The inviscid Burgers equation with fractional Brownian initial data: the dimension of regular Lagrangian points

TL;DR

This paper proves that for the inviscid Burgers equation with fractional Brownian initial data, the Hausdorff dimension of regular Lagrangian points equals the Hurst index H, confirming a longstanding conjecture.

Contribution

It establishes the Hausdorff dimension of regular Lagrangian points as H for the first time, validating the Sinai-Frisch conjecture.

Findings

Hausdorff dimension of regular Lagrangian points is H

Confirms the Sinai-Frisch conjecture from 1992

Provides a mathematical link between fractional Brownian motion and fluid dynamics

Abstract

Fractional Brownian motion, H-FBM , of index 0<H<1 is considered as initial velocity in the inviscid Burgers equation. It is shown that the Hausdorff dimension of regular Lagrangian points at any moment t is equal to H. This fact validates the Sinai-Frisch conjecture known since 1992.