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

TL;DR

This paper proves the existence of global strong solutions for the multi-dimensional compressible Navier-Stokes-Korteweg system with large initial data, using advanced analytical techniques, and without symmetry assumptions.

Contribution

It is the first to establish global strong solutions for the full-space compressible Navier-Stokes equations with large initial data and positive density, without symmetry constraints.

Findings

Global strong solutions exist for large initial data in 2D and 3D.

Positive upper and lower bounds for density are derived.

The results apply to physically relevant cases without symmetry assumptions.

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 $\gamma \ge 1$, and $N=3$ with $1 \le \gamma < 8/3$ 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.