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### Open Access Journals

CPAA

A partial differential equation motivated by electromagnetic field
equations in ferromagnetic media is considered with a relaxed rate
dependent constitutive relation. It is shown that the solutions
converge to the unique solution of the limit parabolic problem with
a rate independent Preisach hysteresis constitutive operator as the
relaxation parameter tends to zero.

DCDS

Rate independent evolutions can be formulated as operators, called
hysteresis operators, between suitable function spaces. In this
paper, we present some results concerning the existence and the
form of directional derivatives and of Hadamard derivatives of
such operators in the scalar case, that is, when the driving
(input) function is a scalar function.

DCDS-S

A structure analysis of the Preisach model in a variational setting is carried out
by means of an auxiliary hyperbolic equation with memory variable playing the role of time,
and amplitude of cycles as spatial variable. Using this representation, we
propose an algorithm and derive error estimates for the identification of the Preisach
density function and for an approximate inversion of the Preisach operator.

DCDS-S

We consider a model for one-dimensional transversal oscillations of
an elastic-ideally plastic beam. It is based on the von Mises model
of plasticity and leads after a dimensional reduction to a
fourth-order partial differential equation with a hysteresis
operator of Prandtl-Ishlinskii type whose weight function is given
explicitly. In this paper, we study the case of clamped beams
involving a kinematic hardening in the stress-strain relation. As
main result, we prove the existence and uniqueness of a weak
solution. The method of proof, based on spatially semidiscrete
approximations, strongly relies on energy dissipation properties of
one-dimensional hysteresis operators.

DCDS-B

Alexei Vadimovich Pokrovskii was an outstanding mathematician, a scientist with very broad mathematical interests, and
a pioneer in the mathematical theory of systems with hysteresis. He died unexpectedly on September 1, 2010 at the age 62.
For the previous nine years he had been Professor and Head of Applied Mathematics at University College Cork in Ireland.

For more information please click the “Full Text” above

For more information please click the “Full Text” above

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DCDS-B

We consider a system of operator equations involving play and Prandtl-Ishlinskii hysteresis operators.
This system generalizes the classical mechanical models of elastoplasticity, friction and fatigue by introducing coupling
between the operators. We show that under quite general assumptions the coupled system is equivalent
to one effective Prandtl-Ishlinskii operator or, more precisely, to
a discontinuous extension of the Prandtl-Ishlinskii operator based on the Kurzweil integral of the derivative
of the state function. This effective operator is described constructively in terms of the parameters
of the coupled system. Our result is based on a substitution formula which we prove for the Kurzweil integral of regulated functions integrated with respect to functions of bounded variation.
This formula allows us to prove the composition rule for the generalized (discontinuous) Prandtl-Ishlinskii
operators. The composition rule, which underpins the analysis of the
coupled model, then establishes that
a composition of generalized Prandtl-Ishlinskii operators is also a generalized Prandtl-Ishlinskii operator provided that a monotonicity condition is satisfied.

DCDS

A phase-field system, non-local in space and non-smooth in time, with heat flux
proportional to the gradient of the inverse temperature, is shown to admit
a unique strong thermodynamically consistent solution on the whole time axis.
The temperature remains globally bounded both from above and from below,
and its space gradient as well as the time derivative of the order parameter
asymptotically vanish in $L^2$-norm as time tends to infinity.

DCDS-B

This paper is concerned with an optimal control problem for a system
of ordinary differential equations with rate independent hysteresis
modelled as a rate independent evolution variational inequality
with a closed convex constraint $Z\subset \mathbb{R}^m$.
We prove existence of optimal solutions as well as necessary optimality
conditions of first order. In particular, under certain regularity
assumptions we completely characterize the jump behaviour of the adjoint.

DCDS-S

We propose an improved
model explaining the occurrence of high stresses due to the
difference in specific volumes during phase transitions between
water and ice. The unknowns of the resulting evolution problem are the absolute temperature,
the volume increment, and the liquid fraction. The main novelty here consists
in including the dependence of the specific heat and of the speed of sound
upon the phase. These additional nonlinearities bring
new mathematical difficulties which require new estimation techniques based on
Moser iteration. We establish the existence of a global solution to
the corresponding initial-boundary value problem, as well as lower and
upper bounds for the absolute temperature. Assuming constant heat conductivity,
we also prove uniqueness and continuous data dependence of the solution.

EECT

We are concerned with a nonstandard phase field model of Cahn-Hilliard type.
The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006),
describes two-species phase segregation
and consists of a system of two highly nonlinearly coupled PDEs.
It has been recently investigated
by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers:
see, in particular, SIAM J. Appl. Math. 2011
and Boll. Unione Mat. Ital. 2012.
In the latter contribution, the authors can treat
the very general case in which the diffusivity coefficient of the parabolic PDE
is allowed to depend nonlinearly on both variables.
In the same framework, this paper investigates the asymptotic limit
of the solutions to the initial-boundary value problems
as the diffusion coefficient $\sigma$
in the equation governing the evolution of the order parameter tends to zero.
We prove that such a limit actually exists and solves the limit problem,
which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics.
In the case of a constant diffusivity, we are able to show uniqueness
and to improve the regularity of the solution.

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