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Global strong solutions with large initial data for the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Korteweg system

Xiangdi Huang, Yongteng Gu, Muxi Lei

TL;DR

This work proves the global existence of strong solutions for the multi-dimensional compressible Navier–Stokes–Korteweg (quantum NSK) system in $\\mathbb{R}^N$ with large initial data, specifically addressing the Cauchy problem for $N=2$ with $\gamma\ge1$ and $N=3$ with $1\le\gamma<8/3$, under strictly positive far-field density. The authors transform to an effective velocity $v = u + \nabla \log\rho$, yielding a parabolic structure, and develop a truncated De Giorgi iteration to obtain a strict density lower bound in the whole space, without symmetry assumptions. They combine Nash–Moser-type iterations for high regularity with Littlewood–Paley techniques to control nonlinearities and derive upper bounds for $\rho$, followed by high-order estimates to achieve global well-posedness. The results constitute the first global strong-solution theory for physically relevant compressible NSK equations in the whole space with large data, advancing understanding of capillarity and quantum-viscous effects in multi-dimensional flows. The methods provide a framework to handle far-field behavior and vacuum tendencies, with potential implications for related capillary models and energy-structure-based analyses.

Abstract

In this paper, we establish global strong solutions for arbitrarily large initial data to the 2D and 3D compressible Navier-Stokes-Korteweg system, also referred to as the quantum Navier-Stokes equations, originally derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985]. Specifically, we prove the existence of global strong solutions for arbitrarily large initial data in the case $N=2$ when $γ\ge 1$, and $N=3$ with $1 \le γ< 8/3$ for the associated Cauchy problem. By employing techniques from Littlewood-Paley theory, range truncation analysis, refined Nash-Moser and De Giorgi iteration methods, we derive positive upper and lower bounds for the density. As a consequence, we are able to treat the whole-space case with strictly positive far-field density. To the best of our knowledge, this is the first result that establishes global strong solutions for physically relevant compressible Navier-Stokes equations in the whole space, without imposing any symmetry or special geometric assumptions on the initial data.

Global strong solutions with large initial data for the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Korteweg system

TL;DR

This work proves the global existence of strong solutions for the multi-dimensional compressible Navier–Stokes–Korteweg (quantum NSK) system in with large initial data, specifically addressing the Cauchy problem for with and with , under strictly positive far-field density. The authors transform to an effective velocity , yielding a parabolic structure, and develop a truncated De Giorgi iteration to obtain a strict density lower bound in the whole space, without symmetry assumptions. They combine Nash–Moser-type iterations for high regularity with Littlewood–Paley techniques to control nonlinearities and derive upper bounds for , followed by high-order estimates to achieve global well-posedness. The results constitute the first global strong-solution theory for physically relevant compressible NSK equations in the whole space with large data, advancing understanding of capillarity and quantum-viscous effects in multi-dimensional flows. The methods provide a framework to handle far-field behavior and vacuum tendencies, with potential implications for related capillary models and energy-structure-based analyses.

Abstract

In this paper, we establish global strong solutions for arbitrarily large initial data to the 2D and 3D compressible Navier-Stokes-Korteweg system, also referred to as the quantum Navier-Stokes equations, originally derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985]. Specifically, we prove the existence of global strong solutions for arbitrarily large initial data in the case when , and with for the associated Cauchy problem. By employing techniques from Littlewood-Paley theory, range truncation analysis, refined Nash-Moser and De Giorgi iteration methods, we derive positive upper and lower bounds for the density. As a consequence, we are able to treat the whole-space case with strictly positive far-field density. To the best of our knowledge, this is the first result that establishes global strong solutions for physically relevant compressible Navier-Stokes equations in the whole space, without imposing any symmetry or special geometric assumptions on the initial data.
Paper Structure (8 sections, 24 theorems, 224 equations)

This paper contains 8 sections, 24 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic facts on Littlewood-Paley theory
  4. Some useful estimates
  5. A priori estimates: Upper bound of $\rho$
  6. Lower bound of $\rho$
  7. High-order estimates
  8. Proof of Theorem \ref{['thm:global_existence_cauchy']}

Key Result

Theorem 1.1

Let $N=2$ or $N=3$. Assume that $\gamma$ satisfies Let $\bar{\rho} > 0$ be the far field behavior of density. Assume that the initial data $(\rho_0, u_0)$ satisfies where $\underline{\varrho}$ and $\bar{\varrho}$ are positive constants. Then the Cauchy problem NSK_simplified-far field admits a unique global strong solution $(\rho, u)$ satisfying for any $0 < T < \infty$ and $(x,t) \in \mathbb{R}

Theorems & Definitions (51)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2: Embedding inequalities
  • Lemma 2.3: Interpolation Inequalities
  • Remark 2.2
  • ...and 41 more