401 research outputs found
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
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
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
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
Collisions in shape memory alloys
We present here a model for instantaneous collisions in a solid made of shape
memory alloys (SMA) by means of a predictive theory which is based on the
introduction not only of macroscopic velocities and temperature, but also of
microscopic velocities responsible of the austenite-martensites phase changes.
Assuming time discontinuities for velocities, volume fractions and temperature,
and applying the principles of thermodynamics for non-smooth evolutions
together with constitutive laws typical of SMA, we end up with a system of
nonlinearly coupled elliptic equations for which we prove an existence and
uniqueness result in the 2 and 3 D cases. Finally, we also present numerical
results for a SMA 2D solid subject to an external percussion by an hammer
stroke
Long-time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth
We investigate the long-time dynamics and optimal control problem of a
diffuse interface model that describes the growth of a tumor in presence of a
nutrient and surrounded by host tissues. The state system consists of a
Cahn-Hilliard type equation for the tumor cell fraction and a
reaction-diffusion equation for the nutrient. The possible medication that
serves to eliminate tumor cells is in terms of drugs and is introduced into the
system through the nutrient. In this setting, the control variable acts as an
external source in the nutrient equation. First, we consider the problem of
`long-time treatment' under a suitable given source and prove the convergence
of any global solution to a single equilibrium as . Then we
consider the `finite-time treatment' of a tumor, which corresponds to an
optimal control problem. Here we also allow the objective cost functional to
depend on a free time variable, which represents the unknown treatment time to
be optimized. We prove the existence of an optimal control and obtain first
order necessary optimality conditions for both the drug concentration and the
treatment time. One of the main aim of the control problem is to realize in the
best possible way a desired final distribution of the tumor cells, which is
expressed by the target function . By establishing the Lyapunov
stability of certain equilibria of the state system (without external source),
we see that can be taken as a stable configuration, so that the
tumor will not grow again once the finite-time treatment is completed
Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in 2D
We study a diffuse interface model for incompressible isothermal mixtures of
two immiscible fluids coupling the Navier--Stokes system with a convective
nonlocal Cahn--Hilliard equation in two dimensions of space. We apply recently
proved well-posedness and regularity results in order to establish existence of
optimal controls as well as first-order necessary optimality conditions for an
associated optimal control problem in which a distributed control is applied to
the fluid flow.Comment: 32 page
Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows
In this paper we investigate the three dimensional general Ericksen-Leslie
(E--L) system with Ginzburg-Landau type approximation modeling nematic liquid
crystal flows. First, by overcoming the difficulties from lack of maximum
principle for the director equation and high order nonlinearities for the
stress tensor, we prove existence of global-in-time weak solutions under
physically meaningful boundary conditions and suitable assumptions on the
Leslie coefficients, which ensures that the total energy of the E--L system is
dissipated. Moreover, for the E--L system with periodic boundary conditions, we
prove the local well-posedness of classical solutions under the so-called
Parodi's relation and establish a blow-up criterion in terms of the temporal
integral of both the maximum norm of the curl of the velocity field and the
maximum norm of the gradient of the liquid crystal director field
Unsaturated deformable porous media flow with phase transition
In the present paper, a continuum model is introduced for fluid flow in a
deformable porous medium, where the fluid may undergo phase transitions.
Typically, such problems arise in modeling liquid-solid phase transformations
in groundwater flows. The system of equations is derived here from the
conservation principles for mass, momentum, and energy and from the
Clausius-Duhem inequality for entropy. It couples the evolution of the
displacement in the matrix material, of the capillary pressure, of the absolute
temperature, and of the phase fraction. Mathematical results are proved under
the additional hypothesis that inertia effects and shear stresses can be
neglected. For the resulting highly nonlinear system of two PDEs, one ODE and
one ordinary differential inclusion with natural initial and boundary
conditions, existence of global in time solutions is proved by means of cut-off
techniques and suitable Moser-type estimates
On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids
We introduce a diffuse interface model describing the evolution of a mixture
of two different viscous incompressible fluids of equal density. The main
novelty of the present contribution consists in the fact that the effects of
temperature on the flow are taken into account. In the mathematical model, the
evolution of the macroscopic velocity is ruled by the Navier-Stokes system with
temperature-dependent viscosity, while the order parameter representing the
concentration of one of the components of the fluid is assumed to satisfy a
convective Cahn-Hilliard equation. The effects of the temperature are
prescribed by a suitable form of the heat equation. However, due to quadratic
forcing terms, this equation is replaced, in the weak formulation, by an
equality representing energy conservation complemented with a differential
inequality describing production of entropy. The main advantage of introducing
this notion of solution is that, while the thermodynamical consistency is
preserved, at the same time the energy-entropy formulation is more tractable
mathematically. Indeed, global-in-time existence for the initial-boundary value
problem associated to the weak formulation of the model is proved by deriving
suitable a-priori estimates and showing weak sequential stability of families
of approximating solutions.Comment: 26 page
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