solution of 2D incompressible NS-like equations as $\kappa$ goes to infinity.