The research of river basin ecological compensation based on water emissions trading mechanism.

By integrating the successful case of the European Union emissions trading system, this study proposes a water emissions trading system, a novel method of reducing water pollution. Assuming that upstream governments allocate initial quotas to upstream businesses as the compensation standard, this approach defines the foundational principles of market trading mechanisms and establishes a robust watershed ecological compensation model to address challenges in water pollution prevention. To be specific, the government establishes a reasonable initial quota for upstream enterprises, which can be used to limit the emissions of upstream pollution. When enterprises exceed their allocated emissions quota, they face financial penalties. Conversely, these emissions rights can be transformed into profitable assets by participating in the trading market as a form of ecological compensation. Numerical simulations demonstrate that various pollutant emissions from upstream businesses will have various effects on the profits of other businesses. Businesses in the upstream region received reimbursement from the assigned emission rights through the market mechanism, demonstrating that ecological compensation for the watershed can be achieved through the market mechanism. This novel market trading system aims at controlling emissions management from the perspectives of individual enterprises and ultimately optimizing the aquatic environment.

A posteriori error estimates of spectral method for nonlinear parabolic optimal control problem.

In this paper, we investigate the spectral approximation of optimal control problem governed by nonlinear parabolic equations. A spectral approximation scheme for the nonlinear parabolic optimal control problem is presented. We construct a fully discrete spectral approximation scheme by using the backward Euler scheme in time. Moreover, by using an orthogonal projection operator, we obtain L2(H1)-L2(L2) a posteriori error estimates of the approximation solutions for both the state and the control. Finally, by introducing two auxiliary equations, we also obtain L2(L2)-L2(L2) a posteriori error estimates of the approximation solutions for both the state and the control.