Eulerian and Lagrangian solutions to the continuity and Euler equations with L1L^1 vorticity

In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in \cite{BouchutCrippa13}. The proof is based on a combination of a stability estimate via optimal transport techniques developed in \cite{Seis16a} and some tools from harmonic analysis introduced in \cite{BouchutCrippa13}. In the second part of the paper, we address a question that arose in \cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in~$L^1$, extending the proof for the $L^p$ case in \cite{CrippaSpirito15}

Bounds on heat transport for two-dimensional buoyancy driven flows between rough boundaries

We consider the two-dimensional Rayleigh-B\'enard convection in a layer of fluid between rough Navier-slip boundaries. The top and bottom boundaries are described by the same height function $h$. We prove rigorous upper bounds on the Nusselt number which capture the dependence on the curvature of the boundary $\kappa$ and the (non-constant) friction coefficient $\alpha$ explicitly. For $h\in W^{2,\infty}$ and $\kappa$ satisfying a smallness condition with respect to $\alpha$, we find $\text{Nu}\lesssim \text{Ra}^{\frac 12}+\|\kappa\|_{\infty}$, which agrees with the predicted Spiegel-Kraichnan scaling when $\kappa=0$. This bound is obtained via local regularity estimates in a small strip at the boundary. When $h\in W^{3,\infty}$ and the functions $\kappa$ and $\alpha$ are sufficiently small in $L^{\infty}$, we prove upper bounds using the background field method, which interpolate between $\text{Ra}^{\frac 12}$ and $\text{Ra}^{\frac{5}{12}}$ with non-trivial dependence on $\alpha$ and $\kappa$. These bounds agree with the result in [DNN22] for flat boundaries and constant friction coefficient. Furthermore, in the regime $\text{Pr}\geq \text{Ra}^{\frac 57}$, we improve the $\text{Ra}^{\frac 12}$-upper bound, showing $\text{Nu}\lesssim_{\alpha,\kappa}\text{Ra}^{\frac 37}$, where $\lesssim_{\alpha,\kappa}$ hides an additional dependency of the implicit constant on $\alpha$ and $\kappa$.Comment: 36 pages, 5 figure

Rayleigh-Bénard convection: bounds on the Nusselt number

We examine the Rayleigh–Bénard convection as modelled by the Boussinesq equation. Our aim is at deriving bounds for the heat enhancement factor in the vertical direction, the Nusselt number, which reproduce physical scalings. In the first part of the dissertation, we examine the the simpler model when the acceleration of the fluid is neglected (Pr=∞) and prove the non-optimality of the temperature background field method by showing a lower bound for the Nusselt number associated to it. In the second part we consider the full model (Pr<∞) and we prove a new upper bound which improve the existing ones (for large Pr numbers) and catches a transition at Pr~Ra^(1/3)

Large time behaviour of the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions

The goal of this paper is to study the large-time bahaviour of a buoyancy driven fluid without thermal diffusion and Navier-slip boundary conditions in a bounded domain with Lipschitz-continuous second derivatives. After showing global well-posedness and regularity of classical solutions, we study their large-time asymptotics. Specifically we prove that, in suitable norms, the solutions converge to the hydrostatic equilibrium. Moreover, we prove linear stability for the hydrostatic equilibrium when the temperature is an increasing affine function of the height, i.e. the temperature is vertically stably stratified. This work is inspired by results in [Doe+18] for free-slip boundary conditions.Comment: 37 page

Limitations of the background field method applied to Rayleigh-Bénard convection

We consider Rayleigh-Bénard convection as modeled by the Boussinesq equations, in case of infinite Prandtl number and with no-slip boundary condition. There is a broad interest in bounds of the upwards heat flux, as given by the Nusselt number $Nu$, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime $Ra \gg 1$. In several works, the background field method applied to the temperature field has been used to provide upper bounds on $Nu$ in terms of $Ra$. In these applications, the background field method comes in form of a variational problem where one optimizes a stratified temperature profile subject to a certain stability condition; the method is believed to capture marginal stability of the boundary layer. The best available upper bound via this method is $Nu \lesssim Ra^{1/3}(\ln Ra)^{1/15}$. it proceeds via the construction of a stable temperature background profile that increases logarithmically in the bulk. In this paper, we show that the background temperature field method cannot provide a tighter upper bound in terms of the power of the logarithm. However, by another method one does obtain the tighter upper bound $Nu \lesssim Ra^{1/3}(\ln \ln Ra)^{1/3}$, so that the result of this paper implies that the background temperature field method is unphysical in the sense that it cannot provide the optimal bound

Prescription appropriateness of anti-diabetes drugs in elderly patients hospitalized in a clinical setting: evidence from the REPOSI Register

Diabetes is an increasing global health burden with the highest prevalence (24.0%) observed in elderly people. Older diabetic adults have a greater risk of hospitalization and several geriatric syndromes than older nondiabetic adults. For these conditions, special care is required in prescribing therapies including anti- diabetes drugs. Aim of this study was to evaluate the appropriateness and the adherence to safety recommendations in the prescriptions of glucose-lowering drugs in hospitalized elderly patients with diabetes. Data for this cross-sectional study were obtained from the REgistro POliterapie-Società Italiana Medicina Interna (REPOSI) that collected clinical information on patients aged ≥ 65 years acutely admitted to Italian internal medicine and geriatric non-intensive care units (ICU) from 2010 up to 2019. Prescription appropriateness was assessed according to the 2019 AGS Beers Criteria and anti-diabetes drug data sheets.Among 5349 patients, 1624 (30.3%) had diagnosis of type 2 diabetes. At admission, 37.7% of diabetic patients received treatment with metformin, 37.3% insulin therapy, 16.4% sulfonylureas, and 11.4% glinides. Surprisingly, only 3.1% of diabetic patients were treated with new classes of anti- diabetes drugs. According to prescription criteria, at admission 15.4% of patients treated with metformin and 2.6% with sulfonylureas received inappropriately these treatments. At discharge, the inappropriateness of metformin therapy decreased (10.2%, P &lt; 0.0001). According to Beers criteria, the inappropriate prescriptions of sulfonylureas raised to 29% both at admission and at discharge. This study shows a poor adherence to current guidelines on diabetes management in hospitalized elderly people with a high prevalence of inappropriate use of sulfonylureas according to the Beers criteria