On admissibility criteria for weak solutions of the Euler equations

We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution. As a byproduct we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique in more than one space dimension.Comment: 33 pages, 1 figure; v2: 35 pages, corrected typos, clarified proof

Características fenotípicas de 44 progênies de Maytenus ilicifolia Mart. ex Reiss. cultivadas no município de Ponta Grossa, PR.

Resumo

Singular and regular solutions of a non-linear parabolic system

We study a dissipative nonlinear equation modelling certain features of the Navier-Stokes equations. We prove that the evolution of radially symmetric compactly supported initial data does not lead to singularities in dimensions $n\leq 4$. For dimensions $n>4$ we present strong numerical evidence supporting existence of blow-up solutions. Moreover, using the same techniques we numerically confirm a conjecture of Lepin regarding existence of self-similar singular solutions to a semi-linear heat equation.Comment: 16 page

The aetiologies of epilepsy

The identification of the aetiology of a patient's epilepsy is instrumental in the diagnosis, prognostic counselling and management of the epilepsies. Indeed, the aetiology can be important for determining the recurrence risk of single seizures and so for making a diagnosis of epilepsy. Here, we divide the aetiologies into six categories: structural, genetic, infectious, metabolic, immune (all of which are part of the International League Against Epilepsy [ILAE] classification system) and neurodegenerative (which we have considered separately because of its growing importance in epilepsy). These are not mutually exclusive categories and many aetiologies fall into more than one category. Indeed, genetic factors probably play a role, to varying degrees, in the risk of seizures in all people with epilepsy. In each of the categories, we discuss what we regard as the most important aetiologies; importance being determined not only by prevalence but also by clinical significance. The introduction contains information suitable for level 1 competency (entry level), whilst the subsequent sections contain information aimed at level 2 competency (proficiency level) as part of the new ILAE competency-based curriculum. As we move towards precision medicine and targeted therapies, so aetiologies will play an even greater role in the management of epilepsy

A Cognitive Model of an Epistemic Community: Mapping the Dynamics of Shallow Lake Ecosystems

We used fuzzy cognitive mapping (FCM) to develop a generic shallow lake ecosystem model by augmenting the individual cognitive maps drawn by 8 scientists working in the area of shallow lake ecology. We calculated graph theoretical indices of the individual cognitive maps and the collective cognitive map produced by augmentation. The graph theoretical indices revealed internal cycles showing non-linear dynamics in the shallow lake ecosystem. The ecological processes were organized democratically without a top-down hierarchical structure. The steady state condition of the generic model was a characteristic turbid shallow lake ecosystem since there were no dynamic environmental changes that could cause shifts between a turbid and a clearwater state, and the generic model indicated that only a dynamic disturbance regime could maintain the clearwater state. The model developed herein captured the empirical behavior of shallow lakes, and contained the basic model of the Alternative Stable States Theory. In addition, our model expanded the basic model by quantifying the relative effects of connections and by extending it. In our expanded model we ran 4 simulations: harvesting submerged plants, nutrient reduction, fish removal without nutrient reduction, and biomanipulation. Only biomanipulation, which included fish removal and nutrient reduction, had the potential to shift the turbid state into clearwater state. The structure and relationships in the generic model as well as the outcomes of the management simulations were supported by actual field studies in shallow lake ecosystems. Thus, fuzzy cognitive mapping methodology enabled us to understand the complex structure of shallow lake ecosystems as a whole and obtain a valid generic model based on tacit knowledge of experts in the field.Comment: 24 pages, 5 Figure

Lack of uniqueness for weak solutions of the incompressible porous media equation

In this work we consider weak solutions of the incompressible 2-D porous media equation. By using the approach of De Lellis-Sz\'ekelyhidi we prove non-uniqueness for solutions in $L^\infty$ in space and time.Comment: 23 pages, 2 fugure

Nox2 underpins microvascular inflammation and vascular contributions to cognitive decline

\ua9 The Author(s) 2022. Chronic microvascular inflammation and oxidative stress are inter-related mechanisms underpinning white matter disease and vascular cognitive impairment (VCI). A proposed mediator is nicotinamide adenine dinucleotide phosphate (NADPH) oxidase 2 (Nox2), a major source of reactive oxygen species (ROS) in the brain. To assess the role of Nox2 in VCI, we studied a tractable model with white matter pathology and cognitive impairment induced by bilateral carotid artery stenosis (BCAS). Mice with genetic deletion of Nox2 (Nox2 KO) were compared to wild-type (WT) following BCAS. Sustained BCAS over 12 weeks in WT mice induced Nox2 expression, indices of microvascular inflammation and oxidative damage, along with white matter pathology culminating in a marked cognitive impairment, which were all protected by Nox2 genetic deletion. Neurovascular coupling was impaired in WT mice post-BCAS and restored in Nox2 KO mice. Increased vascular expression of chemoattractant mediators, cell-adhesion molecules and endothelial activation factors in WT mice post-BCAS were ameliorated by Nox2 deficiency. The clinical relevance was confirmed by increased vascular Nox2 and indices of microvascular inflammation in human post-mortem subjects with cerebral vascular disease. Our results support Nox2 activity as a critical determinant of VCI, whose targeting may be of therapeutic benefit in cerebral vascular disease