Blowup of Solutions of the Hydrostatic Euler Equations

In this paper we prove that for a certain class of initial data, smooth solutions of the hydrostatic Euler equations blow up in finite time.Comment: 7 pages; added 1 reference in section 1, paraphrased lemma 2.2, but all mathematical details remain unchange

Axisymmetric flow of ideal fluid moving in a narrow domain: a study of the axisymmetric hydrostatic Euler equations

In this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain. After providing a formal derivation of the model, we will prove the well-posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition. Finally, a blowup result regarding this model will be discussed as well.Comment: 33 page

On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions

We find a new class of data for which the Prandtl boundary layer equations and the hydrostatic Euler equations are locally in time well-posed. In the case of the Prandtl equations, we assume that the initial datum $u_0$ is monotone on a number of intervals (on some strictly increasing on some strictly decreasing) and analytic on the complement and show that the local existence and uniqueness hold. The same is true for the hydrostatic Euler equations except that we assume this for the vorticity $\omega_0=\partial_y u_0$

Long time Solutions for a Burgers-Hilbert Equation via a Modified Energy Method

We consider an initial value problem for a quadratically nonlinear inviscid Burgers-Hilbert equation that models the motion of vorticity discontinuities. We use a modified energy method to prove the existence of small, smooth solutions over cubically nonlinear time-scales

Bioluminescence Imaging of Heme Oxygenase-1 Upregulation in the Gua Sha Procedure

Gua Sha is a traditional Chinese folk therapy that employs skin scraping to cause subcutaneous microvascular blood extravasation and bruises. The protocol for bioluminescent optical imaging of HO-1-luciferase transgenic mice reported in this manuscript provides a rapid in vivo assay of the upregulation of the heme oxygenase-1 (HO-1) gene expression in response to the Gua Sha procedure. HO-1 has long been known to provide cytoprotection against oxidative stress. The upregulation of HO-1, assessed by the bioluminescence output, is thought to represent an antioxidative response to circulating hemoglobin products released by Gua Sha. Gua Sha was administered by repeated strokes of a smooth spoon edge over lubricated skin on the back or other targeted body part of the transgenic mouse until petechiae (splinter hemorrhages) or ecchymosis (bruises) indicative of extravasation of blood from subcutaneous capillaries was observed. After Gua Sha, bioluminescence imaging sessions were carried out daily for several days to follow the dynamics of HO-1 expression in multiple internal organs

Medium-term Outcomes of Myocarditis and Pericarditis following BNT162b2 Vaccination among Adolescents in Hong Kong

In this study, we examined the clinical and electrophysiological outcomes of adolescents in Hong Kong who developed myocarditis or pericarditis following BNT162b2 vaccination for COVID-19, and followed-up for 60 to 180 days after their initial diagnosis. Clinical assessments included electrocardiogram (ECG) and echocardiogram at the initial admission and follow-up were compared. Treadmill testing was also performed in some cases. Between 14 June 2021 and 16 February 2022, 53 subjects were approached to participate in this follow-up study, of which 28 patients were followed up for >60 days with a median follow-up period of 100 days (range, 61-178 days) and were included in this study. On admission, 23 patients had ECG abnormalities but no high-grade atrioventricular block. Six patients had echocardiogram abnormalities, including reduced contractility, small rim pericardial effusions, and hyperechoic ventricular walls. All patients achieved complete recovery on follow-up. After discharge, 10 patients (35.7%) reported symptoms, including occasional chest pain, shortness of breath, reduced exercise tolerance, and recurrent vasovagal near-syncope. At follow-up, assessments, including ECGs, were almost all normal. Among the three patients with possible ECG abnormalities, all their echocardiograms or treadmill testings were normal. Sixteen patients (57.1%) underwent treadmill testing at a median of 117 days post-admission, which were also normal. However, at follow-up, there was a significant mean bodyweight increase of 1.81kg (95%CI 0.47-3.1 kg, p=0.01), possibly due to exercise restriction. In conclusion, most adolescents experiencing myocarditis and pericarditis following BNT162b2 vaccination achieved complete recovery. Some patients developed non-specific persistent symptoms, and bodyweight changes shall be monitored