Global weak solutions of the Navier-Stokes-Korteweg Equations in one dimension

TL;DR

This paper establishes the global existence of weak solutions for the one-dimensional Navier-Stokes-Korteweg equations with degenerate viscosity and capillarity coefficients, extending previous results and removing upper bounds on viscosity exponents.

Contribution

It proves the existence of weak solutions under very degenerate conditions, allowing viscosity coefficients to be zero on sets with positive measure, without upper bounds on the exponent.

Findings

Proved global weak solutions for degenerate viscosity and capillarity.

Extended previous results by removing upper bounds on viscosity exponents.

Handled very degenerate viscosity coefficients effectively.

Abstract

We prove the global existence of weak solutions of the one-dimensional Navier-Stokes-Korteweg (NSK) equations when the viscosity and the capillarity coefficients are power functions of the density, which may be zero on a set with positive measure. The proofs are based on a truncation argument combined with the Energy estimate and BD Entropy. Notably, we do not require any upper bound on the exponent of the power of the viscosity coefficient. In particular, we are able to consider very degenerate viscosity coefficient and to substantially improve previous results.