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

In this paper, we consider the compressible fluid model of Korteweg type in a critical case where the derivative of pressure equals 0 at a given constant state. We show that the system admits a unique, global strong solution for small initial data in the maximal L-p-L-q regularity class. Consequently, we also prove the decay estimates of the solutions to the nonlinear problem. 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 low frequencies.