Three-phase boundary motions under constant velocities .1. The vanishing surface tension limit

In this paper, we deal with the dynamics of material interfaces such as solid-liquid, grain or antiphase boundaries. We concentrate on the situation in which these internal surfaces separate three regions in the material with different physical attributes (e.g. grain boundaries in a polycrystal). The basic two-dimensional model proposes that the motion of an interface Gamma(ij) between regions i and j (i, j = 1, 2, 3, i not equal j) is governed by the equation (0.1) V-ij = mu(ij)(f(ij)k(ij) + F-ij). Here V-ij, k(ij), mu(ij) and f(ij) denote, respectively, the normal velocity, the curvature, the mobility and the surface tension of the interface and the numbers F-ij stand for the (constant) difference in bulk energies. At the point where the three phases coexist, local equilibrium requires that (0.2) the curves meet at prescribed angles. In case the material constants f(ij) are small, f(ij) = <(epsilon f)over cap (ij)> and epsilon much less than 1, previous analyses based on the parabolic nature of the equations (0.1) do not provide good qualitative information on the behaviour of solutions. In this case, it is more appropriate to consider the singular case with f(ij) = 0. It turns out that this problem, (0.1) with f(ij) = 0, admits infinitely many solutions. Here, we present results that strongly suggest that, in all cases, a unique solution-'the vanishing surface tension (VST) solution'-is selected by letting epsilon --> 0. Indeed, a formal analysis of this limiting process motivates us to introduce the concept of weak viscosity solution for the problem with epsilon = 0. As we show, this weak solution is unique and is therefore expected to coincide with the VST solution. To support this statement, we present a perturbation analysis and a construction of self-similar solutions; a rigorous convergence result is established in the case of symmetric configurations. Finally, we use the weak formulation to write down a catalogue of solutions showing that, in several cases of physical relevance, the VST solution differs from results proposed previously.