A uniqueness and regularity criterion for Q-tensor models with Neumann boundary conditions

We give a regularity criterion for a $Q$-tensor system modeling a nematic Liquid Crystal, under homogeneous Neumann boundary conditions for the tensor $Q$. Starting of a criterion only imposed on the velocity field ${\bf u}$ two results are proved; the uniqueness of weak solutions and the global in time weak regularity for the time derivative $(\partial_t {\bf u},\partial_t Q)$. This paper extends the work done in [F. Guill\'en-Gonz\'alez, M.A. Rodr\'iguez-Bellido \& M.A. Rojas-Medar, Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model, Math. Nachr. 282 (2009), no. 6, 846-867] for a nematic Liquid Crystal model formulated in $({\bf u},{\bf d})$, where ${\bf d}$ denotes the orientation vector of the liquid crystal molecules.Comment: 13 page

Density-dependent incompressible fluids with non-Newtonian viscosity

We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and non-constant viscosity. Indeed, one considers a stress tensor being a nonlinear function of the symmetric velocity gradient, verifying the properties of pcoercivity and (p − 1)-growth, for a given parameter p > 1. The existence of Dirichlet weak solutions was obtained in [2], in the cases p > 12/5 if d = 3 or p > 2 if d = 2, d being the dimension of the domain. In this paper, with help of some new estimates (which lead to point-wise convergence of the velocity gradient), we obtain the existence of space-periodic weak solutions for all p > 2. In addition, we obtain regularity properties of weak solutions whenever p > 20/9 (if d = 3) or p > 2 (if d = 2). Further, some extensions of these results to more general stress tensors or to Dirichlet boundary conditions (with a Newtonian tensor large enough) are obtained.Comisión Interministerial de Ciencia y Tecnologí

Optimal bilinear control problem related to a chemo-repulsion system in 2D domains

In this paper we study a bilinear optimal control problem associated to a chemo-repulsion model with linear production term. We analyze the existence, uniqueness and regularity of pointwise strong solutions in a bidimensional domain. We prove the existence of an optimal solution and, using a Lagrange multipliers theorem, we derive first-order optimality conditions

Approximation of Smectic-A liquid crystals

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.Ministerio de Economía y CompetitividadMinistry of Education, Youth and Sports of the Czech Republi

Superconvergence in velocity and pressure for the 3D time-dependent Navier-Stokes equations

This work is devoted to the superconvergence in space approximation of a fully discrete scheme for the incompressible time-dependent Navier-Stokes Equations in three-dimensional domains. We discrete by Inf-Sup-stable Finite Element in space and by a semi-implicit backward Euler (linear) scheme in time. Using an extension of the duality argument in negative-norm for elliptic linear problems (see for instance [1] Brennet, S., Scott, L. The Mathematical Theory of Finite Element Methods, Springer, 2008) to the mixed velocity-pressure formulation of the Stokes problem, we prove some superconvergence in space results for the velocity with respect to the energy-norm, and for a weaker norm of L2 (0, T;L 2 (Ω)) type (this latter holds only for the case of Taylor-Hood approximation). On the other hand, we also obtain optimal error estimates for the pressure without imposing constraints on the time and spatial discrete parameters, arriving at superconvergence in the H1 (Ω)-norm again for Taylor-Hood approximations. These results are numerically verified by several computational experiments, where two splitting in time schemes are also considered.Ministerio de Ciencia e Innovación (España

Numerical methods for solving the Cahn-Hilliard equation and its applicability to related Energy-based models

In this paper, we review some numerical methods presented in the literature in the last years to approximate the Cahn-Hilliard equation. Our aim is to compare the main properties of each one of the approaches to try to determine which one we should choose depending on which are the crucial aspects when we approximate the equations. Among the properties that we consider desirable to control are the time accuracy order, energy-stability, unique solvability and the linearity or nonlinearity of the resulting systems. In particular, we concern about the iterative methods used to approximate the nonlinear schemes and the constraints that may arise on the physical and computational parameters. Furthermore, we present the connections of the Cahn-Hilliard equation with other physically motivated systems (not only phase field models) and we state how the ideas of efficient numerical schemes in one topic could be extended to other frameworks in a natural way.Ministry of Education, Youth and Sports of the Czech RepublicMinisterio de Economía y Competitivida

On the stability of approximations for the Stokes problem using different finite element spaces for each component of the velocity

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.Ministerio de Economía y CompetitividadJunta de Andalucí