Regularity Analysis Of Solutions Of Hamilton-Jacobi Equations

Hamilton-Jacobi equation is a partial differential equation used to find regular solutions in analytical mechanics.It is a first-order nonlinear partial differential equation plays a very important role in fluid mechanics,optics.It is also one of the important mathematics in optimal control theory and Hamilton dynamics.Firstly,in this paper,we introduce the development of partial differential equation and the research status of Hamilton-Jacobi equation.Secondly,considering the Cauchy problem of the one dimensional Hamilton-Jacobi equation,through several lemmas,the smoothness of the solution and the limitation of the number of shock waves are analyzed when the initial value belongs to a certain category set,and the local finiteness of the solution is obtained.Finally,in the case of the first derivative properties are studied.That is,in any smooth region,the new upper bound of the higher order derivative is obtained,in this case,we can analyze the higher order piecewise smoothness of the solution.