We introduce a family of impulselike methods for the integration of highly oscillatory second-order differential equations whose forces can be split into a fast part and a slow part. Methods of this family are specified by two weight functions φ, ψ; one is used to average positions and the other to mollify the force. When the fast forces are conservative and φ = ψ, the methods here coincide wit...

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ∥u(t, ·) − u(t, ·) ∥∥ L = O(1)(1 + t) · √ε| ln ε| on the distance between an exact BV solution u and a viscous approximation u, letting the viscosity coefficient ε → 0. In the proof, starting from u we construct an approximation of the viscous solution u by taking a mollification u ∗ φ√ ε a...

In this work, we introduce a variant of the standard mollifier technique that is valid up to the boundary of a Lipschitz domain in Rn. A version of Friedrichs’s lemma is derived that gives an estimate up to the boundary for the commutator of the multiplication by a Lipschitz function and the modified mollification. We use this version of Friedrichs’s lemma to prove the density of smooth functio...

We introduce an intrinsic notion of Hölder-Zygmund regularity for Colombeau generalized functions. In case of embedded distributions belonging to some Zygmund-Hölder space this is shown to be consistent. The definition is motivated by the well-known use of Littlewood-Paley decomposition in characterizing Hölder-Zygmund regularity for distributions. It is based on a simple interplay of different...