29,230 research outputs found
Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in 3D
In this paper we study a distributed optimal control problem for a nonlocal
convective Cahn--Hilliard equation with degenerate mobility and singular
potential in three dimensions of space. While the cost functional is of
standard tracking type, the control problem under investigation cannot easily
be treated via standard techniques for two reasons: the state system is a
highly nonlinear system of PDEs containing singular and degenerating terms, and
the control variable, which is given by the velocity of the motion occurring in
the convective term, is nonlinearly coupled to the state variable. The latter
fact makes it necessary to state rather special regularity assumptions for the
admissible controls, which, while looking a bit nonstandard, are however quite
natural in the corresponding analytical framework. In fact, they are
indispensable prerequisites to guarantee the well-posedness of the associated
state system. In this contribution, we employ recently proved existence,
uniqueness and regularity results for the solution to the associated state
system in order to establish the existence of optimal controls and appropriate
first-order necessary optimality conditions for the optimal control problem
Generalized gradient flow structure of internal energy driven phase field systems
In this paper we introduce a general abstract formulation of a variational
thermomechanical model, by means of a unified derivation via a generalization
of the principle of virtual powers for all the variables of the system,
including the thermal one. In particular, choosing as thermal variable the
entropy of the system, and as driving functional the internal energy, we get a
gradient flow structure (in a suitable abstract setting) for the whole
nonlinear PDE system. We prove a global in time existence of (weak) solutions
result for the Cauchy problem associated to the abstract PDE system as well as
uniqueness in case of suitable smoothness assumptions on the functionals
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
Well-posed Vector Optimization Problems and Vector Variational Inequalities
In this paper we introduce notions of well-posedness for a vector optimization problem and for a vector variational inequality of differential type, we study their basic properties and we establish the links among them. The proposed concept of well-posedness for a vector optimization problem generalizes the notion of well-setness for scalar optimization problems, introduced in [2]. On the other side, the introduced definition of well-posedness for a vector variational inequality extends the one given in [13] for the scalar case.Keywords: vector optimization, vector variational inequality, well-posedness
On a 3D isothermal model for nematic liquid crystals accounting for stretching terms
The present contribution investigates the well-posedness of a PDE system
describing the evolution of a nematic liquid crystal flow under kinematic
transports for molecules of different shapes. More in particular, the evolution
of the {\em velocity field} \ub is ruled by the Navier-Stokes incompressible
system with a stress tensor exhibiting a special coupling between the transport
and the induced terms. The dynamic of the {\em director field} \bd is
described by a variation of a parabolic Ginzburg-Landau equation with a
suitable penalization of the physical constraint |\bd|=1. Such equation
accounts for both the kinematic transport by the flow field and the internal
relaxation due to the elastic energy. The main aim of this contribution is to
overcome the lack of a maximum principle for the director equation and prove
(without any restriction on the data and on the physical constants of the
problem) the existence of global in time weak solutions under physically
meaningful boundary conditions on \bd and \ub
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