Extensions for Systems of Conservation Laws

Extensions are central in the theory of conservation laws by providing intrinsic selection criteria for weak solutions. Given a system u _t + f(u) _x = 0 the extensions solve certain second order PDEs which are typically overdetermined. Determining the size of the set of extensions can be a challenging task.Instead of analyzing this second order system directly, we consider the equations satisfied by the lengths β ⁱ of the eigenvectors r _i of the Jacobian matrix Df, measured with 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. The β ⁱ solve a first order algebraic-differential system (the β-system) to which standard integrability theorems can be applied. The number of extensions is determined by determining the number of free constants and functions in the general solution to the β-system. We provide a complete breakdown for 3 × 3-systems, and for rich frames whose β-system has trivial algebraic part.Our framework is motivated by the work [16] where the authors consider conservative systems with prescribed eigen-frames. It is natural to ask how many extensions the resulting systems have, and the answer depends in an essential manner on the prescribed frame.