Global Existence for Phase Transition Problems via a Variational Scheme

Abstract

We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress-strain function which contains a nonlocal convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.

Phase transitions and sharp-interface limits for the 1d-elasticity system with non-local energy

“…Since we are not interested in the case for positive but fixed value for ε we do not give the proof of this statement but focus on the limit process ε → 0. We note that the existence of weak solutions for (1.5), (1.6) under Assumption 4.1 has been shown in [39].…”

“…Systems with capillarity terms like D ε given by (1.7) have been analyzed by many authors ( [2,8,29,39]), also in the closely related case of liquid-vapour phase transitions in Van-der-Waals fluids ( [3,9,19,20,26,42]). In particular with the ε-scaling given by (1.5), (1.7) solutions of the Cauchy problem for (1.5) have been shown to converge to weak solutions of the Cauchy problem for the sharpinterface limit system (1.1) which contain dynamical phase boundaries ( [29]).…”

Global existence and uniqueness of solutions for a viscoelastic two-phase model with nonlocal capillarity

“…For non-negative φ , the existence of weak solutions to system (1.1) for given initial values (on the spatial domain R) was proven for arbitrary Lipschitz continuous σ with a variational technique (see [13], Theorem 5.4) relying on the ideas in [4]. This result is extended in this contribution in various directions: initial boundary value problems are addressed, partly negative kernels are treated, and optimal regularity is proven under quite rough initial conditions.…”