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The aim of this paper is to prove the existence of ${\mathcal R}$-boundedsolution operator families for a resolvent problem on the upper half-spacearising from a compressible fluid model of Korteweg type with free boundarycondition. Such a compressible fluid model was derived by Dunn and Serrin(1985) and studied by Kotschote (2008) as a boundary value problem withnon-slip boundary condition.

Compressible fluid model of Korteweg type with free boundary condition: model problem