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

The Oseen and Navier-Stokes equations in a non-solenoidal framework

The very weak solution for the Stokes, Oseen and Navier-Stokes equations has been studied by several authors in the last decades in domains of Rn, n ≥ 2. The authors studied the Oseen and Navier-Stokes problems assuming a solenoidal convective velocity in a bounded domain Ω ⊂ R3 of class C1,1 for v ∈ Ls (Ω) for s ≥ 3 in some previous papers. The results for the Navier-Stokes equations were obtained by using a fixed-point argument over the Oseen problem. These results improve those of Galdi et al. , Farwig et al. and Kim for the Navier-Stokes equations, because a less regular domain Ω ⊂ R3 and more general hypothesis on the data are considered. In particular, the external forces must not be small. In this work, existence of weak, strong, regularised and very weak solution for the Oseen problem are proved, mainly assuming that v ∈ L3(Ω) and its divergence ∇ · v is sufficiently small in the W−1,3(Ω)-norm. In this sense, one extends the analysis made by the authors for a given solenoidal v in some previous papers. As a consequence, the existence of very weak solution for the Navier-Stokes problem (u, π) ∈ L3(Ω) × W−1,3(Ω)/R for a non-zero divergence condition is obtained in the 3D case.Ministerio de Ciencia e Innovación (España) MTM2009-12927Ministerio de Economía y Competitividad MTM2012-3232

On the very weak solution for the Oseen and Navier-Stokes equations

We study the existence of very weak solutions regularity for the Stokes, Oseen and NavierStokes system when non-smooth Dirichlet boundary data for the velocity are considered in domains of class C1,1. In the Navier-Stokes case, the results will be valid for external forces non necessarily small. Regularity results for more regular data will be also discussed.Ministerio de Educación y CienciaJunta de Andalucí

Very weak solutions for the stationary Stokes equations

The concept of very weak solution introduced by Giga [9] for the stationary Stokes equations has been intensively studied in the last years for the stationary Navier-Stokes equations. We give here a new and simpler proof of the existence of very weak solution for the stationary Navier-Stokes equations, based on density arguments and an adequate functional framework in order to define more rigourously the traces of non regular vector fields. We also obtain regularity results in fractional Sobolev spaces. All these results are obtained in the case of a bounded open set, connected of class C1,1 of R3 and can be extended to the Laplace’s equation and to other dimensions.Ministerio de Educación y CienciaJunta de Andalucí

Weak solutions for the Oseen system in 2D and when the given velocity is not sufficiently regular

The aim of this work is twofold: proving the existence of solution (u,π)∈H1(Ω)×L2(Ω) in bounded domains of R2 and the whole plane for the Oseen problem (O) for solenoidal vector fields v in L2(Ω), and analyzing the same problem in bounded domains of Rn for n=2,3 when h=0, g=0 and the solenoidal field v belongs to Ls(Ω) for s<n

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

Analysis of a chemo-repulsion model with nonlinear production: The continuous problem and unconditionally energy stable fully discrete schemes

We consider the following repulsive-productive chemotaxis model: Let $p\in (1,2)$, find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, satisfying \begin{equation}\label{C5:Am} \left\{ \begin{array} [c]{lll} \partial_t u - \Delta u - \nabla\cdot (u\nabla v)=0 \ \ \mbox{in}\ \Omega,\ t>0,\\ \partial_t v - \Delta v + v = u^p \ \ \mbox{in}\ \Omega,\ t>0, \end{array} \right. \end{equation} in a bounded domain $\Omega\subseteq \mathbb{R}^d$, $d=2,3$. By using a regularization technique, we prove the existence of solutions of this problem. Moreover, we propose three fully discrete Finite Element (FE) nonlinear approximations, where the first one is defined in the variables $(u,v)$, and the second and third ones by introducing ${\boldsymbol\sigma}=\nabla v$ as an auxiliary variable. We prove some unconditional properties such as mass-conservation, energy-stability and solvability of the schemes. Finally, we compare the behavior of the schemes throughout several numerical simulations and give some conclusions.Comment: arXiv admin note: substantial text overlap with arXiv:1807.0111

On the singular times of fluids with nonlinear viscosity

Comisión Interministerial de Ciencia y TecnologíaPatronato de la Fundación Cámara de la Universidad de Sevill

Convergence and error estimates of two iterative methods for the strong solution of the incompressible korteweg model

We show the existence of strong solutions for a fluid model with Korteweg tensor, which is obtained as limit of two iterative linear schemes. The different unknowns are sequentially decoupled in the first scheme and in parallel form in the second one. In both cases, the whole sequences are bounded in strong norms and convergent towards the strong solution of the system, by using a generalization of the Banach’s Fixed Point Theorem. Moreover, we explicit a priori and a posteriori error estimates (respect to the weak norms), which let us to compare both schemes.Dirección General de Investigación (Ministerio de Educación y Ciencia)Junta de Andalucí

A review on the improved regularity for the primitive equations

In this work we will study, some types of regularity properties of solutions for the geophysical model of hydrostatic Navier-Stokes equations, so-called the Primitive Equations (P E). Also, we will present some results about uniqueness and asymptotic behavior in time.Ministerio de Ciencia y Tecnologí