4,970 research outputs found
Regularity of solutions to a model for solid-solid phase transitions driven by configurational forces
In a previous work, we prove the existence of weak solutions to an initial-boundary value problem, with H 1(Ω) initial data, for a system of partial differential equations, which consists of the equations of linear elasticity and a nonlinear, degenerate parabolic equation of second order. This problem models the behavior in time of materials with martensitic phase transitions. This model with diffusive phase interfaces was derived from a model with sharp interfaces, whose evolution is driven by configurational forces, and can be regarded as a regularization of that model. Assuming in this article the initial data is in H 2(Ω), we investigate the regularity of weak solutions that is difficult due to the gradient term which plays a role of a weight. Our proof, in which the difficulties are caused by the weight in the principle term, is only valid in one space dimension
A degenerating PDE system for phase transitions and damage
In this paper, we analyze a PDE system arising in the modeling of phase
transition and damage phenomena in thermoviscoelastic materials. The resulting
evolution equations in the unknowns \theta (absolute temperature), u
(displacement), and \chi (phase/damage parameter) are strongly nonlinearly
coupled. Moreover, the momentum equation for u contains \chi-dependent elliptic
operators, which degenerate at the pure phases (corresponding to the values
\chi=0 and \chi=1), making the whole system degenerate. That is why, we have to
resort to a suitable weak solvability notion for the analysis of the problem:
it consists of the weak formulations of the heat and momentum equation, and,
for the phase/damage parameter \chi, of a generalization of the principle of
virtual powers, partially mutuated from the theory of rate-independent damage
processes. To prove an existence result for this weak formulation, an
approximating problem is introduced, where the elliptic degeneracy of the
displacement equation is ruled out: in the framework of damage models, this
corresponds to allowing for partial damage only. For such an approximate
system, global-in-time existence and well-posedness results are established in
various cases. Then, the passage to the limit to the degenerate system is
performed via suitable variational techniques
Dynamic Transitions for Quasilinear Systems and Cahn-Hilliard equation with Onsager mobility
The main objectives of this article are two-fold. First, we study the effect
of the nonlinear Onsager mobility on the phase transition and on the
well-posedness of the Cahn-Hilliard equation modeling a binary system. It is
shown in particular that the dynamic transition is essentially independent of
the nonlinearity of the Onsager mobility. However, the nonlinearity of the
mobility does cause substantial technical difficulty for the well-posedness and
for carrying out the dynamic transition analysis. For this reason, as a second
objective, we introduce a systematic approach to deal with phase transition
problems modeled by quasilinear partial differential equation, following the
ideas of the dynamic transition theory developed recently by Ma and Wang
Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities
We consider a diffuse interface model for an incompressible isothermal
mixture of two viscous Newtonian fluids with different densities in a bounded
domain in two or three space dimensions. The model is the nonlocal version of
the one recently derived by Abels, Garcke and Gr\"{u}n and consists in a
Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard
equation. The density of the mixture depends on an order parameter. For this
nonlocal system we prove existence of global dissipative weak solutions for the
case of singular double-well potentials and non degenerate mobilities. To this
goal we devise an approach which is completely independent of the one employed
by Abels, Depner and Garcke to establish existence of weak solutions for the
local Abels et al. model.Comment: 43 page
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