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In this paper we propose two fully discrete Finite Elements (FE) schemes for a repulsive chemotaxis model with a quadratic production term. The first one, which corresponds to the backward Euler in time with FE in space, is energy-stable in the primitive variables of the model, under a “compatibility” condition on the FE spaces. The second one, which is obtained modifying the scheme proposed in [13] by applying a regularization procedure, has an “approximated positivity” property which is obtained from discrete energy estimates and an additional estimate for a singular functional. These properties are not available in previous approaches. Additionally, we study the well-posedness and the long time behaviour of the schemes, obtaining exponential convergence to constant states as in the continuous problem. Finally, we compare the numerical schemes throughout several numerical simulations, which are in agreement with the theoretical results.

Models to represent the interactions of liquid crystalline phases with vesicles mem- branes have been widely studied in recent times due to its connection with biological materials. In this work we propose a new model to represent a vesicle membrane with internal nematic order whose equilibrium states depend on the competition between the bending, elastic and anchoring energies. Moreover we present a new unconditionally energy stable numerical scheme to approximate the model. Additionally, we present several numerical simulations in order to illustrate the good performance of the proposed scheme and the influence of the nematic order in the dynamics of the system.

Models of flows containing vesicles membranes with liquid crystalline phases have been widely studied in recent times due to its connection with biological applications. In this work we propose a new model to represent the interaction between flows and vesicle membranes with internal nematic order and preferential orientation of their molecules in the membrane. In fact, the dynamics of the system is determined by the dissipation of an energy that regulates the competition between different effects, through the kinetic, bending, elastic and anchoring energies. Moreover we introduce a new numerical scheme to approximate the model, that is unconditionally energy stable. Additionally, we present several numerical results in order to show the well behavior of the proposed scheme and the dynamics of this type of vesicle membranes

This paper studies the stability of velocity-pressure mixed approximations of the Stokes problem when different finite element (FE) spaces for each component of the velocity field are considered. We consider some new combinations of continuous FE reducing the number of degrees of freedom in some velocity components. Although the resulting FE combinations are not stable in general, by using the Stenberg’s macro-element technique, we show their stability in a wide family of meshes (namely, in uniformly unstructured meshes). Moreover, a post-processing is given in order to convert any mesh family in an uniformly unstructured mesh family. Finally, some 2D and 3D numerical simulations are provided agree with the previous analysis.

In this paper, we present energy-stable numerical schemes for a Smectic-A liquid crystal model. This model involve the hydrodynamic velocity-pressure macroscopic variables (u, p) and the microscopic order parameter of Smectic-A liquid crystals, where its molecules have a uniaxial orientational order and a positional order by layers of normal and unitary vector n. We start from the formulation given in [E’97] by using the so-called layer variable φ such that n = ∇φ and the level sets of φ describe the layer structure of the Smectic-A liquid crystal. Then, a strongly non-linear parabolic system is derived coupling velocity and pressure unknowns of the Navier-Stokes equations (u, p) with a fourth order parabolic equation for φ. We will give a reformulation as a mixed second order problem which let us to define some new energy-stable numerical schemes, by using second order finite differences in time and C 0 - finite elements in space. Finally, numerical simulations are presented for 2D-domains, showing the evolution of the system until it reachs an equilibrium configuration. Up to our knowledge, there is not any previous numerical analysis for this type of models.

The coupled Navier-Stokes and Q-Tensor system is considered in a bounded three-dimensional domain under homogeneous Dirichlet boundary conditions for the velocity u and either nonhomogeneous Dirichlet or homogeneous Neumann boundary conditions for the tensor Q. The corresponding initial-value problem in the whole space R3 was analyzed in [Paicu & Zarnescu, 2012]. In this paper, three main results concerning weak solutions will be proved; the existence of global in time weak solutions (bounded up to infinite time), a uniqueness criteria and a maximum principle for Q. Moreover, we identify how to modify the system to deduce symmetry and traceless for Q, for any weak solution. The presence of a stretching term in the Q-system plays a crucial role in all the analysis.

In this paper, we focus on a smectic-A liquid crystal model in 3D domains, and obtain three main results: the proof of an adequate Lojasiewicz-Simon inequality by using an abstract result; the rigorous proof (via a Galerkin approach) of the existence of global intime weak solutions that become strong (and unique) in long-time; and its convergence to equilibrium of the whole trajectory as time goes to in nity. Given any regular initial data, the existence of a unique global in-time regular solution (bounded up to in nite time) and the convergence to an equilibrium have been previously proved under the constraint of a su ciently high level of viscosity. Here, all results are obtained without imposing said constraint.

In this work, we propose finite element schemes for the numerical approximation of nematic liquid crystal flows, based on a saddle-point formulation of the director vector sub-problem. It introduces a Lagrange multiplier that allows to enforce the sphere condition. In this setting, we can consider the limit problem (without penalty) and the penalized problem (using a Ginzburg-Landau penalty function) in a unified way. Further, the resulting schemes have an stable behavior with respect to the value of the penalty parameter, a key difference with respect to the existing schemes. Two different methods have been considered for the time integration. First, we have considered an implicit algorithm that is unconditionally stable and energy preserving. The linearization of the problem at every time step value can be performed using a quasi-Newton method that allows to decouple fluid velocity and director vector computations for every tangent problem. Then, we have designed a linear semi-implicit algorithm (i.e. it does not involve nonlinear iterations) and proved that it is unconditionally stable, verifying a discrete energy inequality. Finally, some numerical simulations are provided.

In this paper two main results are obtained for a nematic liquid crystal model with timedependent boundary Dirichlet data for the orientation of the crystal molecules. First, the initial-boundary problem is considered, obtaining the existence of global in time (up to infinity time) weak solution, the existence of global regular solution for viscosity coefficient big enough, and the weak/strong uniqueness. Second, using these previous results and the existence of time-periodic weak solutions proved in [2] B. Climent-Ezquerra, F. Guillén-González, M.A. Rojas-Medar Reproductivity for a nematic liquid crystal model, Z. Angew. Math. Phys., 576 (2006) no. 6, 984-998, the regularity of any time-periodic weak solution is deduced for viscosity coefficient big enough.

We consider a Navier-Stokes type system with a Korteweg stress tensor, coupled with a concentration equation without diffusion. We give a result concerning the local in time existence (and uniqueness) of strong solution for any data (and the global in time existence defined in a interval (0, T) for T < +∞ fixed if data are small enough), in the case of a bounded domain Ω ⊂ R3.

In this note, we prove that given u a weak solution of the Primitive Equations, imposing an additional condition on the vertical derivative of the velocity u (concretely ∂zu ∈ L∞(0, T;L2(Ω)) ∩ L2(0, T; H1(Ω))), then two different results hold; namely, uniqueness of weak solution (any weak solution associated to the same data that u must coincide with u) and global in time strong regularity for u (without “smallness assumptions” on the data). Both results are proved when either Dirichlet or Robin type conditions on the bottom are considered. In the last case, a domain with a strictly bounded from below depth has to be imposed, even for the uniqueness result.

The main subject of this work is to study the concept of very weak solution for the hydrostatic Stokes system with mixed boundary conditions (non-smooth Neumann conditions on the rigid surface and homogeneous Dirichlet conditions elsewhere on the boundary). In the Stokes framework, this concept has been studied by Conca [Rev. Mat. Apl. 10 (1989)] imposing non-smooth Dirichlet boundary conditions. In this paper, we introduce the dual problem that turns out to be a hydrostatic Stokes system with non-free divergence condition. First, we obtain strong regularity for this dual problem (which can be viewed as a generalisation of the regularity results for the hydrostatic Stokes system with free divergence condition obtained by Ziane [Appl. Anal. 58 (1995)]). Afterwards, we prove existence and uniqueness of very weak solution for the (primal) problem. As a consequence of this result, the existence of strong solution for the non-stationary hydrostatic Navier-Stokes equations is proved, weakening the hypothesis over the time derivative of the wind stress tensor imposed by Guillén-González, Masmoudi and Rodríguez-Bellido [Differential Integral Equations 50 (2001)]., Le but principal de ce travail est d’étudier le concept de solution très faible pour le système de Stokes hydrostatique avec conditions aux limites mixtes (condition de Neumann non régulière sur la surface rigide et condition de Dirichlet homogène dans le reste). Dans le cas du problème de Stokes, ce sujet a été étudié par Conca [Rev. Mat. Apl. 10 (1989)] en imposant une condition aux limites de Dirichlet non homogène et peu régulière. Dans ce papier, on introduit le problème dual qui est aussi un système de Stokes hydrostatique mais avec une condition de divergence non nulle. D’abord, on obtient la régularité forte pour le problème dual (ce résultat peut être consideré comme une généralisation des résultats de régularité pour le système de Stokes hydrostatique avec la condition de divergence nulle, obtenus par Ziane [Appl. Anal. 58 (1995)]). On montre ensuite l’existence et unicité de solution très faible pour le problème primal. Comme conséquence de ce résultat, on montre l’existence de solution forte pour le problème de Navier-Stokes hydrostatique non-stationnaire, avec une hypothèse sur la dérivée par rapport au temps du tenseur du vent plus faible que celle qui était imposée par Guillén-González, Masmoudi et Rodríguez-Bellido [Differential Integral Equations 50 (2001)].

We prove existence of a global weak solution for a nematic liquid crystal problem by means of a penalization method using a simplified Ericksen-Leslie model and a new compactness property for the gradient of the director field.