Identification of a convolution kernel in a control problem for the heat equation with a boundary memory term

We consider the evolution of the temperature $u$ in a material with thermal memory characterized by a time-dependent convolution kernel $h$. The material occupies a bounded region $\Omega$ with a feedback device controlling the external temperature located on the boundary $\Gamma$. Assuming both $u$ and $h$ unknown, we formulate an inverse control problem for an integrodifferential equation with a nonlinear and nonlocal boundary condition. Existence and uniqueness results of a solution to the inverse problem are proved

On a 3D isothermal model for nematic liquid crystals accounting for stretching terms

The present contribution investigates the well-posedness of a PDE system describing the evolution of a nematic liquid crystal flow under kinematic transports for molecules of different shapes. More in particular, the evolution of the {\em velocity field} \ub is ruled by the Navier-Stokes incompressible system with a stress tensor exhibiting a special coupling between the transport and the induced terms. The dynamic of the {\em director field} \bd is described by a variation of a parabolic Ginzburg-Landau equation with a suitable penalization of the physical constraint |\bd|=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 \bd and \ub

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

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 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 $t\to+\infty$. 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 $\phi_\Omega$. By establishing the Lyapunov stability of certain equilibria of the state system (without external source), we see that $\phi_{\Omega}$ can be taken as a stable configuration, so that the tumor will not grow again once the finite-time treatment is completed

Non-isothermal viscous Cahn--Hilliard equation with inertial term and dynamic boundary conditions

We consider a non-isothermal modified Cahn--Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and a viscous term and it is coupled with a hyperbolic heat equation. The resulting system was studied in the case of no-flux boundary conditions. Here we analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with bounded energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Lojasiewicz--Simon inequality

Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows

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

Global strong solutions of the full Navier-Stokes and QQ-tensor system for nematic liquid crystal flows in 2D2D: existence and long-time behavior

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 $\xi$ 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 3D isothermal model for nematic liquid crystals 1 with delay terms

In this paper we consider a model describing the evolution of a nematic liquid crystal flow with delay external forces. We analyze the evolution of the velocity fi eld u which is ruled by the 3D incompressible Navier-Stokes system containing a delay term and with a stress tensor expressing the coupling between the transport and the induced terms. The dynamics of the director eld d is described by a modifi ed Allen-Cahn equation with a suitable penalization of the physical constraint jdj = 1. We prove the existence of global in time weak solutions under appropriate assumptions, which in some cases requires the delay term to be small with respect to the viscosity parameter

Pointwise gradient bounds for entire solutions of elliptic equations with non-standard growth conditions and general nonlinearities

We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and degenerate nonlinear case with non-standard growth conditions. The sourcing term is also allowed to have a very general form, depending on the space variables, on the solution itself, on its gradient, and possibly on higher order derivatives if additional structural conditions are satisfied