The Formation Of The Shock Wave About Inhomogeneous Equation

In this paper, we have a good research on the formation of shock waves. For such research,Courant gives a brief introduction in the paper [3]，and Chen elaborate the formation and constructionof shock for hyperbolic conservation laws in the paper [1]. Further, the formation and construction ofshock for P-Systems are elaborated by Chen and Dong in the paper [2], in2000. Based on theseresearch, we give the detailed introduction on the formation of shock for inhomogeneous hyperbolicconservation laws equation.Firstly, in this paper, the envelope starting point problem in Chen’s article is described. shockformation comes from the envelope of characteristics. In many existing papers, people often assumethe formation time of envelope is zero. Is this feasible? In this paper we will prove the feasibility. Wefind between the initial value on characteristic line and the initial value of envelope there existone-to-one mapping when t0. Therefore, it can be converted to the condition that t0bytransformation of coordinates.In our work, we mainly discuss the formation of shock for inhomogeneous hyperbolic equation.Specifically, firstly, the smoothing solution begins to blow up when t t*， and a shock isproduced from (t*,x*). then, We will construct a solution u (t,x)，there exist smooth solutionwhen t t*，however it satisfied Hugoniot condition on the shock when t t*.Finally, theconstruction process of solution is explained and the existence of solution is proved by the PeanoTheorem. During the proof, the method of taylor expansion for one-variable functions is expanded tothe version for multi-variable functions.