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On the well-posedness for the Euler-Korteweg model in several space dimensions. (English) Zbl 1125.76060
The Cauchy problem for an Euler-Korteweg model, taking into account capillarity effects in isothermal fluids, is analyzed in arbitrary space dimension $$N$$. A well-posedness result is proved for $$H^{s+1}\times H^s$$ perturbations (for $$s>N/2+1$$) of either a constant state, or a travelling wave profile. Moreover “almost-global” existence is proved for small perturbations, and a blow-up result is shown.

##### MSC:
 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35Q35 PDEs in connection with fluid mechanics
##### Keywords:
capillary fluids; Korteweg stress; commutator estimate
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