Abstract

We discuss examples of constructing solutions for conservation laws wi-thout spatial BV bounds. The convergence of spectral viscosity and finite volume approximationsare prototype examples for such BV-free scalar constructions. The first example involves hyper-viscosity limits for 1D scalar equationswith convex fluxes : existence of entropy solutions follows by compensatedcompactness arguments, based on one entropy production bound. A second example deals with solutions of 2D scalar equations : we showthat a judicious choice of two entropies entropy bounds will suffice. We then raise several open questions : here comes a third ’example’ ofEuler alignment system.

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