Concentration in vanishing adiabatic exponent limit of solutions to the Aw–Rascle traffic model

In this paper, we study the phenomenon of concentration and the formation of delta shock wave in vanishing adiabatic exponent limit of Riemann solutions to the Aw–Rascle traffic model. It is proved that as the adiabatic exponent vanishes, the limit of solutions tends to a special delta-shock rather than the classical one to the zero pressure gas dynamics. In order to further study this problem, we consider a perturbed Aw–Rascle model and proceed to investigate the limits of solutions. We rigorously proved that, as the adiabatic exponent tends to one, any Riemann solution containing two shock waves tends to a delta-shock to the zero pressure gas dynamics in the distribution sense. Moreover, some representative numerical simulations are exhibited to confirm the theoretical analysis.

Delta Shock Index During Emergency Department Stay Is Associated With in Hospital Mortality in Critically Ill Patients

Background: Delta shock index (SI; i.e., change in SI over time) has been shown to predict mortality and need for surgical intervention among trauma patients at the emergency department (ED). However, the usefulness of delta SI for prognosis assessment in non-traumatic critically ill patients at the ED remains unknown. The aim of this study was to analyze the association between delta SI during ED management and in-hospital outcomes in patients admitted to the intensive care unit (ICU).Method: This was a retrospective study conducted in two tertiary medical centers in Taiwan from January 1, 2016, to December 31, 2017. All adult non-traumatic patients who visited the ED and who were subsequently admitted to the ICU were included. We calculated delta SI by subtracting SI at ICU admission from SI at ED triage, and we analyzed its association with in-hospital outcomes. SI was defined as the ratio of heart rate to systolic blood pressure (SBP). The primary outcome was in-hospital mortality, and the secondary outcomes were hospital length of stay (HLOS) and early mortality. Early mortality was defined as mortality within 48 h of ICU admission.Result: During the study period, 11,268 patients met the criteria and were included. Their mean age was 64.5 ± 15.9 years old. Overall, 5,830 (51.6%) patients had positive delta SI. Factors associated with a positive delta SI were multiple comorbidities (51.2% vs. 46.3%, p < 0.001) and high Simplified Acute Physiology Score [39 (29–51) vs. 37 (28–47), p < 0.001). Patients with positive delta SI were more likely to have tachycardia, hypotension, and higher SI at ICU admission. In the regression analysis, high delta SI was associated with in-hospital mortality [aOR (95% CI): 1.21 (1.03–1.42)] and early mortality [aOR (95% CI): 1.26 (1.07–1.48)], but not for HLOS [difference (95% CI): 0.34 (−0.48 to 1.17)]. In the subgroup analysis, high delta SI had higher odds ratios for both mortality and early mortality in elderly [aOR (95% CI): 1.59 (1.11–2.29)] and septic patients [aOR (95% CI): 1.54 (1.13–2.11)]. It also showed a higher odds ratio for early mortality in patients with triage SBP <100 mmHg [aOR (95% CI): 2.14 (1.21–3.77)] and patients with triage SI ≥ 0.9 [aOR (95% CI): 1.62 (1.01–2.60)].Conclusion: High delta SI during ED stay is correlated with in-hospital mortality and early mortality in patients admitted to the ICU via ED. Prompt resuscitation should be performed, especially for those with old age, sepsis, triage SBP <100 mmHg, or triage SI ≥ 0.9.

Four-Quadrant Riemann Problem for a 2×2 System II

In previous work, we considered a four-quadrant Riemann problem for a 2×2 hyperbolic system in which delta shock appears at the initial discontinuity without assuming that each jump of the initial data projects exactly one plane elementary wave. In this paper, we consider the case that does not involve a delta shock at the initial discontinuity. We classified 18 topologically distinct solutions and constructed analytic and numerical solutions for each case. The constructed analytic solutions show the rich structure of wave interactions in the Riemann problem, which coincide with the computed numerical solutions.

Riemann problem and limits of solutions to the isentropic relativistic Euler equations for isothermal gas with flux approximation

We are concerned with the vanishing flux-approximation limits of solutions to the isentropic relativistic Euler equations governing isothermal perfect fluid flows. The Riemann problem with a two-parameter flux approximation including pressure term is first solved. Then, we study the limits of solutions when the pressure and two-parameter flux approximation vanish, respectively. It is shown that, any two-shock-wave Riemann solution converges to a delta-shock solution of the pressureless relativistic Euler equations, and the intermediate density between these two shocks tends to a weighted δ-measure that forms a delta shock wave. By contract, any two-rarefaction-wave solution tends to a two-contact-discontinuity solution of the pressureless relativistic Euler equations, and the intermediate state in between tends to a vacuum state.