Search

A simple qualitative model of dynamic combustion $$\left\{ \begin{gathered} \frac{{\partial (u + qz)}}{{\partial t}} + \frac{{\partial f(u)}}{{\partial x}} = \varepsilon \frac{{\partial ^2 u}}{{\partial x^2 }}, \hfill \\ \frac{{\partial z}}{{\partial t}} = - F(u,z) \hfill \\ \end{gathered} \right.$$ is considered, where constantq>0 represents the binding energy. The existence and uniqueness of classical solution for the initial value problem and for the first boundary value problem of system (1) are proved and then it is proved that ase→0 classical solutions of the initial value problem of system (1) converge to a pair of limit functions, which is an admissible solution of the initial value problem of system (1) withe=0.

Publisher Summary If fluid flow is accompanied by chemical reaction, then very complicated wave motion phenomena occur. It is an interesting problem as to how a mathematical model can be applied to these phenomena, and one may investigate them by the theory of partial differential equations. Some authors have investigated the travelling wave solutions and some Riemann initial value problems for these problems, but up to now, these investigations are not deep as that for the shock waves. The chapter presents some hypotheses and two definitions of weak solutions: Problem P and Problem Q and discusses the strong discontinuous curve and the Riemann problem.