129 research outputs found
Convergence properties for a generalization of the Caginalp phase field system
We are concerned with a phase field system consisting of two partial
differential equations in terms of the variables thermal displacement, that is
basically the time integration of temperature, and phase parameter. The system
is a generalization of the well-known Caginalp model for phase transitions,
when including a diffusive term for the thermal displacement in the balance
equation and when dealing with an arbitrary maximal monotone graph, along with
a smooth anti-monotone function, in the phase equation. A Cauchy-Neumann
problem has been studied for such a system in arXiv:1107.3950v2 [math.AP], by
proving well-posedness and regularity results, as well as convergence of the
problem as the coefficient of the diffusive term for the thermal displacement
tends to zero. The aim of this contribution is rather to investigate the
asymptotic behaviour of the problem as the coefficient in front of the
Laplacian of the temperature goes to 0: this analysis is motivated by the types
III and II cases in the thermomechanical theory of Green and Naghdi. Under
minimal assumptions on the data of the problems, we show a convergence result.
Then, with the help of uniform regularity estimates, we discuss the rate of
convergence for the difference of the solutions in suitable norms.Comment: Key words: phase field model, initial-boundary value problem,
regularity of solutions, convergence, error estimate
Optimal control of a phase field system of Caginalp type with fractional operators
In their recent work `Well-posedness, regularity and asymptotic analyses for
a fractional phase field system' (Asymptot. Anal. 114 (2019), 93-128; see also
the preprint arXiv:1806.04625), two of the present authors have studied phase
field systems of Caginalp type, which model nonconserved, nonisothermal phase
transitions and in which the occurring diffusional operators are given by
fractional versions in the spectral sense of unbounded, monotone, selfadjoint,
linear operators having compact resolvents. In this paper, we complement this
analysis by investigating distributed optimal control problems for such
systems. It is shown that the associated control-to-state operator is Fr\'echet
differentiable between suitable Banach spaces, and meaningful first-order
necessary optimality conditions are derived in terms of a variational
inequality and the associated adjoint state variables.Comment: 38 pages. Key words: fractional operators, phase field system,
nonconserved phase transition, optimal control, first-order necessary
optimality condition
Optimal control of a nonconserved phase field model of Caginalp type with thermal memory and double obstacle potential
In this paper, we investigate optimal control problems for a nonlinear state system which constitutes a version of the Caginalp phase field system modeling nonisothermal phase transitions with a nonconserved order parameter that takes thermal memory into account. The state system, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. It consists of two nonlinearly coupled partial differential equations that govern the phase dynamics and the universal balance law for internal energy, written in terms of the phase variable and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. We extend recent results obtained for optimal control problems in which the free energy governing the phase transition was differentiable (i.e., of regular or logarithmic type) to the nonsmooth case of a double obstacle potential. As is well known, in this nondifferentiable case standard methods to establish the existence of appropriate Lagrange multipliers fail. This difficulty is overcome utilizing of the so-called deep quench approach. Namely, the double obstacle potential is approximated by a family of (differentiable) logarithmic ones for which the existence of optimal controls and first-order necessary conditions of optimality in terms of the adjoint state variables and a variational inequality are known. By proving appropriate bounds for the adjoint states of the approximating systems, we can pass to the limit in the corresponding first-order necessary conditions, thereby establishing meaningful first-order necessary optimality conditions also for the case of the double obstacle potential
Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions
We prove well-posedness results for the solution to an initial and
boundary-value problem for an Allen-Cahn type equation describing the
phenomenon of phase transitions for a material contained in a bounded and
regular domain. The dynamic boundary conditions for the order parameter have
been recently proposed by some physicists to account for interactions with the
walls. We show our results using suitable regularizations of the nonlinearities
of the problem and performing some a priori estimates which allow us to pass to
the limit thanks to compactness and monotonicity arguments.Comment: Key words: Allen-Cahn equation, dynamic boundary conditions, maximal
monotone graphs, initial boundary value problem, existence and uniqueness
result
On a Cahn--Hilliard system with source term and thermal memory
A nonisothermal phase field system of Cahn--Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn--Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter is lost. We provide several mathematical results under general assumptions on the source term and the double-well nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem
Optimal control of a phase field system of Caginalp type with fractional operators
In their recent work ``Well-posedness, regularity and asymptotic analyses for a fractional phase field system'' (Asymptot. Anal. 114 (2019), 93--128), two of the present authors have studied phase field systems of Caginalp type, which model nonconserved, nonisothermal phase transitions and in which the occurring diffusional operators are given by fractional versions in the spectral sense of unbounded, monotone, selfadjoint, linear operators having compact resolvents. In this paper, we complement this analysis by investigating distributed optimal control problems for such systems. It is shown that the associated control-to-state operator is Fréchet differentiable between suitable Banach spaces, and meaningful first-order necessary optimality conditions are derived in terms of a variational inequality and the associated adjoint state variables
Singular limit of an integrodifferential system related to the entropy balance
A thermodynamic model describing phase transitions with thermal memory, in
terms of an entropy equation and a momentum balance for the microforces, is
adressed. Convergence results and error estimates are proved for the related
integrodifferential system of PDE as the sequence of memory kernels converges
to a multiple of a Dirac delta, in a suitable sense.Comment: Key words: entropy equation, thermal memory, phase field model,
nonlinear partial differential equations, asymptotics on the memory ter
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
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