Very weak solutions and large uniqueness classes of stationary Navier–Stokes equations in bounded domains of R2

AbstractExtending the notion of very weak solutions, developed recently in the three-dimensional case, to bounded domains Ω⊂R2 we obtain a new class of unique solutions u in Lq(Ω), q>2, to the stationary Navier–Stokes system −Δu+u⋅∇u+∇p=f, divu=k, u|∂Ω=g with data f,k,g of low regularity. As a main consequence we obtain a new uniqueness class also for classical weak or strong solutions. Indeed, such a solution is unique if its Lq-norm is sufficiently small or the data satisfy the uniqueness condition of a very weak solution

Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations

We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point $z$ if either the scaled $L^{p,q}_{x,t}$-norm of the velocity with $3/p+2/q\leq 2$, $1\leq q\leq \infty$, or the $L^{p,q}_{x,t}$-norm of the vorticity with $3/p+2/q\leq 3$, $1 \leq q < \infty$, or the $L^{p,q}_{x,t}$-norm of the gradient of the vorticity with $3/p+2/q\leq 4$, $1 \leq q$, $1 \leq p$, is sufficiently small near $z$

Analysis of the velocity tracking control problem for the 3D evolutionary Navier-Stokes equations

The velocity tracking problem for the evolutionary Navier–Stokes equations in three dimensions is studied. The controls are of distributed type and are submitted to bound constraints. The classical cost functional is modified so that a full analysis of the control problem is possible. First and second order necessary and sufficient optimality conditions are proved. A fully discrete scheme based on a discontinuous (in time) Galerkin approach, combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, τ and h, respectively, satisfy τ ≤ Ch2, the L2(ΩT ) error estimates of order O(h) are proved for the difference between the locally optimal controls and their discrete approximations. Finally, combining these techniques and the approach of Casas, Herzog, and Wachsmuth [SIAM J. Optim., 22 (2012), pp. 795–820], we extend our results to the case of L1(ΩT ) type functionals that allow sparse controls.This author was partially supported by the Spanish Ministerio de Economía y Competitividad under projects MTM2011-22711 and MTM2014-57531-

Southern American University Undergraduates\u27 Attitudes toward Intrauterine Insemination Undertaken by Women of Differing Age, Marital Status and Sexual Orientation

Undergraduate college students in the southern U.S. were presented with vignettes about a fictional woman seeking to become pregnant via intrauterine insemination (IUI). Participants were randomly assigned to conditions in which the woman described was 26 or 41 years old, and single, married to a man, or married to a woman. After reading the vignettes, participants rated their expectations of the prospective mother’s preparedness for parenting, ability to provide quality of life for a child, risk for pregnancy complications and achieving a healthy pregnancy. Results yielded marginally significantly (p = .05) lower expectations of achieving a healthy pregnancy when the mother was over 40, and significantly (p \u3c .05) lower anticipation of preparedness for parenting and ability to provide quality of life when she was designated as single or married to a woman. We discuss findings in terms of bias favoring traditional families with a mother and father begun when the parents were in their twenties

Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor

In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, i.e., the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three-dimensional Navier-Stokes equations in the whole space, as well as for the case of periodic boundary conditions

Risk groups for clinical complications of norovirus infections: an outbreak investigation.

Norovirus infections have been described as self-limiting diseases of short duration. An investigation of a norovirus outbreak in a university hospital provided evidence for severe clinical features in patients with several underlying diseases. Clinical outcomes of norovirus infection were defined. Risk-factor analysis targeting underlying diseases and medication was performed using multivariate analyses. In five outbreak wards, 84 patients and 60 nurses were infected (an overall attack rate of 32% in patients, and 76% in nurses). The causative agent was the new variant Grimsby virus. Severe clinical features, including acute renal failure, arrhythmia and signs of acute graft organ rejection in renal transplant patients, were observed in seven (8.3%) patients. In multivariate analyses, cardiovascular disease (OR 17.1, 95% CI 2.17-403) and renal transplant (OR 13.0, 95% CI 1.63-281) were risk-factors for a potassium decrease of >20%. Age >65 years (OR 11.6, 95% CI 1.89-224) was a risk-factor for diarrhoea lasting >2 days. Immunosuppression (OR 5.7, 95% CI 1.78-20.1) was a risk-factor for a creatinine increase of >10%. Norovirus infections in patients with underlying conditions such as cardiovascular disease, renal transplant and immunosuppressive therapy may lead to severe consequences typified by decreased potassium levels, increased levels of C-reactive protein and creatine phosphokinase. In the elderly, norovirus infection may lead to an increased duration of diarrhoea. Therefore patients at risk should be hospitalised early and monitored frequently. Strict preventional measures should be implemented as early as possible to minimise the risk of nosocomial outbreaks

Self-propelled micro-swimmers in a Brinkman fluid

We prove an existence, uniqueness, and regularity result for the motion of a self-propelled micro-swimmer in a particulate viscous medium, modelled as a Brinkman fluid. A suitable functional setting is introduced to solve the Brinkman system for the velocity field and the pressure of the fluid by variational techniques. The equations of motion are written by imposing a self-propulsion constraint, thus allowing the viscous forces and torques to be the only ones acting on the swimmer. From an infinite-dimensional control on the shape of the swimmer, a system of six ordinary differential equations for the spatial position and the orientation of the swimmer is obtained. This is dealt with standard techniques for ordinary differential equations, once the coefficients are proved to be measurable and bounded. The main result turns out to extend an analogous result previously obtained for the Stokes system

A geometric condition implying energy equality for solutions of 3D Navier-Stokes equation

We prove that every weak solution $u$ to the 3D Navier-Stokes equation that belongs to the class $L^3L^{9/2}$ and \n u belongs to $L^3L^{9/5}$ localy away from a 1/2-H\"{o}lder continuous curve in time satisfies the generalized energy equality. In particular every such solution is suitable.Comment: 10 page

On the existence of traveling waves in the 3D Boussinesq system

We extend earlier work on traveling waves in premixed flames in a gravitationally stratified medium, subject to the Boussinesq approximation. For three-dimensional channels not aligned with the gravity direction and under the Dirichlet boundary conditions in the fluid velocity, it is shown that a non-planar traveling wave, corresponding to a non-zero reaction, exists, under an explicit condition relating the geometry of the crossection of the channel to the magnitude of the Prandtl and Rayleigh numbers, or when the advection term in the flow equations is neglected.Comment: 15 pages, to appear in Communications in Mathematical Physic