# Onsager's conjecture in bounded domains for the conservation of entropy and other companion laws

## TL;DR

This paper proves that weak solutions of conservation laws in bounded domains conserve entropy and related laws under fractional differentiability and boundary flux conditions, extending previous results.

## Contribution

It extends existing results by establishing entropy conservation for weak solutions in bounded domains with fractional differentiability and boundary flux decay.

## Key findings

- Weak solutions conserve entropy if they have fractional differentiability of order 1/3.
- Boundary flux conditions are crucial for entropy conservation.
- Results apply to general conservation laws in bounded domains.

## Abstract

We show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy, and other respective companion laws, if they possess a certain fractional differentiability of order 1/3 in the interior of the domain, and if the normal component of the corresponding fluxes tend to zero as one approaches the boundary. This extends various recent results of the authors.