On the Optimality of Velocity Averaging Lemmas

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Velocity averaging, kinetic formulations, and regularizing effects in quasi‐linear PDEs (2007, background): “…v) ∈ L q loc(R d x×Rv) are real-valued functions of the spatial variables x = (x1, . . . , xd) ∈ Rd and an additional parameter v ∈ R, called velocity by analogy with the kinetic framework, and L(∇x, v) is a differential operator on Rdx of order ≤ k, whose coefficients are smooth functions of v.”
Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws (2014, background): “10) is essentially sharp for quasi-solutions, a fact which is not true for entropy solutions; see, for example, De Lellis and Westdickenberg [16] and De Lellis, Otto and Westdickenberg [15].”