Dispersive estimates for linearized water wave type equations in Rd\mathbb R^d

We derive a $L^1_x (\mathbb R^d)-L^{\infty}_x ( \mathbb R^d)$ decay estimate of order $\mathcal O \left( t^{-d/2}\right)$ for the linear propagators $$\exp \left( {\pm it \sqrt{ |D|\left(1+ \beta |D|^2\right) \tanh |D | } }\right), \qquad \beta \in \{0, 1\}. \quad D = -i\nabla,$$ with a loss of $3d/4$ or $d/4$-derivatives in the case $\beta=0$ or $\beta=1$, respectively. These linear propagators are known to be associated with the linearized water wave equations, where the parameter $\beta$ measures surface tension effects. As an application we prove low regularity well-posedness for a Whitham-Boussinesq type system in $\mathbb R^d$, $d\ge 2$. This generalizes a recent result by Dinvay, Selberg and the third author where they proved low regularity well-posedness in $\mathbb R$ and $\mathbb R^2$.Comment: 18 page

Improved Lower Bound for the Radius of Analyticity of Solutions to the fifth order KdV-BBM model

We show that the uniform radius of spatial analyticity $\sigma(t)$ of solutions at time $t$ to the fifth order KdV-BBM equation cannot decay faster than $1/ \sqrt{t}$ for large $t$, given initial data that is analytic with fixed radius $\sigma_0$. This improves a recent result by Belayneh, Tegegn and the third author, where they obtained a $1/t$ decay of $\sigma(t)$ for large time $t$.Comment: 1

Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE right-hand sides belong to the Sobolev (Bessel potential) space Hs−2(Ω ) or H˜s−2(Ω ) , 12<s<32, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators.EPSR

Prioritizing Health Care Strategies to Reduce Childhood Mortality

IMPORTANCE: Although child mortality trends have decreased worldwide, deaths among children younger than 5 years of age remain high and disproportionately circumscribed to sub-Saharan Africa and Southern Asia. Tailored and innovative approaches are needed to increase access, coverage, and quality of child health care services to reduce mortality, but an understanding of health system deficiencies that may have the greatest impact on mortality among children younger than 5 years is lacking. OBJECTIVE: To investigate which health care and public health improvements could have prevented the most stillbirths and deaths in children younger than 5 years using data from the Child Health and Mortality Prevention Surveillance (CHAMPS) network. DESIGN, SETTING, AND PARTICIPANTS: This cross-sectional study used longitudinal, population-based, and mortality surveillance data collected by CHAMPS to understand preventable causes of death. Overall, 3390 eligible deaths across all 7 CHAMPS sites (Bangladesh, Ethiopia, Kenya, Mali, Mozambique, Sierra Leone, and South Africa) between December 9, 2016, and December 31, 2021 (1190 stillbirths, 1340 neonatal deaths, 860 infant and child deaths), were included. Deaths were investigated using minimally invasive tissue sampling (MITS), a postmortem approach using biopsy needles for sampling key organs and fluids. MAIN OUTCOMES AND MEASURES: For each death, an expert multidisciplinary panel reviewed case data to determine the plausible pathway and causes of death. If the death was deemed preventable, the panel identified which of 10 predetermined health system gaps could have prevented the death. The health system improvements that could have prevented the most deaths were evaluated for each age group: stillbirths, neonatal deaths (aged <28 days), and infant and child deaths (aged 1 month to <5 years). RESULTS: Of 3390 deaths, 1505 (44.4%) were female and 1880 (55.5%) were male; sex was not recorded for 5 deaths. Of all deaths, 3045 (89.8%) occurred in a healthcare facility and 344 (11.9%) in the community. Overall, 2607 (76.9%) were deemed potentially preventable: 883 of 1190 stillbirths (74.2%), 1010 of 1340 neonatal deaths (75.4%), and 714 of 860 infant and child deaths (83.0%). Recommended measures to prevent deaths were improvements in antenatal and obstetric care (recommended for 588 of 1190 stillbirths [49.4%], 496 of 1340 neonatal deaths [37.0%]), clinical management and quality of care (stillbirths, 280 [23.5%]; neonates, 498 [37.2%]; infants and children, 393 of 860 [45.7%]), health-seeking behavior (infants and children, 237 [27.6%]), and health education (infants and children, 262 [30.5%]). CONCLUSIONS AND RELEVANCE: In this cross-sectional study, interventions prioritizing antenatal, intrapartum, and postnatal care could have prevented the most deaths among children younger than 5 years because 75% of deaths among children younger than 5 were stillbirths and neonatal deaths. Measures to reduce mortality in this population should prioritize improving existing systems, such as better access to antenatal care, implementation of standardized clinical protocols, and public education campaigns

ON THE PERSISTENCE OF SPATIAL ANALYTICITY FOR THE BEAM EQUATION

Persistence of spatial analyticity is studied for solution of the beam equation utt + (m + Δ2) u + |u| p−1u = 0 on Rn × R. In particular, for a class of analytic initial data with a uniform radius of analyticity σ0, we obtain an asymptotic lower bound σ(t) c/√t on the uniform radius of analyticity σ(t) of solution u(·,t), as t → ∞. © 2022 Elsevier Inc. All rights reserve

Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Mixed BVP in 2D

Humboldt Foundation, ISP (Sweden), EPSRC, UK