Well-posedness of the Hele–Shaw–Cahn–Hilliard system

Wang, Xiaoming; Zhang, Zhifei

doi:10.1016/j.anihpc.2012.06.003

Wang, Xiaoming ; Zhang, Zhifei

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 3, pp. 367-384

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Résumé

We study the well-posedness of the Hele–Shaw–Cahn–Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. For initial data in $H^{s}$ , $s > \frac{d}{2} + 1$ , the existence and uniqueness of solution in $C ([0, T]; H^{s}) \cap L^{2} (0, T; H^{s + 2})$ that is global in time in the two dimensional case ( $d = 2$ ) and local in time in the three dimensional case ( $d = 3$ ) are established. Several blow-up criterions in the three dimensional case are provided as well. One of the tools that we utilized is the Littlewood–Paley theory in order to establish certain key commutator estimates.

MR Zbl

DOI : 10.1016/j.anihpc.2012.06.003

Keywords: Hele–Shaw–Cahn–Hilliard, Well-posedness, Blow-up criterion

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     author = {Wang, Xiaoming and Zhang, Zhifei},
     title = {Well-posedness of the {Hele{\textendash}Shaw{\textendash}Cahn{\textendash}Hilliard} system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {367--384},
     publisher = {Elsevier},
     volume = {30},
     number = {3},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.06.003},
     mrnumber = {3061427},
     zbl = {1291.35240},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.06.003/}
}

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Wang, Xiaoming; Zhang, Zhifei. Well-posedness of the Hele–Shaw–Cahn–Hilliard system. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 3, pp. 367-384. doi: 10.1016/j.anihpc.2012.06.003

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