A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

In this paper, we first prove the local existence of strong solutions to the 3D Boussinesq equations in a bounded domain with Navier boundary conditions. Then we show the global stability of strong large solutions under a suitable integral condition.

On the global stability of large solutions for the Boussinesq equations with Navier boundary conditions

The main objective of this paper is concerned with the convergence of the boundary parameter for the large-scale, three-dimensional, viscous primitive equations. Such equations are often used for weather prediction and climate change. Under the assumptions of some boundary conditions, we obtain a prior bounds for the solutions of the equations by using the differential inequality technology and method of the energy estimates, and the convergence of the equations on the boundary parameter is proved.

/pdf/convergence-of-the-boundary-parameter-for-the-three-1vz2w3w1.pdf

Convergence of the Boundary Parameter for the Three-Dimensional Viscous Primitive Equations of Large-Scale

Autonomous vehicles must be able to understand the surrounding traffic flows and predict the future traffic conditions for planning a safe maneuver. During prediction, the action of autonomous vehicles should be considered, as it influences the interaction between vehicles sharing the same traffic scene and thus influences the future traffic flow. From this perspective, not only should the prediction be considered for planning, but also the action of autonomous vehicles generated by planning should be considered for traffic scene prediction. Therefore, prediction and planning must work interactively at every time step, considering results of each other. In this paper, we present a novel learning-based framework that simultaneously forecasts a nearby traffic scene and plans a maneuver of autonomous vehicle at every time step. Through experiments, we demonstrated that the proposed method exhibits better planning performance than baselines in complex traffic conditions involving various surrounding vehicles.

STFP - Simultaneous Traffic Scene Forecasting and Planning for Autonomous Driving.

In this paper, we consider the solvability, regularity and vanishing viscosity limit of the 3D viscous Boussinesq equations with a Navier-slip boundary condition. We also obtain the rate of convergence of the solution of viscous Boussinesq equations to the corresponding ideal Boussinesq equations.

/pdf/on-the-boussinesq-system-local-well-posedness-of-the-strong-5ee3p0lrod.pdf

On the Boussinesq system: local well-posedness of the strong solution and inviscid limits

The aim of this paper is to show some applications of Sobolev inequalities in partial differential equations. With the aid of some well-known inequalities, we derive the existence of global solution for the quasilinear parabolic equations. When the blow-up occurs, we derive the lower bound of the blow-up solution.

/pdf/the-applications-of-sobolev-inequalities-in-proving-the-28xzxwjrbc.pdf

The applications of Sobolev inequalities in proving the existence of solution of the quasilinear parabolic equation

In this paper, we consider the a posteriori error estimates of the mixed finite element method for the optimal control problems governed by fourth order hyperbolic equations. The state is discretized by the order k Raviart–Thomas mixed elements and control is discretized by piecewise polynomials of degree k. We adopt the mixed elliptic reconstruction to derive the a posteriori error estimates for both the state and the control approximations.

/pdf/a-posteriori-error-estimates-for-fourth-order-hyperbolic-2u80f43ai3.pdf

A posteriori error estimates for fourth order hyperbolic control problems by mixed finite element methods

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On the Boussinesq system: local well-posedness of the strong solution and inviscid limits

The applications of Sobolev inequalities in proving the existence of solution of the quasilinear parabolic equation

A posteriori error estimates for fourth order hyperbolic control problems by mixed finite element methods