Global strong solution for the Korteweg system with quantum pressure in dimension $N\geq 2$

TL;DR

This paper proves the existence of global strong solutions for a quantum pressure model of capillary fluids in dimensions two and higher, extending previous local results and handling vacuum states using De Giorgi's method.

Contribution

It establishes global strong solutions for the Korteweg system with quantum pressure, including cases with vacuum, using novel energy estimates and De Giorgi's regularity technique.

Findings

Global strong solutions exist in dimensions N≥2.

Solutions can be extended beyond initial lifespan under specific conditions.

The method handles vacuum states via De Giorgi's regularity approach.

Abstract

This work is devoted to prove the existence of global strong solution in dimension $N\geq 2$ for a isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) (see \cite{fDS}), which can be used as a phase transition model. We will restrict us to the case of the so called compressible Navier-Stokes system with quantum pressure which corresponds to consider the capillary coefficient $\kappa(\rho)=\frac{\kappa_1}{\rho}$ with $\kappa_1>0$. In a first part we prove the existence of strong solution in finite time for large initial data with a precise bound by below on the life span $T^*$. This one depends on the norm of the initial data $(\rho_0,v_0)$. The second part consists in proving the existence of global strong solution with particular choice on the capillary coefficient ( where $\kappa_1=\mu^2$) and on the viscosity tensor which corresponds to the viscous shallow water case $-2\mu{\rm div}(\rho Du)$. To do this we derivate different energy estimate on the density and the effective velocity $v$ which ensures that the strong solution can be extended beyond $T^*$. The main difficulty consists in controlling the vacuum or in other words to estimate the $L^\infty$ norm of $\frac{1}{\rho}$. The proof relies mostly on a method introduced by De Giorgi \cite{DG} (see also Ladyzhenskaya et al in \cite{La} for the parabolic case) to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument.