426 research outputs found
Finite dimensional reduction and convergence to equilibrium for incompressible Smectic-A liquid crystal flows
We consider a hydrodynamic system that models the Smectic-A liquid crystal
flow. The model consists of the Navier-Stokes equation for the fluid velocity
coupled with a fourth-order equation for the layer variable \vp, endowed with
periodic boundary conditions. We analyze the long-time behavior of the
solutions within the theory of infinite-dimensional dissipative dynamical
systems. We first prove that in 2D, the problem possesses a global attractor
in certain phase space. Then we establish the existence of an
exponential attractor which entails that the global attractor
has finite fractal dimension. Moreover, we show that each
trajectory converges to a single equilibrium by means of a suitable
Lojasiewicz--Simon inequality. Corresponding results in 3D are also discussed
Attractors for the semiflow associated with a class of doubly nonlinear parabolic equations
A doubly nonlinear parabolic equation of the form , complemented with initial and either Dirichlet or Neumann
homogeneous boundary conditions, is addressed. The two nonlinearities are given
by the maximal monotone function and by the derivative of a
smooth but possibly nonconvex potential ; is a known source. After
defining a proper notion of solution and recalling a related existence result,
we show that from any initial datum emanates at least one solution which gains
further regularity for . Such "regularizing solutions" constitute a
semiflow for which uniqueness is satisfied for strictly positive times and
we can study long time behavior properties. In particular, we can prove
existence of both global and exponential attractors and investigate the
structure of -limits of single trajectories
Analysis of a variational model for nematic shells
We analyze an elastic surface energy which was recently introduced by G.
Napoli and L.Vergori to model thin films of nematic liquid crystals. We show
how a novel approach that takes into account also the extrinsic properties of
the surfaces coated by the liquid crystal leads to considerable differences
with respect to the classical intrinsic energy. Our results concern three
connected aspects: i) using methods of the calculus of variations, we establish
a relation between the existence of minimizers and the topology of the surface;
ii) we prove, by a Ginzburg-Landau approximation, the well-posedness of the
gradient flow of the energy; iii) in the case of a parametrized axisymmetric
torus we obtain a stronger characterization of global and local minimizers,
which we supplement with numerical experiments.Comment: Revised version. Includes referee's comments. Some proofs are
changed. To appear on Mathematical Models and Methods in Applied Sciences
(M3AS
Equilibrium configurations of nematic liquid crystals on a torus
The topology and the geometry of a surface play a fundamental role in
determining the equilibrium configurations of thin films of liquid crystals. We
propose here a theoretical analysis of a recently introduced surface Frank
energy, in the case of two-dimensional nematic liquid crystals coating a
toroidal particle. Our aim is to show how a different modeling of the effect of
extrinsic curvature acts as a selection principle among equilibria of the
classical energy, and how new configurations emerge. In particular, our
analysis predicts the existence of new stable equilibria with complex windings.Comment: 9 pages, 6 figures. This version is to appear on Phys. Rev.
Global attractors for gradient flows in metric spaces
We develop the long-time analysis for gradient flow equations in metric
spaces. In particular, we consider two notions of solutions for metric gradient
flows, namely energy and generalized solutions. While the former concept
coincides with the notion of curves of maximal slope, we introduce the latter
to include limits of time-incremental approximations constructed via the
Minimizing Movements approach.
For both notions of solutions we prove the existence of the global attractor.
Since the evolutionary problems we consider may lack uniqueness, we rely on the
theory of generalized semiflows introduced by J.M. Ball. The notions of
generalized and energy solutions are quite flexible and can be used to address
gradient flows in a variety of contexts, ranging from Banach spaces to
Wasserstein spaces of probability measures.
We present applications of our abstract results by proving the existence of
the global attractor for the energy solutions both of abstract doubly nonlinear
evolution equations in reflexive Banach spaces, and of a class of evolution
equations in Wasserstein spaces, as well as for the generalized solutions of
some phase-change evolutions driven by mean curvature
Well-posedness and long-time behavior for a class of doubly nonlinear equations
This paper addresses a doubly nonlinear parabolic inclusion of the form
. Existence of a solution is proved under suitable
monotonicity, coercivity, and structure assumptions on the operators and
, which in particular are both supposed to be subdifferentials of
functionals on . Moreover, under additional hypotheses on ,
uniqueness of the solution is proved. Finally, a characterization of
-limit sets of solutions is given and we investigate the convergence of
trajectories to limit points
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
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