Dynamics of Compressible Non-isothermal Fluids of Non-Newtonian Korteweg Type

Abstract

The equations of motion for compressible fluids of Korteweg type as derived by Dunn and Serrin in 1985 are studied in their full generality: the Korteweg tensor is assumed to be an arbitrary function of the form $\mathcal{K} := \left( - \rho^2 \partial_{\rho} \psi + \rho \nabla \cdot ( \kappa \nabla \rho) \right) \mathcal{I} - \kappa \nabla \rho \otimes \nabla \rho, \quad \kappa := 2 \rho \partial_{\phi} \psi(\rho,\theta,\phi), \quad \phi:=|\nabla \rho|^2,$ where $\psi$ denotes Helmholtz free energy density and the capillarity $\kappa$ is subject only to the natural positivity conditions $\kappa(\rho,\theta,\phi) >0, \quad \kappa(\rho,\theta,\phi) + 2 \phi \partial_{\phi} \kappa(\rho,\theta,\phi) > 0, \quad \rho, \theta, \phi \ge 0.$ The viscous stress is supposed to be of generalized Newtonian type. The main result of the paper establishes well-posedness on domains with compact boundaries; the proof is based on refined methods of maximal regularity.