Energy conservation for the Euler equations on T2×R+\mathbb{T}^2\times \mathbb{R}_+ for weak solutions defined without reference to the pressure

We study weak solutions of the incompressible Euler equations on $\mathbb{T}^2\times \mathbb{R}_+$; we use test functions that are divergence free and have zero normal component, thereby obtaining a definition that does not involve the pressure. We prove energy conservation under the assumptions that $u\in L^3(0,T;L^3(\mathbb{T}^2\times \mathbb{R}_+))$, $$\lim_{|y|\to 0}\frac{1}{|y|}\int^T_0\int_{\mathbb{T}^2}\int^\infty_{x_3>|y|} |u(x+y)-u(x)|^3\mathrm{d} x\, \mathrm{d} t=0,$$ and an additional continuity condition near the boundary: for some $\delta>0$ we require $u\in L^3(0,T;C^0(\mathbb{T}^2\times [0,\delta])))$. We note that all our conditions are satisfied whenever $u(x,t)\in C^\alpha$, for some $\alpha>1/3$, with H\"older constant $C(x,t)\in L^3(\mathbb{T}^2\times\mathbb{R}^+\times(0,T))$.Comment: 21 page

Risk of COVID-19 after natural infection or vaccinationResearch in context

Background: While vaccines have established utility against COVID-19, phase 3 efficacy studies have generally not comprehensively evaluated protection provided by previous infection or hybrid immunity (previous infection plus vaccination). Individual patient data from US government-supported harmonized vaccine trials provide an unprecedented sample population to address this issue. We characterized the protective efficacy of previous SARS-CoV-2 infection and hybrid immunity against COVID-19 early in the pandemic over three-to six-month follow-up and compared with vaccine-associated protection. Methods: In this post-hoc cross-protocol analysis of the Moderna, AstraZeneca, Janssen, and Novavax COVID-19 vaccine clinical trials, we allocated participants into four groups based on previous-infection status at enrolment and treatment: no previous infection/placebo; previous infection/placebo; no previous infection/vaccine; and previous infection/vaccine. The main outcome was RT-PCR-confirmed COVID-19 >7–15 days (per original protocols) after final study injection. We calculated crude and adjusted efficacy measures. Findings: Previous infection/placebo participants had a 92% decreased risk of future COVID-19 compared to no previous infection/placebo participants (overall hazard ratio [HR] ratio: 0.08; 95% CI: 0.05–0.13). Among single-dose Janssen participants, hybrid immunity conferred greater protection than vaccine alone (HR: 0.03; 95% CI: 0.01–0.10). Too few infections were observed to draw statistical inferences comparing hybrid immunity to vaccine alone for other trials. Vaccination, previous infection, and hybrid immunity all provided near-complete protection against severe disease. Interpretation: Previous infection, any hybrid immunity, and two-dose vaccination all provided substantial protection against symptomatic and severe COVID-19 through the early Delta period. Thus, as a surrogate for natural infection, vaccination remains the safest approach to protection. Funding: National Institutes of Health

Risk of COVID-19 after natural infection or vaccinationResearch in context

Summary: Background: While vaccines have established utility against COVID-19, phase 3 efficacy studies have generally not comprehensively evaluated protection provided by previous infection or hybrid immunity (previous infection plus vaccination). Individual patient data from US government-supported harmonized vaccine trials provide an unprecedented sample population to address this issue. We characterized the protective efficacy of previous SARS-CoV-2 infection and hybrid immunity against COVID-19 early in the pandemic over three-to six-month follow-up and compared with vaccine-associated protection. Methods: In this post-hoc cross-protocol analysis of the Moderna, AstraZeneca, Janssen, and Novavax COVID-19 vaccine clinical trials, we allocated participants into four groups based on previous-infection status at enrolment and treatment: no previous infection/placebo; previous infection/placebo; no previous infection/vaccine; and previous infection/vaccine. The main outcome was RT-PCR-confirmed COVID-19 >7–15 days (per original protocols) after final study injection. We calculated crude and adjusted efficacy measures. Findings: Previous infection/placebo participants had a 92% decreased risk of future COVID-19 compared to no previous infection/placebo participants (overall hazard ratio [HR] ratio: 0.08; 95% CI: 0.05–0.13). Among single-dose Janssen participants, hybrid immunity conferred greater protection than vaccine alone (HR: 0.03; 95% CI: 0.01–0.10). Too few infections were observed to draw statistical inferences comparing hybrid immunity to vaccine alone for other trials. Vaccination, previous infection, and hybrid immunity all provided near-complete protection against severe disease. Interpretation: Previous infection, any hybrid immunity, and two-dose vaccination all provided substantial protection against symptomatic and severe COVID-19 through the early Delta period. Thus, as a surrogate for natural infection, vaccination remains the safest approach to protection. Funding: National Institutes of Health