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

Yongteng Gu, Xiangdi Huang, Weili Meng, Huitao Zhou

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.

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 and , and that the Korteweg stress tensor complies with a generalized Bohm identity of the form , 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 does not exceed the viscosity constant . 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.
Paper Structure (11 sections, 20 theorems, 397 equations)

This paper contains 11 sections, 20 theorems, 397 equations.

Table of Contents

  1. Introduction
  2. Main strategy
  3. Upper bounds for the density
  4. Positive lower bounds for the density
  5. Derivation of higher-order estimates
  6. Energy estimates
  7. Upper bound for the density
  8. Positive lower bound for the density
  9. Higher-order estimates
  10. Proof of Theorem \ref{['Thm 1.1']}
  11. Appendix

Key Result

Theorem 1.1

Let $N\in \{2,3\}$ and $\beta=\sqrt{1-\frac{\varepsilon^2}{\nu^2}}$. Assume that $(\alpha,\gamma,\beta)$ satisfies where $\beta_N^-(x)$ is the positive root of the following quadratic equation: Moreover, assume that the initial data $(\rho_0,u_0)$ satisfy where $\underline{\rho_0}$ and $\overline{\rho_0}$ are positive constants. Then the problem Equ1--ini data admits a unique global strong solu

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 37 more