Asymptotic Theory for the Probability Density Functions in Burgers Turbulence

A systematic analysis is carried out for the randomly forced Burgers equation in the infinite Reynolds number (inviscid) limit. No closure approximations are made. Instead the probability density functions of velocity and velocity gradient are related to the statistics of quantities defined along the shocks. This method allows one to compute the dissipative anomalies, as well as asymptotics for the structure functions and the probability density functions. It is shown that the left tail for the probability density function of the velocity gradient has to decay faster than $|\xi {|}^{-3\mathrm{}}$. A further argument confirms the prediction of E et al. [Phys. Rev. Lett. 78, 1904 (1997)] that it should decay as $|\xi {|}^{-7/2\mathrm{}}$.