The Riemann Problem for Systems

Abstract

We return to the conservation law (1.2), but now study the case of systems, i.e.,

$$\displaystyle u_{t}+f(u)_{x}=0,$$

(5.1)

where $u=u(x,t)=(u_{1},\dots,u_{n})$ and $f=f(u)=(f_{1},\dots,f_{n})\in C^{2}$ are vectors in $\mathbb{R}^{n}$. (We will not distinguish between row and column vectors, and use whatever is more convenient.) Furthermore, in this chapter we will consider only systems on the line; i.e., the dimension of the underlying physical space is still one. In Chapt. 2 we proved existence, uniqueness, and stability of the Cauchy problem for the scalar conservation law in one space dimension, i.e., well-posedness in the sense of Hadamard. However, this is a more subtle question in the case of systems of hyperbolic conservation laws. We will here first discuss the basic concepts for systems: fundamental properties of shock waves and rarefaction waves. In particular, we will discuss various entropy conditions to select the right solutions of the Rankine–Hugoniot relations.

Using these results, we will eventually be able to prove well-posedness of the Cauchy problem for systems of hyperbolic conservation laws with small variation in the initial data.

Diese Untersuchung macht nicht darauf Anspruch, der experimentellen Forschung nützliche Ergebnisse zu liefern; der Verfasser wünscht sie nur als einen Beitrag zur Theorie der nicht linearen partiellen Differentialgleichungen betrachtet zu sehen.The present work does not claim to lead to results in experimental research; the author asks only that it be considered as a contribution to the theory of nonlinear partial differential equations. — G. F. B. Riemann [156] Diese Untersuchung macht nicht darauf Anspruch, der experimentellen Forschung nützliche Ergebnisse zu liefern;der Verfasser wünscht sie nur als einen Beitrag zur Theorie der nicht linearen partiellen Differentialgleichungen betrachtet zu sehen. — G. F. B. Riemann [156] The present work does not claim to lead to results in experimental research; the author asks only that it be considered as a contribution to the theory of nonlinear partial differential equations.

Diese Untersuchung macht nicht darauf Anspruch, der experimentellen Forschung nützliche Ergebnisse zu liefern;der Verfasser wünscht sie nur als einen Beitrag zur Theorie der nicht linearen partiellen Differentialgleichungen betrachtet zu sehen.

— G. F. B. Riemann 156;

The present work does not claim to lead to results in experimental research; the author asks only that it be considered as a contribution to the theory of nonlinear partial differential equations.