Asymptotic analysis for Cahn–Hilliard type phase‐field systems related to tumor growth in general domains

Abstract

This article considers a limit system by passing to the limit in the following Cahn–Hilliard type phase‐field system related to tumor growth as β↘0: α∂tμβ+∂tφβ−normalΔμβ=pfalse(σβ−μβfalse)in1emnormalΩ×false(0,Tfalse),μβ=β∂tφβ+false(−normalΔ+1false)φβ+ξβ+πfalse(φβfalse),1emξβ∈Bfalse(φβfalse)in1emnormalΩ×false(0,Tfalse),∂tσβ−normalΔσβ=−pfalse(σβ−μβfalse)in1emnormalΩ×false(0,Tfalse), in a bounded or an unbounded domain normalΩ⊂RN with smooth‐bounded boundary. Here, N∈double-struckN, T > 0, α > 0, β > 0, p ≥ 0, B is a maximal monotone graph, and π is a Lipschitz continuous function. In the case that Ω is a bounded domain, p and −Δ + 1 are replaced with p(φβ) and −Δ, respectively, and p is a Lipschitz continuous function; Colli, Gilardi, Rocca, and Sprekels (Discrete Contin Dyn Syst Ser S 2017; 10:37–54) have proved existence of solutions to the limit problem with this approach by applying the Aubin–Lions lemma for the compact embedding H1(Ω)↪L2(Ω) and the continuous embedding L2(Ω)↪(H1(Ω))∗. However, the Aubin–Lions lemma cannot be applied directly when Ω is an unbounded domain. The present work establishes existence of weak solutions to the limit problem along with uniqueness and error estimates in terms of the parameter β↘0. To this end, we construct an applicable theory by noting that the embedding H1(Ω)↪L2(Ω) is not compact in the case that Ω is an unbounded domain.

Convergence of the Finite Volume Method on Unstructured Meshes for a 3D Phase Field Model of Solidification

“…Throughout the years, the model derived using the groundwork laid out by Cahn and Hilliard gained more concrete form suitable for particular applications [1,27,28,34]. More recently, the functional theory of phase transition has been formalized further leading to a more rigorous treatment [6,21] and many new applications were found [29,36,33]. Our main focus is on a phase field model formulation that finds use in the modeling of dendritic growth and grain evolution during solidification [26,27,28,43,39,41].…”

Time discretization of a nonlinear phase field system in general domains

“…The case of unbounded domains has the difficult mathematical point that compactness methods cannot be applied directly (related discussions can be found, e.g., in [24,[30][31][32][33]). It would be interesting to construct an applicable theory for the case of unbounded domains and to set assumptions for the case of unbounded domains by trying to keep some typical features in previous works, that is, in the case of bounded domains.…”

Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme

“…Moreover, in [26] the analyses of the same model without the relaxation terms α∂ t µ β and β∂ t ϕ β , has been performed considering regular potentials and allowing P to possess polynomial growth. Besides, as long-time behavior of solutions are concerned, we also mention [41], where the author extends the well-posedness results proved in [11,13], as β ց 0, to the case of unbounded domains. Moreover, we refer to [45], where the authors investigate the long-time behavior of the non-relaxed version of system (1.2)-(1.6), i.e.…”