Book review: "Nonlinear Partial Differential Equations with Applications" by Tomáš Roubiček
Book review: "Elliptic Equations: An Introductory Course" by M. Chipot
Book review: "A Stability Technique for Evolution Partial Differential Equations" by V.A. Galaktionov, J. L. Vázques
Book review: "Methods of nonlinear analysis. Applications to Differential Equations" by P. Drábek, J. Milota
Optimal control of dynamical processes in two-phase systems of solid-liquid type
A thermodynamic approach to nonisothermal phase-field models
The goal of this paper is to work out a thermodynamical setting for nonisothermal phase-field models with conserved and nonconserved order parameters in thermoelastic materials. Our approach consists in exploiting the second law of thermodynamics in the form of the entropy principle according to I. Müller and I. S. Liu, which leads to the evaluation of the entropy inequality with multipliers. As the main result we obtain a general scheme of phase-field models which involves an...
Existence and uniqueness of solutions for a three-dimensional thermoelastic system
A thermodynamically consistent model of shape memory alloys in three dimensions is studied. The thermoelastic system, based on the strain tensor, its gradient and the absolute temperature, is a generalization of the well-known one-dimensional Falk model. The key assumptions concerning the form of constitutive relations are discussed. The detailed and selfcontained proof of the global-in-time existence and uniqueness of solutions is presented.
On diffused-interface models of shape memory alloys
Existence and uniqueness for the three-dimensional thermoelasticity system in shape memory problems
A thermodynamically consistent model of shape memory alloys in three dimensions is studied. The thermoelasticity system, based on the strain tensor, its gradient and the absolute temperature, generalizes the well-known one-dimensional Falk model. Under simplifying structural assumptions we prove global in time existence and uniqueness of the solution.
Measure-valued solutions of a heterogeneous Cahn-Hilliard system in elastic solids
The paper is concerned with the existence of measure-valued solutions to the Cahn-Hilliard system coupled with elasticity. The system under consideration is anisotropic and heterogeneous in the sense of admitting the elasticity and gradient energy tensors dependent on the order parameter. Such dependences introduce additional nonlinearities to the model for which the existence of weak solutions is not known so far.
Global existence and uniqueness of weak solutions to Cahn-Hilliard-Gurtin system in elastic solids
In this paper we study the Cahn-Hilliard-Gurtin system describing the phase-separation process in elastic solids. The system has been derived by Gurtin (1996) as an extension of the classical Cahn-Hilliard equation. For a version with viscosity we prove the existence and uniqueness of a weak solution on an infinite time interval and derive an absorbing set estimate.
Long time behaviour of a Cahn-Hilliard system coupled with viscoelasticity
The long-time behaviour of a unique regular solution to the Cahn-Hilliard system coupled with viscoelasticity is studied. The system arises as a model of the phase separation process in a binary deformable alloy. It is proved that for a sufficiently regular initial data the trajectory of the solution converges to the ω-limit set of these data. Moreover, it is shown that every element of the ω-limit set is a solution of the corresponding stationary problem.
Page 1