48 research outputs found
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