Stochastic pseudomonotone parabolic obstacle problem: well-posedness &\& Lewy-Stampacchia's inequalities

We consider obstacle problems for nonlinear stochastic evolution equations. More precisely, the leading operator in our equation is a nonlinear, second order pseudomonotone operator of Leray-Lions type. The multiplicative noise term is given by a stochastic integral with respect to a Q-Wiener process. We show well-posedness of the associated initial value problem for random initial data on a bounded domain with a homogeneous Dirichlet boundary condition. Therefore we consider a singular perturbation of our problem by a higher order operator. Through the a priori estimates for the approximate solutions of the singular perturbation, only weak convergence is obtained. This convergence is not compatible with the nonlinearities in the equation. Therefore we use the theorems of Prokhorov and Skorokhod to establish existence of martingale solutions. Then, path-wise uniqueness follows from a L1-contraction principle and we may apply the method of Gy\"ongy-Krylov to obtain stochastically strong solutions. These well-posedness results serve as a basis for the study of variational inequalities and Lewy-Stampacchia's inequalities for our problem.Comment: 36 page

The Impacts Of Neighbourhood Traffic Management

A major traffic-related problem faced by residents is speeding, which not only causes safety concerns, but also noise issues. Traffic calming is a much favoured traffic management tool employed by road controlling authorities to primarily reduce vehicle speed, hence improve community liveability. This research aimed to investigate the impacts of traffic calming on speed, safety and traffic noise. The objectives included developing models for the prediction of speed and noise on traffic-calmed streets, and providing guidance for good design practices. Speeds of individual vehicles as they approached and crossed traffic calming devices were observed in order to identify the behaviour of individual drivers. Results indicated that the speed hump and the raised angled slow point produced the largest speed reductions and least variation in speeds, while mid-block narrowings had no significant speed changes. Inter-device speed was found to be mainly controlled by the separation between devices. 85th percentile speeds at distances from calming devices were 40 – 45 km/h for vertical deflections and 45 – 55 km/h for horizontal deflections. Speeds on approach to speed humps were found to be influenced by the distance available on the approaches, while operating speed at the speed humps were partly influenced by the hump width relative to the road width. There was evidence of safety benefits of traffic calming overall, despite mid-block crashes increasing post-calming. However, there was no association between the traffic calming and the crashes, which appeared to probably be due to other factors, human factors in particular. Noise levels produced by light vehicles across speed humps were in fact lower than on a flat section of road, given their respective mean speeds. At a reference speed of 25 km/h, noise levels produced over the 100 mm hump were 3.6 dBA higher than those produced by the 75 mm hump