Existence of weak solution for compressible fluid models of Korteweg type

TL;DR

This paper proves the existence of global weak solutions for a class of isothermal capillary fluid models, extending previous results to lower dimensions and specific capillary coefficients, with initial data near equilibrium.

Contribution

It establishes the existence of global weak solutions in 2D and 1D for certain capillary fluid models, improving prior results by considering specific coefficients and initial conditions.

Findings

Existence of global weak solutions in 2D for initial data near equilibrium.

Existence of global weak solutions in 1D with large initial data.

Results apply to capillary coefficients close to constant values.

Abstract

This work is devoted to prove existence of global weak solutions for a general isothermal model of capillary fluids derived by J.- E Dunn and J. Serrin (1985) [6], which can be used as a phase transition model. We improve the results of [5] by showing the existence of global weak solution in dimension two for initial data in the energy space, close to a stable equilibrium and with specific choices on the capillary coefficients. In particular we are interested in capillary coefficients approximating a constant capillarity coefficient. To finish we show the existence of global weak solution in dimension one for a specific type of capillary coefficients with large initial data in the energy space.