2 x 2 hyperbolic systems of conservation laws in classes of functions of bounded p-variation

TL;DR

This paper proves the existence of entropy solutions for 2x2 hyperbolic conservation laws with characteristic fields satisfying certain conditions, within fractional BV spaces of functions of bounded p-variation, for small initial data.

Contribution

It extends the existence theory of entropy solutions to hyperbolic systems in fractional BV spaces for a broader class of characteristic fields.

Findings

Existence of entropy solutions in fractional BV spaces ${W}_p( eal)$ for $p \, \in \, [1, \frac{3}{2}]$.

Applicable to systems with characteristic fields satisfying $r_i\cdot \nabla \lambda_i \geq 0$.

Results hold for small initial data.

Abstract

In this paper, we consider $2 \times 2$ hyperbolic systems of conservation laws in one space dimension with characteristic fields satisfying a condition that encompasses genuine nonlinearity and linear degeneracy as well as intermediate cases, namely, with standard notations, $r_i\cdot \nabla \lambda_i \geq 0$. We prove the existence of entropy solutions in the fractional $BV$ spaces ${W}_p(\mathbb{R})$ of functions of bounded $p$-variation, $p \in [1,\frac{3}{2}]$, for small initial data.