Global existence of a solution to a phase field model for supercooling

Global solution to a phase field model with irreversible and constrained phase evolution

“…in SI, (2 0 <6 and 0 < dtx < A a.e. in Q, (2)(3)(4)(5)(6)(7)(8)(9)(10)(11) and such that the following equations hold a.e. in Q:…”

“…On the other hand, in [9], these authors, together with Pierluigi Colli, are able to show an existence result for a system consisting of (1.7) and a modification of (1.8), where the term -A\ is missing (it corresponds to setting v = 0 in (1)(2)(3)(4)(5)(6)(7)(8)). The latter is the typical inclusion of the phase relaxation model with irreversible evolution (introduced in [10]).…”

“…(1.6) oO Actually, it is not difficult to show [8] that in this setting the Second Principle of Thermodynamics is satisfied in the form of the Clausius-Duhem inequality. The balance and constitutive laws-together with (1.3) and (1.4)-yield the following system: the subdifferential of jT[0,i] (resp.…”

“…Anyway, if we rewrite (1.11) as csdt0 -kA9 = dtX + vdtx + , (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) a simple comparison in (1.12) leads to…”

“…in fi, (1.16) dn9 = dnx = 0 a.e. on dtt x (0, T), (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) where n stands for the outer unit normal vector to the boundary dfl. Actually, the boundary condition on x seems to be natural and may be rigorously motivated in the framework of the derivation of the model.…”