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.

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

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