Structure of shocks in Burgers turbulence with L\'evy noise initial data

We study the structure of the shocks for the inviscid Burgers equation in dimension 1 when the initial velocity is given by L\'evy noise, or equivalently when the initial potential is a two-sided L\'evy process $\psi_0$. When $\psi_0$ is abrupt in the sense of Vigon or has bounded variation with $\limsup_{|h| \downarrow 0} h^{-2} \psi_0(h) = \infty$, we prove that the set of points with zero velocity is regenerative, and that in the latter case this set is equal to the set of Lagrangian regular points, which is non-empty. When $\psi_0$ is abrupt we show that the shock structure is discrete. When $\psi_0$ is eroded we show that there are no rarefaction intervals.Comment: 22 page

Milrinone acts as a vasodilator but not an inotrope in children after cardiac surgery - Insights from wave intensity analysis

Objectives: Milrinone is an inodilator widely used in the postoperative management of children undergoing cardiac surgery. The literature supporting its inotropic effect is sparse. We sought to study the effect of milrinone on the vasculature and its effects on the ventricular function using wave intensity analysis. We also intended to evaluate the feasibility of using wave intensity analysis by the bedside. Design: prospective single-center observational study. Setting: PICU of a tertiary children's hospital. Patients: Children (< 18 yr) admitted to PICU following cardiac surgery who required to be commenced on a milrinone infusion. Interventions: Echocardiography and Doppler ultrasound assessments for wave intensity analysis were performed prior to commencing milrinone and 4-6 hours after milrinone infusion. Measurements and Main Results: Wave intensity analysis was successfully performed and analyzed in 15 of 16 patients (94%). We identified three waves - a forward compression wave, backward compression wave, and forward decompression wave. The waves were described with their cumulative intensity and wave-related pressure change. There was a 26% reduction in backward compression wave cumulative intensity following the introduction of milrinone. Other variables (backward compression wave cumulative intensity/forward compression wave cumulative intensity ratio, backward compression wave wave-related pressure change, backward compression wave wave-related pressure change/forward compression wave wave-related pressure change ratio) consistent with vasodilation also decreased after milrinone. It also decreased the vascular wavespeed by 7.1% and increased the distensibility of the vessels by 14.6%. However, it did not increase forward compression wave cumulative intensity, a variable indicating the systolic force generated by the ventricle. Forward decompression wave cumulative intensity indicating ventricular early diastolic relaxation also did not change. Conclusions: In a cohort of children recovering in PICU after having undergone cardiac surgery, we found that milrinone acted as a vasodilator but did not demonstrate an improvement in the contractility or an improved relaxation of the left ventricle as assessed by wave intensity analysis. We were able to demonstrate the feasibility and utility of wave intensity analysis to further understand ventriculo-vascular interactions in an intensive care setting

Gestational age and risk of mortality in term-born critically ill neonates admitted to PICUs in Australia and New Zealand*

Objectives: Gestational age at birth is declining, probably because more deliveries are being induced. Gestational age is an important modifiable risk factor for neonatal mortality and morbidity. We aimed to investigate the association between gestational age and mortality in hospital for term-born neonates (>= 37 wk') admitted to PICUs in Australia and New Zealand. Design: Observational multicenter cohort study. Setting: PICUs in Australia and New Zealand. Patients: Term-born neonates (>= 37 wk) admitted to PICUs. Interventions: None Measurements and Main Results: We studied 5,073 infants born with a gestational age greater than or equal to 37 weeks and were less than 28 days old when admitted to a PICU in Australia or New Zealand between 2007 and 2016. The association between gestational age and mortality was estimated using a multivariable logistic regression model, adjusting for age, sex, indigenous status, Pediatric Index of Mortality version 2, and site. The median gestational age was 39.1 weeks (interquartile range, 38.2-40 wk) and mortality in hospital was 6.6%. Risk of mortality declined log-linearly with gestational age. The adjusted analysis showed a 20% (95% CI, 11-28%) relative reduction in mortality for each extra week of gestation beyond 37 weeks. The effect of gestation was stronger among those who received extracorporeal life support: each extra week of gestation was associated with a 44% (95% CI, 25-57%) relative reduction in mortality. Longer gestation was also associated with reduced length of stay in hospital: each week increase in gestation, the average length of stay decreased by 4% (95% CI, 2-6%). Conclusions: Among neonates born at "term" who are admitted to a PICU, increasing gestational age at birth is associated with a substantial reduction in the risk of dying in hospital. The maturational influence on outcome was more strongly noted in the sickest neonates, such as those requiring extracorporeal life support. This information is important in view of the increasing proportion of planned births in both high- and low-/middle-income countries