Existence of weak-renormalized solution for a nonlinear system

We prove an existence result for a coupled system of the reactiondiffusion kind. The fact that no growth condition is assumed on some nonlinear terms motivates the search of a weak-renormalized solution

Existence and uniqueness results for a coupled problem related to the stationary Navier-Stokes System

In this paper, we consider some systems which are close to the stationary Navier-Stokes equations. The structure of these systems is the following: An N-dimensional equation for motion, the incompressibility condition and a scalar equation involving an additional unknown, k = k(x). Among other things, they serve to model the behavior of certain turbulent flows. Our main interest concerns existence and uniqueness. The main difficulties are due to the structure of the scalar equation; in partic- ular, the right side is typically in L1 and, furthermore, there are nonlinear terms of the kind ∇ · (μ(k)∇k) and ∇ · (B(k)), where μ and B are general continuous functions.Partially supported by D.G.I.C.Y.T. (Spain), Proyecto PB92– 0696

Some existence and uniqueness results for a time-dependent coupled problem of the Navier-Stokes kind

In this paper, we consider some systems which are close to the instationary Navier-Stokes equations. The structure of these systems is the following: An (N +1)-dimensional equation for motion (including the incompressibility condition) and a scalar equation involving an additional unknown, k = k(x; t). Among other things, they serve to model the behavior of certain turbulent ows. We are mainly concerned with existence and uniqueness results. The main di culties are due to the scalar equation. In particular, the right side is typically in L1; furthermore, there are nonlinear terms of the kind r ( (k)rk) and r (B(k)), where and B are general continuous functions (no growth condition at in nity is imposed). Following the previous work of other authors, it is crucial to introduce the notion of weak-renormalized solution. Our results provide existence in the two-dimensional case, as well as the uniqueness of regular solution in both the two and three-dimensional cases

Convergence to equilibrium for smectic-A liquid crystals in 3D domains without constraints for the viscosity

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

Global in time solutions for the Poiseuille flow of Oldroyd type in 3D domains

A Poiseuille flow in a 3D cylindrical domain is considered for a non-newtonian fluid of Oldroyd type. We prove existence (and uniqueness) of a global (in time) weak solution. Moreover, this weak solution is an strong solution when data are more regular. These results has already been obtained in the case of two concentrical cylinders . Now, we consider an extension to an unique cylinder. Then, a mixed parabolic-hyperbolic PDE's system appears but the parabolic equation is of degenerate type. The key of the proofs is to estimate in appropriate Sobolev weighted spaces (and to obtain strong convergence in weak norms by means of a Cauchy argument)

Global in time solution and time-periodicity for a smectic-A liquid crystal model

In this paper some results are obtained for a smectic-A liquid crystal model with time-dependent boundary Dirichlet data for the so-called layer variable φ (the level sets of φ describe the layer structure of the smectic-A liquid crystal). First, the initial-boundary problem for arbitrary initial data is considered, obtaining the existence of weak solutions which are bounded up to infinity time. Second, the existence of time-periodic weak solutions is proved. Afterwards, the problem of the global in time regularity is attacked, obtaining the existence and uniqueness of regular solutions (up to infinity time) for both problems, i.e. the initial-valued problem and the time-periodic one, but assuming a dominant viscosity coefficient in the linear part of the diffusion tensor.Ministerio de Educación y Ciencia MTM2006–07932Junta de Andalucía P06-FQM-0237

Regular time-reproductive solutions for generalized Boussinesq model with Neumann boundary conditions for temperature

The aim of this work is to prove existence of regular time reproductive solutions for a generalized Boussinesq model (with nonlinear diffusion for velocity and temperature). The main idea is to obtain higher regularity (of H3 type) for temperature than for velocity (of H2 type), using specifically the Neumann boundary condition for temperature.Ministerio de Ciencia y Tecnología BFM2003-06446-C02-01Conselho Nacional de Desenvolvimento Científico e Tecnológico 301354/03-

On a double penalized smectic-A model

In smectic-A liquid crystals a unity director vector n appear, modeling an average preferential direction of the molecules and also the normal vector of the layer configuration. In the E’s model [5] W. E. Nonlinear Continuum Theory of Smectic-A Liquid Crystals, Arch. Rat. Mech. Anal., 137, 2 (2010), 1473-1493, the Ginzburg-Landau penalization related to the constraint |n| = 1 is considered and, assuming the constraint ∇ × n = 0, n is replaced by the so-called layer variable ϕ such that n = ∇ϕ. In this paper, a double penalized problem is introduced related to a smectic-A liquid crystal flows, considering a Cahn-Hilliard system to model the behavior of n. Then, the issue of the global in time behavior of solutions is attacked, including the proof of the convergence of the whole trajectory towards a unique equilibrium state.Dirección General de Investigación (Ministerio de Educación y Ciencia

Convergence to equilibrium of global weak solutions for a Cahn-Hilliard-Navier-Stokes vesicle model

In this paper, we introduce a model describing the dynamic of vesicle membranes within an incompressible viscous fluid in 3D domains. The system consists of the NavierStokes equations, with an extra stress tensor depending on the membrane, coupled with a Cahn-Hilliard phase-field equation associated to a bending energy plus a penalization term related to the area conservation. This problem has a dissipative in time free-energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a modified Lojasiewicz-Simon’s result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase.Ministerio de Economía y Competitividad (MINECO). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER