Qualitative Analysis Of The Solution To Hamilton-Jacobi Equation

Hamilton-Jacobi equation is one of the most important mathematical models in atmospheric dynamic,fluid mechanics,ocean wave mechanics and optics.It plays an important role in Hamilton dynamics,the optimal control theory and the theory differential game theory.Firstly,in this paper,we introduce Hamilton-Jacobi equation and its research tendency,discusses the points where Hamilton-Jacobi equation differs from mechanical dynamics of different formulas,thus deriving the different forms of Hamilton-Jacobi equations in different coordinate systems.Secondly,We are concerned with the viscous solution for the Cauchy problems of high-dimensional Hamilton-Jacobi equation and build the relations between the characteristic line and diffeomorphism to obtain the global(local)existence of smooth solution.Thirdly,we analyzed asymptotic behavior of one dimensional Hamilton-Jacobi equations and shock wave,especially the gradual expression.Finally,we summarize consequence of the Cauchy problem for the Hamilton-Jacobi equation in this paper.