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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
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