In this paper we investigate the three dimensional general Ericksen-Leslie (E-L) system with Ginzburg-Landau type approximation modeling nematic liquid crystal flows. First, by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor, we prove existence of global-in-time weak solutions under physically meaningful boundary conditions and suitable assumptions on the Leslie coefficients, which ensures that the total energy of the E-L system is dissipated. Moreover, for the E-L system with periodic boundary conditions, we prove the local well-posedness of classical solutions under the so-called Parodi's relation and establish a blow-up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field.

In the present contribution we study a PDE system describing the evolution of a nematic liquid crystals flow under kinematic transports for molecules of different shapes. More in particular, the evolution of the velocity field u is ruled by the Navier-Stokes incompressible system with a stress tensor exhibiting a special coupling between the transport and the induced terms. The dynamics of the director field d is described by a variation of a parabolic Ginzburg-Landau equation with a suitable penalization of the physical constraint |d| = 1. Such equation accounts for both the kinematic transport by the flow field and the internal relaxation due to the elastic energy. The main aim of this contribution is to overcome the lack of a maximum principle for the director equation and prove (without any restriction on the data and on the physical constants of the problem) the existence of global in time weak solutions under physically meaningful boundary conditions on d and u.

We investigate the long-time dynamics and optimal control problem of a 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-Hilliard type equation for the tumor cell fraction and a reaction-diffusion 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 "long-time treatment" under a suitable given source and prove the convergence of any global solution to a single equilibrium as t → +∞. Then we consider the "finite-time treatment" of a tumor, which corresponds to an optimal control problem. Here we also 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 φ Ω . By establishing the Lyapunov stability of certain equilibria of the state system (without external source), we see that φ Ω can be taken as a stable configuration, so that the tumor will not grow again once the finite-time treatment is completed.

We consider phase-field systems of Caginalp type on a three-dimensional bounded domain. The order parameter ψ fulfills a dynamic boundary condition, while the (relative) temperature θ is subject to a boundary condition of Dirichlet, Neumann or Robin type. Moreover, the two equations are nonlinearly coupled through a quadratic growth function. Here we extend a number of results which have been proven by some of the authors for the linear coupling. More precisely, we demonstrate the existence and uniqueness of global solutions. Then we analyze the associated dynamical system and we establish the existence of global as well as exponential attractors. We also discuss the convergence of given solutions to single equilibria.Without loss of generality, from now on we let δ = ε = 1. We denote by · p and · p,Γ , the norms on L p (Ω) and L p (Γ) , respectively. In the case p = 2, ·, · 2 (or ·, · 2,Γ ) stands for the usual scalar product which induces the L 2 norm (even for vector-valued functions). The norms on H s (Ω) and H s (Γ) are indicated by · H s (Ω) and · H s (Γ) , respectively, for any s > 0. In order to account for all the cases, we also introduce the family of linear operators inequalities and Hölder's inequality, from (4.16) we deduce