457 research outputs found
Modeling and analysis of a phase field system for damage and phase separation processes in solids
In this work, we analytically investigate a multi-component system for
describing phase separation and damage processes in solids. The model consists
of a parabolic diffusion equation of fourth order for the concentration coupled
with an elliptic system with material dependent coefficients for the strain
tensor and a doubly nonlinear differential inclusion for the damage function.
The main aim of this paper is to show existence of weak solutions for the
introduced model, where, in contrast to existing damage models in the
literature, different elastic properties of damaged and undamaged material are
regarded. To prove existence of weak solutions for the introduced model, we
start with an approximation system. Then, by passing to the limit, existence
results of weak solutions for the proposed model are obtained via suitable
variational techniques.Comment: Keywords: Cahn-Hilliard system, phase separation, elliptic-parabolic
systems, doubly nonlinear differential inclusions, complete damage, existence
results, energetic solutions, weak solutions, linear elasticity,
rate-dependent system
Weighted Energy-Dissipation principle for gradient flows in metric spaces
This paper develops the so-called Weighted Energy-Dissipation (WED)
variational approach for the analysis of gradient flows in metric spaces. This
focuses on the minimization of the parameter-dependent global-in-time
functional of trajectories \mathcal{I}_\varepsilon[u] = \int_0^{\infty}
e^{-t/\varepsilon}\left( \frac12 |u'|^2(t) + \frac1{\varepsilon}\phi(u(t))
\right) \dd t, featuring the weighted sum of energetic and dissipative
terms. As the parameter is sent to~, the minimizers
of such functionals converge, up to subsequences, to curves of
maximal slope driven by the functional . This delivers a new and general
variational approximation procedure, hence a new existence proof, for metric
gradient flows. In addition, it provides a novel perspective towards
relaxation
Optimal control of the sweeping process over polyhedral controlled sets
The paper addresses a new class of optimal control problems governed by the
dissipative and discontinuous differential inclusion of the sweeping/Moreau
process while using controls to determine the best shape of moving convex
polyhedra in order to optimize the given Bolza-type functional, which depends
on control and state variables as well as their velocities. Besides the highly
non-Lipschitzian nature of the unbounded differential inclusion of the
controlled sweeping process, the optimal control problems under consideration
contain intrinsic state constraints of the inequality and equality types. All
of this creates serious challenges for deriving necessary optimality
conditions. We develop here the method of discrete approximations and combine
it with advanced tools of first-order and second-order variational analysis and
generalized differentiation. This approach allows us to establish constructive
necessary optimality conditions for local minimizers of the controlled sweeping
process expressed entirely in terms of the problem data under fairly
unrestrictive assumptions. As a by-product of the developed approach, we prove
the strong -convergence of optimal solutions of discrete
approximations to a given local minimizer of the continuous-time system and
derive necessary optimality conditions for the discrete counterparts. The
established necessary optimality conditions for the sweeping process are
illustrated by several examples
Variational convergence of gradient flows and rate-independent evolutions in metric spaces
We study the asymptotic behaviour of families of gradient flows in a general
metric setting, when the metric-dissipation potentials degenerate in the limit
to a dissipation with linear growth. We present a general variational
definition of BV solutions to metric evolutions, showing the different
characterization of the solution in the absolutely continuous regime, on the
singular Cantor part, and along the jump transitions. By using tools of metric
analysis, BV functions and blow-up by time rescaling, we show that this
variational notion is stable with respect to a wide class of perturbations
involving energies, distances, and dissipation potentials. As a particular
application, we show that BV solutions to rate-independent problems arise
naturally as a limit of -gradient flows, , when the exponents
converge to 1
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