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

Small business incubators: An emerging phenomenon in South Africa’s SMME economy

In South Africa much policy attention is focused on the potential of the small, medium and micro-enterprise (SMME) economy for job creation. Nevertheless, despite government support for the SMME economy, high mortality rates are experienced by start-up enterprises. In common with international experience South Africa has adopted business incubation as a strategic tool for assisting the survival as well as building the competitiveness of SMMEs. This article analyses the state of business incubation in South Africa drawing attention to marked differences between the groups of public sector business incubators as opposed to those business incubators which have been initiated by the private sector

On a hyperbolic system arising in liquid crystal modelling

We consider a model of liquid crystals, based on a nonlinear hyperbolic system of differential equations, that represents an inviscid version of the model proposed by Qian and Sheng. A new concept of dissipative solution is proposed, for which a global-in-time existence theorem is shown. The dissipative solutions enjoy the following properties: (i) they exist globally in time for any nite energy initial data; (ii) dissipative solutions enjoying certain smoothness are classical solutions; (iii) a dissipative solution coincides with a strong solution originating from the same initial data as long as the latter exists.Romanian National Authority for Scienti c Research and Innovation, CNCS-UEFISCDI, project number PN-II-RUTE-2014-4-065

Organic dye for dye-sensitized solar cells

Organic dye for a dye-sensitized solar cell (DSSC) comprising at least one electron-acceptor unit and at least one π-conjugated unit. Said organic dye is particularly useful in a dye-sensitized photoelectric transformation element which, in its turn, can be used in a dye-sensitized solar cell (DSSC)

Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics

We consider a coupled bulk-surface system of partial differential equations with nonlinear coupling modelling receptor-ligand dynamics. The model arises as a simplification of a mathematical model for the reaction between cell surface resident receptors and ligands present in the extra-cellular medium. We prove the existence and uniqueness of solutions. We also consider a number of biologically relevant asymptotic limits of the model. We prove convergence to limiting problems which take the form of free boundary problems posed on the cell surface. We also report on numerical simulations illustrating convergence to one of the limiting problems as well as the spatio-temporal distributions of the receptors and ligands in a realistic geometry

On the long-time behavior of some mathematical models for nematic liquid crystals

A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of two basic state variables: the velocity field u and the director field d, representing the preferred orientation of molecules in a neighborhood of any point in a reference domain. After recalling a known existence result, we investigate the long-time behavior of weak solutions. In particular, we show that any solution trajectory admits a non-empty \u3c9-limit set containing only stationary solutions. Moreover, we give a number of sufficient conditions in order that the \u3c9-limit set contains a single point. Our approach improves and generalizes existing results on the same problem

A new approach to non-isothermal models for nematic liquid crystals

We introduce a new class of non-isothermal models describing the evolution of nematic liquid crystals and prove their consistency with the fundamental laws of classical thermodynamics. The resulting system of equations captures all essential features of physically relevant models; in particular, the effect of stretching of the director field is taken into account. In addition, the associated initial-boundary value problem admits global-in-time weak solutions without any essential restrictions on the size of the initial data

On the existence of strong solutions to the Cahn--Hilliard--Darcy system with mass source

We study a diffuse interface model describing the evolution of the flow of a binary fluid in a Hele-Shaw cell. The model consists of a Cahn--Hilliard--Darcy type system with transport and mass source. A relevant physical application is related to tumor growth dynamics, which in particular justifies the occurrence of a mass inflow. We study the initial-boundary value problem for this model and prove global existence and uniqueness of strong solutions in two space dimensions as well as local existence in three space dimensions