Some properties on the Q-Tensor system

We study the coupled Navier-Stokes and Q-Tensor system (analyzed in cf. [Paicu, M., and Zarnescu, A. Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system. Arch. Ration. Mech. Anal. 203 (2012), 45–67] in the whole R3) in a bounded three-dimensional domain for several boundary conditions, rewriting the system in a way that properties as symmetry and null-trace for the tensor Q can be proved. We show some analytical results such as: the existence of global in time weak solution, a maximum principle for the Q-tensor, local in time strong solution (which is global assuming an additional regularity criterion for the velocity in the space-periodic boundary condition case), global in time strong solution imposing dominant viscosity (for the space-periodic or homogeneous Neumann boundary condition cases) and regularity criteria for uniqueness of weak solutions

On the regularity of the Q-Tensor depending on the data

The coupled Navier-Stokes and Q-Tensor system is one of the models used to describe the behavior of the nematic liquid crystals, an intermediate phase between crystalline solids and isotropic fluids. These equations model the dynamics of the fluid via velocity and pressure (u, p) and the orientation of the molecules via a tensor Q. A review on the existence of weak solutions, maximum principle and a uniqueness criteria can be seen in [Guillen-González, F., and Rodríguez-Bellido, M. Á. Some properties on the Q-tensor system. Monogr. Mat. García Galdeano 39 (2014), 133–145] (the corresponding Cauchy problem in the whole R3 is analyzed by Zarnescu. However, the regularity of such solutions is only analyzed under some restrictive conditions: large viscosity or periodic boundary conditions. In this work, we study two different types of regularity for the Q-Tensor model: one inherited from the usual strong solution for the Navier-Stokes equations, and another one where (u, Q) and (∂tu, ∂tQ) have weak regularity (weak-t). This latter regularity is introduced due to the impossibility of obtaining local in time strong estimates for nonperiodic boundary conditions, where only the existence (and uniqueness) of local weak-t solution is obtained. Some regularity criteria for (u, Q) will also be given. In the particular case of Neumann boundary conditions for Q, the regularity criteria only must be imposed for the velocity u)

Continuation of Gerver's supereight choreography

Ejemplar dedicado a: Actas de las IX Jornadas de Mecánica CelesteIn [6] we developed a continuation technique for periodic orbits in reversible systems having some first integrals and corresponding symmetries. One of the applications was the continuation of Gerver’s supereight choreography when one or several of the masses are varied. In this note we give a more complete description of the families of periodic orbits which can be obtained in this way.Spanish Ministry of Education BFM2003-00336Spanish Ministry of Education MTM2006-00847University of Seville SAB2005-018

Management Mathematics for european schools

Beyond the topics, the Mathematics folklore says that pupils think that this field is a boring, non-practical mental exercise, far apart from daily life. This thought is influencing the entire education system and even society, motivating important lack of mathematical skills among students. This work presents experiences, developed within the MaMaEuSch project oriented to stimulate and enforce the study of Mathematics among the students in the High-School education level

Well-balanced Finite Volume schemes: some stability and convergence results

We report a stability and convergence analysis for some simplified well-balanced Finite Volume solvers of Hyperbolic Systems of Conservation Laws. These are specific solvers, recently introduced, that balance all steady solutions up to second order of accuracy by means of an additional numerical source term. We prove the stability and convergence of some of these solvers for scalar hyperbolic equations under reasonable conditions on the additional term

Numerical solution of a Laplace equation with data in L1

In this work, we address the numerical solution of the Laplace equation with data in L1 by IP1 finite element schemes. Even if this is a simple problem, its analysis is difficult and requires new tools because finite element schemes are based on variational formulations which do not lend themselves to estimates in the L1 norm. The approach for analyzing this problem consists in applying some of the techniques that are used by Murat (cf. [5]) and Boccardo & Gallouet (cf. [2]) in constructing the renormalized solution of the problem. The key ingredient is the assumption that all the angles of the grid are acute; then the matrix of the system is an M matrix. Interestingly, with this sole assumption, we prove that uh tends to u in mesure in Ω