Weak versus $\mathcal{D}$-solutions to linear hyperbolic first order systems with constant coefficients

TL;DR

This paper compares two notions of generalized solutions for linear hyperbolic PDE systems with constant coefficients, demonstrating their equivalence under certain conditions, and introduces a new duality-free solution concept based on probabilistic limits.

Contribution

It establishes the equivalence of weak solutions and $\\mathcal{D}$-solutions for a broad class of hyperbolic systems, connecting classical and novel solution frameworks.

Findings

Weak solutions and $\mathcal{D}$-solutions coincide for the studied systems.

The $\mathcal{D}$-solutions provide a duality-free alternative to distributions.

The approach links probabilistic methods with PDE solution concepts.

Abstract

We establish a consistency result by comparing two independent notions of generalised solutions to a large class of linear hyperbolic first order PDE systems with constant coefficients, showing that they eventually coincide. The first is the usual notion of weak solutions defined via duality. The second is the new notion of $\mathcal{D}$-solutions introduced in the recent paper [K8], which arose in connection to vectorial Calculus of Variations in $L^\infty$ and fully nonlinear elliptic systems. This new approach is a duality-free alternative to distributions and is based on the probabilistic representation of limits of difference quotients.