Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term

The paper is devoted to a modification of the classical Cahn-Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient which is usually small in comparison to the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is "supercritical" in the case of two and three space dimensions. Thus the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small inertial coefficient and arbitrary growth rate of the nonlinearity. The present contribution studies the long-time behavior of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers. Namely, we prove here that the attractors for energy and strong solutions coincide for both the cases mentioned above. Thus, the energy solutions are asymptotically smooth. In addition, we show that the non-smooth part of any energy solution decays exponentially in time and deduce that the (smooth) exponential attractor for the strong solutions constructed previously is simultaneously the exponential attractor for the energy solutions as well

Evolution of CODYRUN from Thermal Simulation to Coupled Thermal and Daylight Simulation Software

AbstractCODYRUN is a multi-zone software integrating thermal building simulation, airflow, and pollutant transfer. Described in numerous publications, this software was originally used for the passive design of buildings, both for research and teaching purposes. In this context, the data treated were mainly concerned with volumes (zones), surfaces and thicknesses (walls and windows), materials, and systems, with the aim to determine temperatures, heat fluxes, energy consumed, air transfers, and so on.The question thus arose as to the integration of indoor lighting conditions into the simulation. Hence, previous data structures had to be amended to incorporate the spatial positioning of entities (walls, windows, and artificial lighting sources) through vertexes. A set of procedures was also developed for polygons as well as calculating natural and artificial lighting.The results of this new daylighting module were then compared with other results of simulation codes and experimental cases both in artificial and natural environments. Excellent agreements were obtained, such as the values for luminous efficiencies in a tropical and humid climate.A simulation exercise was conducted in a classroom located in Reunion Island (French overseas territory in the Indian Ocean), thus confirming the interest for thermal and daylighting designs in low-energy buildings

Finite-dimensional global and exponential attractors for the reaction-diffusion problem with an obstacle potential

A reaction-diffusion problem with an obstacle potential is considered in a bounded domain of $\R^N$. Under the assumption that the obstacle \K is a closed convex and bounded subset of $\mathbb{R}^n$ with smooth boundary or it is a closed $n$-dimensional simplex, we prove that the long-time behavior of the solution semigroup associated with this problem can be described in terms of an exponential attractor. In particular, the latter means that the fractal dimension of the associated global attractor is also finite

Global attractors for Cahn-Hilliard equations with non constant mobility

We address, in a three-dimensional spatial setting, both the viscous and the standard Cahn-Hilliard equation with a nonconstant mobility coefficient. As it was shown in J.W. Barrett and J.W. Blowey, Math. Comp., 68 (1999), 487-517, one cannot expect uniqueness of the solution to the related initial and boundary value problems. Nevertheless, referring to J. Ball's theory of generalized semiflows, we are able to prove existence of compact quasi-invariant global attractors for the associated dynamical processes settled in the natural "finite energy" space. A key point in the proof is a careful use of the energy equality, combined with the derivation of a "local compactness" estimate for systems with supercritical nonlinearities, which may have an independent interest. Under growth restrictions on the configuration potential, we also show existence of a compact global attractor for the semiflow generated by the (weaker) solutions to the nonviscous equation characterized by a "finite entropy" condition

Longtime behavior of nonlocal Cahn-Hilliard equations

Here we consider the nonlocal Cahn-Hilliard equation with constant mobility in a bounded domain. We prove that the associated dynamical system has an exponential attractor, provided that the potential is regular. In order to do that a crucial step is showing the eventual boundedness of the order parameter uniformly with respect to the initial datum. This is obtained through an Alikakos-Moser type argument. We establish a similar result for the viscous nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In this case the validity of the so-called separation property is crucial. We also discuss the convergence of a solution to a single stationary state. The separation property in the nonviscous case is known to hold when the mobility degenerates at the pure phases in a proper way and the potential is of logarithmic type. Thus, the existence of an exponential attractor can be proven in this case as well

Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design

The aim of this paper is to briefly recall heat transfer modes and explain their integration within a software dedicated to building simulation (CODYRUN). Detailed elements of the validation of this software are presented and two applications are finally discussed. One concerns the modeling of a flat plate air collector and the second focuses on the modeling of Trombe solar walls. In each case, detailed modeling of heat transfer allows precise understanding of thermal and energetic behavior of the studied structures. Recent decades have seen a proliferation of tools for building thermal simulation. These applications cover a wide spectrum from very simplified steady state models to dynamic simulation ones, including computational fluid dynamics modules (Clarke, 2001). These tools are widely available in design offices and engineering firms. They are often used for the design of HVAC systems and still subject to detailed research, particularly with respect to the integration of new fields (specific insulation materials, lighting, pollutants transport, etc.). Available from: http://www.intechopen.com/books/evaporation-condensation-and-heat-transfer/heat-transfer-in-buildings-application-to-solar-air-collector-and-trombe-wall-designComment: Available from: http://www.intechopen.com/books/evaporation-condensation-and-heat-transfer/heat-transfer-in-buildings-application-to-solar-air-collector-and-trombe-wall-desig

Successful Use of Squeezed-Fat Grafts to Correct a Breast Affected by Poland Syndrome

This study attempted to reconstruct deformities of a Poland syndrome patient using autologous fat tissues. All injected fat tissues were condensed by squeezing centrifugation. Operations were performed four times with intervals over 6 months. The total injection volume was 972 ml, and the maintained volume of 628 ml was measured by means of a magnetic resonance image (MRI). The entire follow-up period was 4.5 years. After surgery, several small cysts and minimal calcifications were present but no significant complications. The cosmetic outcomes and volume maintenance rates were excellent despite the overlapped large-volume injections. In conclusion, higher condensation of fat tissues through squeezing centrifugation would help to achieve better results in volume maintenance and reduce complications. It is necessary, however, to perform more comparative studies with many clinical cases for a more scientific analysis. The study experiments with squeezed fat simply suggest a hypothesis that squeezing centrifugation could select healthier cells through pressure disruption of relatively thinner membranes of larger, more vulnerable and more mature fat cells

On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities

We study a non-local variant of a diffuse interface model proposed by Hawkins--Darrud et al. (2012) for tumour growth in the presence of a chemical species acting as nutrient. The system consists of a Cahn--Hilliard equation coupled to a reaction-diffusion equation. For non-degenerate mobilities and smooth potentials, we derive well-posedness results, which are the non-local analogue of those obtained in Frigeri et al. (European J. Appl. Math. 2015). Furthermore, we establish existence of weak solutions for the case of degenerate mobilities and singular potentials, which serves to confine the order parameter to its physically relevant interval. Due to the non-local nature of the equations, under additional assumptions continuous dependence on initial data can also be shown.Comment: 28 page