Solvability via viscosity solutions for a model of phase transitions driven by configurational forces

In the present article, we are interested in an initial boundary value problem for a coupled system of partial differential equations arising in martensitic phase transition theory of elastically deformable solid materials, e.g., steel. This model was proposed and investigated in previous work by Alber and Zhu in which the weak solutions are defined in a standard way, however the key technique is not applicable to multi-dimensional problem. Intending to solve this multi-dimensional problem and to investigate the sharp interface limits of our models, we thus define weak solutions in a different way by using the notion of viscosity solution, then prove the existence of weak solutions to this problem in one space dimension, yet the multi-dimensional problem is still open.Comment: 21 page

Spherically symmetric solutions to a model for phase transitions driven by configurational forces

We prove the global in time existence of spherically symmetric solutions to an initial-boundary value problem for a system of partial differential equations, which consists of the equations of linear elasticity and a nonlinear, non-uniformly parabolic equation of second order. The problem models the behavior in time of materials in which martensitic phase transitions, driven by configurational forces, take place, and can be considered to be a regularization of the corresponding sharp interface model. By assuming that the solutions are spherically symmetric, we reduce the original multidimensional problem to the one in one space dimension, then prove the existence of spherically symmetric solutions. Our proof is valid due to the essential feature that the reduced problem is one space dimensional.Comment: 25 page

Viscosity solutions to a new phase-field model for martensitic phase transformations

summary:We investigate a new phase-field model which describes martensitic phase transitions, driven by material forces, in solid materials, e.g., shape memory alloys. This model is a nonlinear degenerate parabolic equation of second order, its principal part is not in divergence form in multi-dimensional case. We prove the existence of viscosity solutions to an initial-boundary value problem for this model

Asymptotic Behavior of the Solutions to a Landau-Ginzburg System with Viscosity for Martensitic Phase Transitions in Shape Memory Alloys

In this paper, we investigate the system of partial differential equations governing the dynamics of martensitic phase transitions in shape memory alloys under the presence of a (possibly small) viscous stress. The corresponding free energy is assumed in Landau-Ginzburg form and nonconvex as function of the order parameter. Results concerning the asymptotic behavior of the solution as time tends to infinity are proved, and the compactness of the orbit is shown

Stationary waves to viscous heat-conductive gases in half-space: Existence, stability and convergence rate

The main concern of this paper is to study large-time behavior of solutions to an ideal polytropic model of compressible viscous gases in one-dimensional half-space. We consider an outflow problem and obtain a convergence rate of solutions toward a corresponding stationary solution. Here the existence of the stationary solution is proved under a smallness condition on the boundary data with the aid of center manifold theory. We also show the time asymptotic stability of the stationary solution under smallness assumptions on the boundary data and the initial perturbation in the Sobolev space, by employing an energy method. Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. The proof is based on deriving a priori estimates by using a time and space weighted energy method. © 2010 World Scientific Publishing Company

Advances in Nonlinear Complexity Analysis for Partial Differential Equations

1 p. -- [Editorial Material