TY - JOUR

T1 - Three-phase boundary motions under constant velocities. I

T2 - The vanishing surface tension limit

AU - Reitich, Fernando

AU - Soner, H. Mete

N1 - Funding Information:
This work was partially supported by the Army Research Office and the National Science Foundation through the Center for Nonlinear Analysis. The second author

PY - 1996

Y1 - 1996

N2 - 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 Γij between regions i and j (i, j = 1,2,3, i ≠ j) is governed by the equation Vij = μij(fijκij + Fij) (0.1) Here Vij, κij, μij and fij denote, respectively, the normal velocity, the curvature, the mobility and the surface tension of the interface and the numbers Fij stand for the (constant) difference in bulk energies. At the point where the three phases coexist, local equilibrium requires that the curves meet at prescribed angles. (0.2) In case the material constants fij are small, fij = εfij and ε ≪ 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 fij = 0. It turns out that this problem, (0.1) with fij = 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 ε → 0. Indeed, a formal analysis of this limiting process motivates us to introduce the concept of weak viscosity solution for the problem with ε = 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.

AB - 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 Γij between regions i and j (i, j = 1,2,3, i ≠ j) is governed by the equation Vij = μij(fijκij + Fij) (0.1) Here Vij, κij, μij and fij denote, respectively, the normal velocity, the curvature, the mobility and the surface tension of the interface and the numbers Fij stand for the (constant) difference in bulk energies. At the point where the three phases coexist, local equilibrium requires that the curves meet at prescribed angles. (0.2) In case the material constants fij are small, fij = εfij and ε ≪ 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 fij = 0. It turns out that this problem, (0.1) with fij = 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 ε → 0. Indeed, a formal analysis of this limiting process motivates us to introduce the concept of weak viscosity solution for the problem with ε = 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.

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U2 - 10.1017/S0308210500023106

DO - 10.1017/S0308210500023106

M3 - Article

AN - SCOPUS:21344436924

VL - 126

SP - 837

EP - 865

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 4

ER -