Large data global strong solutions to the quantum Navier-Stokes equations with degenerate viscosity

Abstract

This work is devoted to study the Cauchy problem for the quantum Navier-Stokes equations in $\mathbb R^N$, $N=2,3$, with density dependent viscosities $μ(ρ)=ρ,λ(ρ)=0$, quantum capillarity $κ(ρ)=1/ρ$, which was derived by Dunn and Serrin (see Arch. Ration. Mech. Anal. 88(2):95-133, 1985) as a phase transition model. For arbitrarily large initial data satisfying suitable regularity and decay assumptions, with density positive at every finite point and vanishing at infinity, we prove the existence and uniqueness of global strong solutions without symmetry assumptions. The admissible pressure range is $γ\ge 1$ in two dimensions and $1\leγ< 8/3$ in three dimensions. On every finite time interval, the density remains pointwise comparable to its initial profile, and both the effective velocity and the logarithmic density gradient remain bounded. The central estimates require no uniform positive lower bound for the density. When $γ>1$, a suitable combination of the density and effective velocity equations allows the maximum principle to yield an $L^\infty$ bound for the effective velocity using only an upper bound for the density. We then introduce the relative density with respect to a fixed profile determined by the initial data. In the isothermal case $γ=1$, we combine estimates for this relative density with a scalar inequality involving the effective velocity and the logarithmic density to close the coupled estimates and obtain the effective velocity bound. Combining the relative-density bounds with weighted gradient estimates and localized De Giorgi iteration, we control the logarithmic density gradient and close the higher-order estimates.