Time discretization of a nonlinear phase field system in general domains

Abstract

This paper deals with the nonlinear phase field system ∂ t (θ + ℓϕ) − ∆θ = fin Ω × (0, T ),Here N ∈ N, T > 0, ℓ > 0, f is a source term, β is a maximal monotone graph and π is a Lipschitz continuous function. We note that in the above system the nonlinearity β + π replaces the derivative of a potential of double well type. Thus it turns out that the system is a generalization of the Caginalp phase field model and it has been studied by many authors in the case that Ω is a bounded domain. However, for unbounded domains the analysis of the system seems to be at an early stage. In this paper we study the existence of solutions by employing a time discretization scheme and passing to the limit as the time step h goes to 0. In the limit procedure we face with the difficulty that the embedding H 1 (Ω) ֒→ L 2 (Ω) is not compact in the case of unbounded domains. Moreover, we can prove an interesting error estimate of order h 1/2 for the difference between continuous and discrete solutions.2010 Mathematics Subject Classification: 93C20, 34B15, 35A35, 82B26.

Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems

“…We assumed (J2) in reference to assumptions in [12,13,18,19]. Moreover, we set (J3) in reference to assumptions in [7,11]. Then the function R ∋…”

“…as h = h j → +0. Therefore we can conclude that there exists a solution of (P) by combining (4.30), (4.32)-(4.42), (C13) and by observing that f h → f strongly in L 2 (0, T ; H) as h → +0 (see [7,Section 5]). Next we establish uniqueness of solutions to (P).…”

“…where T > 0, L : H → H is a linear positive selfadjoint operator, B : D(B) ⊂ H → H, A j : D(A j ) ⊂ H → H (j = 1, 2) are linear maximal monotone selfadjoint operators, V j (j = 1, 2) are linear subspaces of V satisfying D(A j ) ⊂ V j (j = 1, 2), Φ : D(Φ) ⊂ H → H is a maximal monotone operator, L : H → H is a Lipschitz operator, f : (0, T ) → H and θ 0 ∈ V 1 , ϕ 0 , v 0 ∈ V 2 are given. Moreover, in reference to [6,7], we deal with the problem…”

“…Therefore combining (5.5) and (5.14) means that there exists a constant C 9 = C 9 (T ) > 0 such that for all h > 0 (see [7,Section 5]), we can prove Theorem 1.3 by Lemma 5.1.…”

Temperature dependent extensions of the Cahn-Hilliard equation

“…Although the system was not originally related to the law of thermodynamics, the equations of Caginalp have been extensively investigated in literature, both from an analytical point of view [15,24,27,28], as well as from numerical simulations [2,19]. A variation of this model has been used to study dynamical undercooling at the liquid-solid interface, as well as asymptotically relating to the free boundary model (sharp interface) [11].…”