PDF's Of The Burgers Equation On The Semiline With Fluctuating Flux At The Origin

TL;DR

This paper analyzes the asymptotic behavior of the probability density for shock slopes in a turbulent Burgers system with fluctuating flux at the origin, revealing time-dependent shock dynamics and stationary state characteristics.

Contribution

It introduces a novel analysis of the inhomogeneous shock slopes in Burgers turbulence with fluctuating flux, deriving their asymptotic distribution and shock disappearance time.

Findings

Derived the asymptotic behavior of the one-point probability density.

Calculated the time dependence of the shock disappearance point $x_f$.

Linked the stationary state to the long-time limit of a diffusion equation with a random source.

Abstract

We derive the asymptotic behaviour of the one point probability density for the inhomogeneous shock slopes in the turbulent regime, when a Gaussian fluctuating flux at origin derives the system. We also calculate the time dependence of the $x_{f}$ beyond which there won't exists any velocity shocks as $ x_{f}\cong t^{3/4}{(log{t})}^{1\4}$. We argue that the stationary state of the problem would be equivalent with the long time limit of the diffusion equation with arandom source at origin.