Well-posedness of a nonlinear second-order anisotropic reaction-diffusion problem with nonlinear and inhomogeneous dynamic boundary conditions

The paper is concerned with a qualitative analysis for a nonlinear second-order boundary value problem, endowed with nonlinear and inhomogeneous dynamic boundary conditions, extending the types of bounday conditions already studied. Under certain assumptions on the input data: $f_{_1}(t,x)$, $w(t,x)$ and $u_0(x)$, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $W^{1,2}_p(Q)$. This extends previous works concerned with nonlinear dynamic boundary conditions, allowing to the present mathematical model to better approximate the real physical phenomena (the anisotropy effects, phase change in $\Omega$ and at the boundary $\partial\Omega$, etc.).

Improving predictability of high-ozone episodes through dynamic boundary conditions, emission refresh and chemical data assimilation during the Long Island Sound Tropospheric Ozone Study (LISTOS) field campaign

Although air quality in the United States has improved remarkably in the past decades, ground-level ozone (O3) often rises in exceedance of the national ambient air quality standard in nonattainment areas, including the Long Island Sound (LIS) and its surrounding areas. Accurate prediction of high-ozone episodes is needed to assist government agencies and the public in mitigating harmful effects of air pollution. In this study, we have developed a suite of potential forecast improvements, including dynamic boundary conditions, rapid emission refresh and chemical data assimilation, in a 3 km resolution Community Multiscale Air Quality (CMAQ) modeling system. The purpose is to evaluate and assess the effectiveness of these forecasting techniques, individually or in combination, in improving forecast guidance for two major air pollutants: surface O3 and nitrogen dioxide (NO2). Experiments were conducted for a high-O3 episode (28–29 August 2018) during the Long Island Sound Tropospheric Ozone Study (LISTOS) field campaign, which provides abundant observations for evaluating model performance. The results show that these forecast system updates are useful in enhancing the capability of this 3 km forecasting model with varying effectiveness for different pollutants. For O3 prediction, the most significant improvement comes from the dynamic boundary conditions derived from the NOAA operational forecast system, National Air Quality Forecast Capability (NAQFC), which increases the correlation coefficient (R) from 0.81 to 0.93 and reduces the root mean square error (RMSE) from 14.97 to 8.22 ppbv, compared to that with the static boundary conditions (BCs). The NO2 from all high-resolution simulations outperforms that from the operational 12 km NAQFC simulation, regardless of the BCs used, highlighting the importance of spatially resolved emission and meteorology inputs for the prediction of short-lived pollutants. The effectiveness of improved initial concentrations through optimal interpolation (OI) is shown to be high in urban areas with high emission density. The influence of OI adjustment, however, is maintained for a longer period in rural areas, where emissions and chemical transformation make a smaller contribution to the O3 budget than that in high-emission areas. Following the assessment of individual updates, the forecasting system is configured with dynamic boundary conditions, optimal interpolation of initial concentrations and emission adjustment, to simulate a high-ozone episode during the 2018 LISTOS field campaign. The newly developed forecasting system significantly reduces the bias of surface NO2 prediction. When compared with the NASA Langley GeoCAPE Airborne Simulator (GCAS) vertical column density (VCD), this system is able to reproduce the NO2 VCD with a higher correlation (0.74), lower normalized mean bias (40 %) and normalized mean error (61 %) than NAQFC (0.57, 45 % and 76 %, respectively). The 3 km system captures magnitude and timing of surface O3 peaks and valleys better. In comparison with lidar, O3 profile variability of the vertical O3 is captured better by the new system (correlation coefficient of 0.71) than by NAQFC (correlation coefficient of 0.54). Although the experiments are limited to one pollution episode over the Long Island Sound, this study demonstrates feasible approaches to improve the predictability of high-O3 episodes in contemporary urban environments.

Linear evolution equations on the half-line with dynamic boundary conditions

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.