On the Extension of Onsager's Conjecture for General Conservation Laws

TL;DR

This paper extends Onsager's conjecture to a broader class of conservation laws with generalized entropy, establishing the universal regularity threshold for entropy conservation regardless of system nonlinearity.

Contribution

It proves the Onsager conjecture for conservation laws with generalized entropy, demonstrating the universality of the regularity exponent $rac{1}{3}$ for entropy conservation.

Findings

The Onsager exponent $rac{1}{3}$ is sufficient for entropy conservation.

The result applies regardless of the system's nonlinearity structure.

The work broadens the applicability of Onsager's conjecture to generalized conservation laws.

Abstract

The aim of this work is to extend and prove the Onsager conjecture for a class of conservation laws that possess generalized entropy. One of the main findings of this work is the "universality" of the Onsager exponent, $\alpha > 1/3$, concerning the regularity of the solutions, say in $C^{0,\alpha}$, that guarantees the conservation of the generalized entropy; regardless of the structure of the genuine nonlinearity in the underlying system.