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Some diffusive capillary models of Korteweg type. (Quelques modèles diffusifs capillaires de type Korteweg.) (French. Abridged English version) Zbl 1386.76070
Summary: The purpose of this Note is to propose new diffusive capillary models of Korteweg type and discuss their mathematical properties. More precisely, we introduce viscous models which provide some additional information on the behavior of the density close to vacuum. We actually prove that if some compatibility conditions between diffusion and capillarity are satisfied, some extra regularity information on a quantity involving the density is available. We obtain a non-trivial equality deduced from the special structure of the momentum equation. This note generalizes to some extent the authors’ previous works on the Korteweg model (with constant capillary coefficient) and on the shallow water equation.

MSC:
76D45 Capillarity (surface tension) for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
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