Our goal in this paper is to investigate a Cahn–Hilliard system based on microforce and an unconstrained order parameter with Neumann conditions. Our study begins by establishing a priori estimates, demonstrating the existence and uniqueness of solutions, and proving that the solutions converge to those of the generalized Caginalp system over finite time intervals as a small parameter approaches zero.
BoyerF. (1999). Mathematical study of multi-phase ow under shear through order parameter formula-tion. Asymptotic Analysis, 20, 175–212.
2.
CagilnalpG. (1988). Conserved-phase field system; Implication for kinetic undercooling. Applications and Review-B, 38, 789–791.
3.
CaginalpG. (1986). An analysis of a phase field model of a free boundary. Archive for Rational Mechanics and Analysis, 92, 205–245.
4.
CaginalpG. (1990). The dynamics of a conserved phase-field system; Stefan-like, hele-shaw and cahn-hilliard models as asymptotic limits. IMA Journal of Applied Mathematics , 44, 77–94.
5.
CherfilsL.MiranvilleA. (2007). Some results on the asymptotic behavior of the caginalp system with singularpotentials. Advances in Mathematical Sciences and Applications, 17, 107–129.
6.
CherfilsL.MiranvilleA. (2009). On the caginalp system with dynamic boundary conditions and singular potentials. Applicable Analysis, 54, 89–115.
FriedE.GurtinM. E. (1994). Dynamic solid-solid transitions with phase characterized by an order parameter. Journal of Physics D, 72, 287–308.
10.
GalC. G.GrasselliM. (2008). The nonisothermal allen-cahn equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems A, 22, 1009–1040.
11.
GiacominG.LebowitzJ. (1997). Phase segregation dynamics in particle systems with long range interaction I, macroscopic limits. Journal of Statistical Physics, 87, 37–61.
12.
GiorgiC.GrasseliM.PataV. (1999). Uniform attractors for a phase-field model with memory and quadratic nonlinearity. Indiana University Mathematics Journal, 48, 1395–1446.
13.
GurtinM. (1996). Generalized ginzburg-landau and cahn-hilliard equations based on a microforce balance. Journal of Physics D, 92, 178–192.
14.
MiranvilleA. (2019). The Cahn–Hilliard equation: Recent advances and applications. In CBMS-NSF Regional conference series in Appl. Math. 95, society for industrial and Appl. Math. (SIAM), Philadelphia, PA.
15.
NimiA. C.LangaF. D. R. (2024). Cahn–hilliard system with proliferation term. Asymptotic Analysis, 140, 123–145.
16.
NimiA. C.LangaF. D. R.BissouesseJ. P.MoukokoD.BatchiM. (2023). Robust exponential attractors for the cahn-hilliar-Oono-Navier-Stokes system. Discrete and Continuous Dynamical Systems-Series S, https://doi.org/10.3934/dcdss.2022214
