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

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 $x_f\cong t^{3\mathrm{/}4}{(logt)}^{1\text{4}}$. 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.