Optimal boundary control of a simplified Ericksen--Leslie system for nematic liquid crystal flows in 2D2D

In this paper, we investigate an optimal boundary control problem for a two dimensional simplified Ericksen--Leslie system modelling the incompressible nematic liquid crystal flows. The hydrodynamic system consists of the Navier--Stokes equations for the fluid velocity coupled with a convective Ginzburg--Landau type equation for the averaged molecular orientation. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the molecular orientation is subject to a time-dependent Dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals. We first establish the existence of optimal boundary controls. Then we show that the control-to-state operator is Fr\'echet differentiable between appropriate Banach spaces and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables

Identifying a space dependent coefficient in a reaction-diffusion equation

We consider a reaction-diffusion equation for the front motion [u] in which the reaction term is given by [c(x)g(u)]. We formulate a suitable inverse problem for the unknowns [u] and [c], where [u] satisfies homogeneous Neumann boundary conditions and the additional condition is of integral type on the time interval [[0,T]]. Uniqueness of the solution is proved in the case of a linear [g]. Assuming [g] non linear, we show uniqueness for large [T]

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 thermodynamically consistent 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\u2013Hilliard type equation for the tumor cell fraction and a reaction\u2013diffusion 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 \u201clong-time treatment\u201d under a suitable given mass source and prove the convergence of any global solution to a single equilibrium as t\u2192+ 1e . Second, we consider the \u201cfinite-time treatment\u201d that corresponds to an optimal control problem. Here we 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 \u3d5\u3a9 . By establishing the Lyapunov stability of certain equilibria of the state system (without external source), we show that \u3d5\u3a9 can be taken as a stable configuration, so that the tumor will not grow again once the finite-time treatment is completed

Cahn-Hilliard equations with memory and dynamic boundary conditions

We consider a modified Cahn\u2013Hiliard equation where the velocity of the order parameter u depends on the past history of \u394\u3bc, \u3bc being the chemical potential with an additional viscous term \u3b1ut, \u3b1 650. This type of equation has been proposed by P. Galenko et al. to model phase separation phenomena in special materials (e.g., glasses). In addition, the usual no-flux boundary condition for u is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The resulting boundary value problem is subject to suitable initial conditions and is reformulated in the so-called past history space. Existence of a variational solution is obtained. Then, in the case \u3b1>0, we can also prove uniqueness and construct a strongly continuous semigroup acting on a suitable phase space. We show that the corresponding dynamical system has a (smooth) global attractor as well as an exponential attractor. In the case \u3b1=0, we only establish the existence of a trajectory attractor

Trajectory attractors for the Sun-Liu model for nematic liquid crystals in 3D

In this paper we prove the existence of a trajectory attractor (in the sense of V.V. Chepyzhov and M.I. Vishik) for a nonlinear PDE system coming from a 3D liquid crystal model accounting for stretching effects. The system couples a nonlinear evolution equation for the director d (introduced in order to describe the preferred orientation of the molecules) with an incompressible Navier-Stokes equation for the evolution of the velocity field u. The technique is based on the introduction of a suitable trajectory space and of a metric accounting for the double-well type nonlinearity contained in the director equation. Finally, a dissipative estimate is obtained by using a proper integrated energy inequality. Both the cases of (homogeneous) Neumann and (non-homogeneous) Dirichlet boundary conditions for d are considered.Comment: 32 page

Global strong solutions of the full Navier-Stokes and Q-tensor system for nematic liquid crystal flows in two dimensions

We consider a full Navier-Stokes and Q-tensor system for incompressible liquid crystal flows of nematic type. In the two dimensional periodic case, we prove the existence and uniqueness of global strong solutions that are uniformly bounded in time. This result is obtained without any smallness assumption on the physical parameter. that measures the ratio between tumbling and aligning effects of a shear flow exerting over the liquid crystal directors. Moreover, we show the uniqueness of asymptotic limit for each global strong solution as time goes to infinity and provide an uniform estimate on the convergence rate

A nonlinear model for marble sulphation including surface rugosity: Theoretical and numerical results

We consider an evolution system describing the phenomenon of marble sulphation of a monument, accounting of the surface rugosity. We first prove a local in time well posedness result. Then, stronger assumptions on the data allow us to establish the existence of a global in time solution. Finally, we perform some numerical simulations that illustrate the main feature of the proposed model

Novel frontiers in neuroprotective therapies in glaucoma: Molecular and clinical aspects

In the last years, neuroprotective therapies have attracted the researcher interests as modern and challenging approach for the treatment of neurodegenerative diseases, aimed at protecting the nervous system from injuries. Glaucoma is a neurodegenerative disease characterized by progressive excavation of the optic nerve head, retinal axonal injury and corresponding vision loss that affects millions of people on a global scale. The molecular basis of the pathology is largely uncharacterized yet, and the therapeutic approaches available do not change the natural course of the disease. Therefore, in accordance with the therapeutic regimens proposed for other neurodegenerative diseases, a modern strategy to treat glaucoma includes prescription of drugs with neuroprotective activities. With respect to this, several preclinical and clinical investigations on a plethora of different drugs are currently ongoing. In this review, first, the conceptualization of the rationale for the adoption of neuroprotective strategies for retina is summarized. Second, the molecular aspects highlighting glaucoma as a neurodegenerative disease are reported. In conclusion, the molecular and pharmacological properties of most promising direct neuroprotective drugs used to delay glaucoma progression are examined, including: neurotrophic factors, NMDA receptor antagonists, the α2-adrenergic agonist, brimonidine, calcium channel blockers, antioxidant agents, nicotinamide and statins

Global existence for a nonstandard viscous Cahn--Hilliard system with dynamic boundary condition

In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies

Limiting problems for a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions

This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp.105--118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases