Extensions for Systems of Conservation Laws

الأصل · EN

Extensions (entropies) play a central role in the theory of hyperbolic conservation laws by providing intrinsic selection criteria for weak solutions. For a given hyperbolic system uₜ+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 betaⁱ of the right eigenvectors rᵢ the Jacobian matrix Df, as measured with respect to the inner product defined by an extension. Our geometric formulation provides a natural and systematic approach to existence of extensions. By prescribing the eigen-fields rᵢ our results automatically apply to all systems with the same eigen-frame. The equations for the lengths betaⁱ 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 systems of three equations, as well as for rich hyperbolic systems of any size.