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Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions
No full textTHE SEPARATION PROPERTY FOR 2D CAHN-HILLIARD EQUATIONS: LOCAL, NONLOCAL AND FRACTIONAL ENERGY CASES
No full textWe study the separation property for Cahn-Hilliard type equations with constant mobility and (physically relevant) singular potentials in two dimensions. That is, any solution with initial finite energy stays uniformly away from the pure phases +/- 1 from a certain time on. Beyond its physical interest, this property plays a crucial role to achieve high order Sobolev and analytic regularity of the solutions and to analyze their longtime behavior. In the local case, we streamline known arguments by exploiting the Sobolev inequality to obtain direct entropy estimates. In the nonlocal case, we provide a new proof based on De Giorgi estimates rather than the Alikakos-Moser type argument. Finally, in the spectral-fractional case, we prove nonlinear estimates and the separation property for any fractional index s is an element of (0,1) filling the gap between first-order (local) and zero-order (nonlocal) energy cases. In all of the aforementioned cases, our new proofs neither make use of the Trudinger-Moser inequality nor of any assumptions involving the third derivative of the entropy, as in the previous contributions. In particular, they apply for a more general class of singular potentials than the Flory-Huggins (Boltzmann-Gibbs) logarithmic density. Besides, the new methods present a series of technical advantages, which can be useful to the analysis of important physical systems that couple Cahn-Hilliard equations with other equations (e.g., reaction-diffusion equations and/or Navier-Stokes type systems) as well as their stochastic counterparts
Allen–Cahn–Navier–Stokes–Voigt Systems with Moving Contact Lines
No full textWe consider a diffuse interface model for an incompressible binary fluid flow. The model consists of the Navier-Stokes-Voigt equations coupled with the mass-conserving Allen-Cahn equation with Flory-Huggins potential. The resulting system is subject to generalized Navier boundary conditions for the (volume averaged) fluid velocity u and to a dynamic contact line boundary condition for the order parameter phi. These boundary conditions account for the moving contact line phenomenon. We establish the existence of a global weak solution which satisfies an energy inequality. A similar result is proven for the Allen-Cahn-Navier-Stokes system. In order to obtain some higher-order regularity (w.r.t. time) we propose the Voigt approximation: in this way we are able to prove the validity of the energy identity and of the strict separation property. Thanks to this property, we can show the uniqueness of quasi-strong solutions, even in dimension three. Regularization in finite time of weak solutions is also shown
The non-isothermal Allen-Cahn equation with dynamic boundary conditions
No full textWe consider a model of nonisothermal phase transitions taking
place in a bounded spatial region. The order parameter is governed by an
Allen-Cahn type equation which is coupled with the equation for the temperature.
The former is subject to a dynamic boundary condition recently
proposed by some physicists to account for interactions with the walls. The
latter is endowed with a boundary condition which can be a standard one
(Dirichlet, Neumann or Robin) or a dynamic one of Wentzell type. We thus
formulate a class of initial and boundary value problems whose local existence
and uniqueness is proven by means of a fixed point argument. The local solution
becomes global owing to suitable a priori estimates. Then we analyze the
asymptotic behavior of the solutions within the theory of infinite-dimensional
dynamical systems. In particular, we demonstrate the existence of the global
attractor as well as of an exponential attractor
Multi–component Cahn–Hilliard Systems with Singular Potentials: Theoretical Results
Get PDFWe consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation of an ideal mixture of N >= 2 components occupying a bounded region Omega subset of R-d, d <= 3. Our system is subject to a constant mobility matrix of coefficients, a free energy functional given in terms of singular entropy generated potentials and localized capillarity effects. We prove well-posedness and regularity results which generalize the ones obtained by Elliott and Luckhaus (IMA Preprint Ser 887, 1991). In particular, if d <= 2, we derive the uniform strict separation of solutions from the singular points of the (entropy) nonlinearity. Then, even if d = 3, we prove the existence of a global (regular) attractor as well as we establish the convergence of solutions to single equilibria. If d = 3, this convergence requires the validity of the asymptotic strict separation property. This work constitutes the first part of an extended three-part study involving the phase behavior of multi-component systems, with a second part addressing the presence of nonlocal capillarity effects, and a final part concerning the numerical study of such systems along with some relevant application
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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