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A novel approach for the stabilization of the Discontinuous Galerkin methodbased on the Dafermos entropy rate crition is presented. First, estimates forthe maximal possible entropy dissipation rate of a weak solution are derived.Second, families of conservative Hilbert-Schmidt operators are identified todissipate entropy. Steering these operators using the bounds on the entropydissipation results in high-order accurate shock-capturing DG schemes for theEuler equations, satisfying the entropy rate criterion and an entropyinequality.

Stabilizing Discontinuous Galerkin Methods Using Dafermos' Entropy Rate  Criterion: II -- Systems of Conservation Laws and Entropy Inequality  Predictors