Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

Analysis of a system related to a model for phase transitions in dissipative isochoric thermoviscoelastic materials

“…By using (86), (84), and (87), we easily deduce that the triple .u , , p / satisfies the second and third equations of Problem 2 and that u and p satisfy respectively the initial conditions in (10) and (38). In the same way, convergences (84) for f n g and (85) for f n g allow us to conclude that . , / satisfies the first equation of Problem 2 and that  satisfies the initial condition (8).…”

“…Moreover, their associated local solutions had the same property: for t in the small time interval of the existence of such solutions; thus, all the process occurs in the mushy zone, and strict phase transitions do not happen; that is, the obtained solutions did not allow the possibility of formation of either pure liquid or pure solid regions in cases of potential allowing this formation; the employed techniques also ruled out the possibility of solutions asymptotically approaching such pure states. On the other hand, global or more general results were obtained in the one‐dimensional case; see .…”

Existence of solutions to a phase transition model with microscopic movements

“…Inequality (21) has the following physical interpretation: it implies that the entropy is controlled by the dissipation of the system. Moreover, it is easy to check that (20) together with (2), (9), and (21) give rise to the standard energy balance (1), at least in case all the quantities are sufficiently smooth. Unfortunately, to prove that the solution has this extra regularity in case ε = 0 is out of reach with our method.…”

Long-time behaviour of solutions to a one-dimensional strongly nonlinear model for phase transitions with micro-movements

“…In [1], Berti et al . Phase transition models taking the microscopic movements into account are also proposed for thermoviscoelastic materials [3,28,29] and ferromagnetic materials [2,26]. More precisely, they derived a phase transition model analogous to ours under the assumptions that q = −k 0 θ∇θ and λ(ϕ) = −3ϕ 2 + 8ϕ 3 − 6ϕ 2 , G(ϕ) = 6ϕ 2 (1 − ϕ) 2 in expression of the free energy (1.3).…”

“…As we can see from the literature, when the small perturbation assumption is not taken into account and the standard Fourier heat flux law is considered, the existence and uniqueness of the global solutions to the full models is obtained only in the one-dimensional case (see, for example, [20][21][22]29]). The global well-posedness of classical solutions in the three-dimensional case is still an open problem due to the high nonlinearities of the original models.…”