The global well-posedness of the compressible fluid model of Korteweg type for the critical case

TL;DR

This paper proves the global existence and decay of solutions for a critical compressible Korteweg fluid model with small initial data, using maximal regularity and decay estimates.

Contribution

It establishes the global well-posedness of the compressible Korteweg model in the critical case, a significant extension of previous results.

Findings

Unique global strong solutions for small initial data

Decay estimates for solutions to the nonlinear problem

Decay properties of linearized solutions under low-frequency assumptions

Abstract

In this paper, we consider the compressible fluid model of Korteweg type in a critical case where the derivative of pressure equals to $0$ at the given constant state. It is shown that the system admits a unique, global strong solution for small initial data in the maximal $L_p$-$L_q$ regularity class. As a result, we also prove the decay estimates of the solutions to the nonliner problem. In order to obtain the global well-posedness for the critical case, we show $L_p$-$L_q$ decay properties of solutions to the linearized equations under an additional assumption for a low frequencies.