Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system
Haspot, Boris; Charve, Frédéric (2013), Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system, SIAM Journal on Mathematical Analysis, 45, 2, p. 469-494. http://dx.doi.org/10.1137/120861801
Type
Article accepté pour publication ou publiéExternal document link
http://hal.archives-ouvertes.fr/hal-00635983/fr/Date
2013Journal name
SIAM Journal on Mathematical AnalysisVolume
45Number
2Publisher
SIAM
Pages
469-494
Publication identifier
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Show full item recordAbstract (EN)
In the first part of this paper, we prove the existence of global strong solution for Korteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the physically relevant solutions for a hyperbolic system, we show that the global strong solution of the Korteweg system converges in the case of a $\gamma$ law for the pressure ($P(\rho)=a\rho^{\gamma}$, $\gamma>1$) to entropic solution of the compressible Euler equations. In particular it justifies that the Korteweg system is suitable for selecting the physical solutions in the case where the Euler system is strictly hyperbolic. The problem remains open for a Van der Waals pressure because in this case the system is not strictly hyperbolic and in particular the classical theory of Lax and Glimm (see \cite{Lax,G}) can not be used.Subjects / Keywords
Korteweg system; strong solution; vanishing capillarity-viscosity; compressible Euler systemRelated items
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