A phase-field model of grain boundary motion

Very singular diffusion equations: second and fourth order problems

“…In fact, a global-in-time weak solution is constructed in [72]. If η 2 in front of θ t is replaced by a strictly positive function having a similar property, a global-in-time weak solution is constructed in [71] and the uniqueness is given for one-dimension. Existence and uniqueness of one dimensional model with κη x x replaced by (σ η t + κη) x x has been established in [70].…”

Global solvability of a model for grain boundary motion with constraint

“…By a similar argument to [13,, we can show Theorem 2.2 in the case when [η 0 , θ 0 ] ∈ D 0 . Namely we shall construct a global solution to (P) with good initial data [η 0 , θ 0 ] ∈ D 0 by discussing the convergence of the following approximate problems (P) ε with real parameter ε ∈ (0, 1], as ε ↓ 0:…”

“…Then there is a subsequence {θ n k } of {θ n } and a function θ ∈ W 1,2 (0, T ; H) ∩ L ∞ (0, T ; H 1 0 ) such that θ n k → θ weakly in W 1,2 (0, T ; H) and weakly * in L ∞ (0, T ; Proof. We give the sketch of the proof of Proposition 3.2, which is a similar arguments to [13,Proposition 3.2].…”

“…Recently the authors [13,14] discussed the same system from the theoretical point of view. In fact, assuming that [η 0 , θ 0 ] is the good initial data in H 1 (Ω) × H 1 0 (Ω), the authors [13] showed the existence of a weak solution in any dimension of space and the uniqueness in dimension one.…”

Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system

“…Recently, the existence results relative to the Kobayashi-Warren-Carter system were reported in several literatures [2], [3], [6], [8], [9], [10], and in each of these, the forerunners adopted some approximation problems, configured as gradient systems of the following types of relaxed free-energies:…”