## On some compressible fluid models: Korteweg, lubrication, and shallow water systems.(English)Zbl 1106.76436

Summary: We give some mathematical results for an isothermal model of capillary compressible fluids derived by J. E. Dunn and J. Serrin [Arch. Rat. Mech. Anal. 88, No. 2, 95–133 (1985; Zbl 0582.73004)], which can be used as a phase transition model. We consider a periodic domain $$\Omega = T^d$$ ($$d=2$$ or 3) or a strip domain $$\Omega = (0,1)\times T^{d-1}$$. We look at the dependence of the viscosity $$\mu$$ and the capillarity coefficient $$\kappa$$ with respect to the density $$\rho$$. Depending on the cases we consider, different results are obtained. We prove for instance for a viscosity $$\mu(\rho) = \nu\rho$$ and a surface tension $$\kappa(\rho)=\tilde\kappa=\text{const}$$ the global existence of weak solutions of the Korteweg system without smallness assumption on the data. This model includes a shallow water model and a lubrication model. We discuss the validity of the result for the shallow water equations since the density is less regular than in the Korteweg case.

### MSC:

 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35Q35 PDEs in connection with fluid mechanics 76D45 Capillarity (surface tension) for incompressible viscous fluids

Zbl 0582.73004
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