M\"uller-Zhang truncation for general linear constraints with first or second order potential

TL;DR

This paper extends a theorem on the convergence of differential operator sequences to convex sets, providing uniform convergence under bounded derivatives, and constructs a second-order potential for the linearized Euler system.

Contribution

It generalizes Müller's theorem for gradient sequences to more general linear differential operators and constructs a second-order potential for the Euler system.

Findings

Sequences of differential operator applications can be uniformly approximated to convex sets.

The results hold on various domains including the whole space and bounded open sets.

A second-order potential for the linearized Euler system is explicitly constructed.

Abstract

Let $\mathcal{B}$ be a homogeneous differential operator of order $l=1$ or $l=2$. We show that a sequence of functions of the form $(\mathcal{B}u_j)_j$ converging in the $L^1$-sense to a compact, convex set $K$ can be modified into a sequence converging uniformly to this set provided that the derivatives of order $l$ are uniformly bounded. We prove versions of our result on the whole space, an open domain, and for $K$ varying uniformly continuously on an open, bounded domain. This is a conditional generalization of a theorem proved by S. M\"uller for sequences of gradients, cf. [6]. Moreover, a potential of order two for the linearized isentropic Euler system is constructed.