Abstract: Entropies (convex extensions) play a central role in the theory of hyperbolic conservation laws by providing intrinsic selection criteria for weak solutions and local well-posedness for the Cauchy problem. While many systems occurring in physical models are equipped with extensions, it is well-known that existence of a non-trivial (i.e. non-linear) extension requires the solution of an over-determined system of equations. On the other hand, so-called rich systems are equipped with large sets of entropies. Beyond these general facts little seems to be known about “how many” extensions a particular system of conservation laws has.

For a given hyperbolic system $u_t + f(u)_x = 0$, a standard approach is to analyze directly the second order PDE system for the extensions. Instead we find it advantageous to consider the equations satisfied by the lengths $\beta^i$ of the right eigenvectors $r_i$ of $Df$, as measured with respect to the inner product defined by an extension. For a given eigen-frame $\{r_i\}$ the extensions are determined uniquely, up to trivial affine parts, by these lengths.

This geometric formulation provides a natural and systematic approach to existence of extensions. By considering the eigen-fields $r_i$ as prescribed our results automatically apply to all systems with the same eigen-frame. As a computational benefit we note that the equations for the lengths $\beta^i$ form a first order algebraic-differential system (the $\beta$-system) to which standard integrability theorems can be applied. The size of the set of extensions follows by determining the number of free constants and functions present in the general solution to the $\beta$-system. We provide a complete breakdown of the various possibilities for $3\times 3$-systems, as well as for rich frames in any dimension provided the $\beta$-system has trivial algebraic part. The latter case covers $2\times 2$-systems, strictly hyperbolic rich systems of any size, and any rich system with an orthogonal eigen-frame.

Our analysis is relevant whenever there exists a non-trivial conservative system whose eigen-frame coincides with the given frame. This issue was analyzed by the authors in [25], where the problem was formulated in terms of another algebraic-differential system, the “$\lambda$-system,” whose solutions provide the characteristic speeds (eigenvalues) of the resulting conservative systems. We investigate the relationships between the $\lambda$- and $\beta$-systems and recover standard results for symmetric systems (orthogonal frames). It turns out that despite close structural connections between the $\lambda$- and the $\beta$-system, there is no general relationship between the sizes of their solution sets. We provide a list of examples that illustrate our results.