An invariant in shock clustering and Burgers turbulence

TL;DR

This paper explores the connection between shock clustering in 1-D scalar conservation laws and Burgers turbulence, deriving an invariant analogous to Loitsiansky's from integrable systems theory, and discussing related physical concepts.

Contribution

It introduces a novel derivation of a turbulence invariant within the framework of integrable Hamiltonian systems for scalar conservation laws.

Findings

Derivation of an invariant similar to Loitsiansky's in shock clustering.

Discussion of energy dissipation and spectral properties in this context.

Linking turbulence invariants to integrable systems theory.

Abstract

1-D scalar conservation laws with convex flux and Markov initial data are now known to yield a completely integrable Hamiltonian system. In this article, we rederive the analogue of Loitsiansky's invariant in hydrodynamic turbulence from the perspective of integrable systems. Other relevant physical notions such as energy dissipation and spectrum are also discussed.