Well-posedness of the Hele–Shaw–Cahn–Hilliard system
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 3, p. 367-384
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>d 2+1, the existence and uniqueness of solution in C([0,T];H s )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.
DOI : https://doi.org/10.1016/j.anihpc.2012.06.003
Keywords: Hele–Shaw–Cahn–Hilliard, Well-posedness, Blow-up criterion
@article{AIHPC_2013__30_3_367_0,
     author = {Wang, Xiaoming and Zhang, Zhifei},
     title = {Well-posedness of the Hele--Shaw--Cahn--Hilliard system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {30},
     number = {3},
     year = {2013},
     pages = {367-384},
     doi = {10.1016/j.anihpc.2012.06.003},
     zbl = {1291.35240},
     mrnumber = {3061427},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2013__30_3_367_0}
}
Wang, Xiaoming; Zhang, Zhifei. Well-posedness of the Hele–Shaw–Cahn–Hilliard system. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 3, pp. 367-384. doi : 10.1016/j.anihpc.2012.06.003. http://www.numdam.org/item/AIHPC_2013__30_3_367_0/

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