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

Fei Jiang, Pengfei Li, Jiawei Wang

TL;DR

The paper analyzes the diffuse-interface NSK system in a slab with Navier–slip boundaries and constant viscosity, focusing on the large capillarity regime $\kappa\to\infty$. It proves the global-in-time existence of strong solutions with uniform-in-$\kappa$ energy bounds for small initial perturbations, and shows that, as $\kappa\to\infty$, the 3D NSK dynamics converge to a 2D incompressible NS-like system governing the horizontal velocity, with a precise description of the limit fields. The analysis hinges on refined energy methods, handling a challenging nonlinear term via transport equations and anisotropic estimates, and a compactness framework to pass to the limit. The results reveal a stabilizing elastic-like effect of capillarity and establish a rigorous link between the 3D NSK dynamics and a reduced 2D NS-type model in the high-capillarity regime. These contributions advance understanding of capillarity-driven asymptotics and provide a methodology for extracting reduced-dimensional limits in complex fluid models.

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 $κ^2$, 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 $κ>0$, where the solutions converge to a unique solution of 2D incompressible NS-like equations as $κ$ goes to infinity.

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

TL;DR

The paper analyzes the diffuse-interface NSK system in a slab with Navier–slip boundaries and constant viscosity, focusing on the large capillarity regime . It proves the global-in-time existence of strong solutions with uniform-in- energy bounds for small initial perturbations, and shows that, as , the 3D NSK dynamics converge to a 2D incompressible NS-like system governing the horizontal velocity, with a precise description of the limit fields. The analysis hinges on refined energy methods, handling a challenging nonlinear term via transport equations and anisotropic estimates, and a compactness framework to pass to the limit. The results reveal a stabilizing elastic-like effect of capillarity and establish a rigorous link between the 3D NSK dynamics and a reduced 2D NS-type model in the high-capillarity regime. These contributions advance understanding of capillarity-driven asymptotics and provide a methodology for extracting reduced-dimensional limits in complex fluid models.

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 , 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 , where the solutions converge to a unique solution of 2D incompressible NS-like equations as goes to infinity.

Paper Structure

This paper contains 8 sections, 22 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['thm2']}
  4. Basic estimates
  5. A priori stability estimates
  6. Proof of Theorem \ref{['thm2']}
  7. Proof of Theorem \ref{['thm3']}
  8. Analysis tools

Key Result

Theorem 2.1

Let $\mu$, $\kappa$, $d >0$ be given, and $\bar{\rho}$ be defined by 202501232128. There exist constants $c_1$, $c_2$ and $\chi$, where such that, for any $(\sigma^0,u^0)$ satisfying 202412051008, the initial-boundary value problem 20220202081737--1.8x has a unique global strong solution Moreover the solution satisfies the following estimates and where $\mathcal{P}:=\beta+\kappa \partial_3\

Theorems & Definitions (30)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 20 more