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Global regularity of the multi-dimensional compressible Navier-Stokes-Korteweg system

Xiangdi Huang, Weili Meng, Xueyao Zhang

TL;DR

The paper proves global existence of strong solutions to the NSK system on the periodic torus for N=2,3 with large data away from vacuum, by reformulating with an effective velocity and establishing a novel logarithmic-type bound that couples velocity to the density's reciprocal. This enables simultaneous upper and positive lower density bounds, followed by higher-order estimates to close the a priori framework. The 3D result hinges on the new density bounds, while in 2D global regularity is achieved for any adiabatic exponent greater than 1 via energy methods and De Giorgi iteration. The work extends understanding of NSK with density-dependent viscosity and capillarity, covering both isothermal and non-isothermal regimes and providing a unified global-regularity framework.

Abstract

In this paper, we investigate the 2D and 3D compressible Navier-Stokes-Korteweg system derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985], which is widely used to describe compressible fluids with capillarity effects. Specifically, we prove that strong solutions exist globally for arbitrarily large initial data on the periodic torus. One of the key ingredients in the proof lies in establishing a novel critical control relation between the effective velocity and the lower bound of the density, i.e., the upper bound of the effective velocity is controlled by the square root of the logarithm of the reciprocal of the density's lower bound, which plays a crucial role in deriving the lower bound of the density.

Global regularity of the multi-dimensional compressible Navier-Stokes-Korteweg system

TL;DR

The paper proves global existence of strong solutions to the NSK system on the periodic torus for N=2,3 with large data away from vacuum, by reformulating with an effective velocity and establishing a novel logarithmic-type bound that couples velocity to the density's reciprocal. This enables simultaneous upper and positive lower density bounds, followed by higher-order estimates to close the a priori framework. The 3D result hinges on the new density bounds, while in 2D global regularity is achieved for any adiabatic exponent greater than 1 via energy methods and De Giorgi iteration. The work extends understanding of NSK with density-dependent viscosity and capillarity, covering both isothermal and non-isothermal regimes and providing a unified global-regularity framework.

Abstract

In this paper, we investigate the 2D and 3D compressible Navier-Stokes-Korteweg system derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985], which is widely used to describe compressible fluids with capillarity effects. Specifically, we prove that strong solutions exist globally for arbitrarily large initial data on the periodic torus. One of the key ingredients in the proof lies in establishing a novel critical control relation between the effective velocity and the lower bound of the density, i.e., the upper bound of the effective velocity is controlled by the square root of the logarithm of the reciprocal of the density's lower bound, which plays a crucial role in deriving the lower bound of the density.
Paper Structure (5 sections, 13 theorems, 128 equations)

This paper contains 5 sections, 13 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Upper bound for the density (3D)
  3. Positive lower bound for the density (3D)
  4. High-order estimates (3D)
  5. The proof of Theorem \ref{['Thm 1.1']}

Key Result

Theorem 1.1

Let $N=2$ or $N=3$. Assume that $\gamma$ satisfies and that the initial data $(\varrho_0,u_0)$ satisfies where $\underline{\varrho_0}$ and $\overline{\varrho_0}$ are two positive constants. Then the problem Equ1--ini data admits a unique global strong solution $(\varrho,u)$ satisfying for any $0<T<\infty$ and $(x,t)\in \mathbb{T}^N\times[0,T]$, where the constant $C(T)>0$ depends on the initial

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.1
  • proof
  • ...and 16 more