Riemann Problem for First-Order Partial Equations Without the Convexity of a State Functions

Abstract

In this work, the exact solution of the Riemann problem for first-order nonlinear partial equation with non-convex state function in $Q_T=\{(x,t)|x\in I=\left( -\infty ,\ \ \infty \right) ,\ t\in \left[ 0,T\right) \}\subset R^2$ is found. Here $F\in C^2{(Q}_T)\$ and $\ F^{''}(u)$ change their signs, that is F(u) has convex and concave parts. In particular, the state function $F\left( u\right) =-{\cos u\ }$ on $\ \left[ \frac{\pi }{2},\frac{3\pi }{2}\right]$ and $\ \left[ \frac{\pi }{2},\frac{5\pi }{2}\right]$ is discussed. For this, when it is necessary, the auxiliary problem which is equivalent to the main problem is introduced. The solution of the proposed problem permits constructing the weak solution of the main problem that conserves the entropy condition. In some cases, depending on the nature of the investigated problem a convex or a concave hull is constructed. Thus, the exact solutions are found by using these functions.