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Entropy solutions have been widely accepted as the suitable solutionframework for systems of conservation laws in several space dimensions.However, recent results in \cite{CDL1,CDL2} have demonstrated that entropysolutions may not be unique. In this paper, we present numerical evidence thatdemonstrates that state of the art numerical schemes \emph{may not} necessarilyconverge to an entropy solution of systems of conservation laws as the mesh isrefined. Combining these two facts, we argue that entropy solutions may not besuitable as a solution framework for systems of conservation laws, particularlyin several space dimensions.  Furthermore, we propose a more general notion, that of \emph{entropy measurevalued solutions}, as an appropriate solution paradigm for systems ofconservation laws. To this end, we present a detailed numerical procedure,which constructs stable approximations to entropy measure valued solutions andprovide sufficient conditions that guarantee that these approximations convergeto an entropy measure valued solution as the mesh is refined, thus providing aviable numerical framework for systems of conservation laws in several spacedimensions. A large number of numerical experiments that illustrate theproposed schemes are presented and are utilized to examine several interestingproperties of the computed entropy measure valued solutions.

Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws