On the statistical solution of the Riemann equation and its implications for Burgers turbulence

The statistics of the multivalued solutions of the forced Riemann equation, ut+uux=f, is considered. An exact equation for the signed probability density function of these solutions and their gradient ξ=ux is derived, and some properties of this equation are analyzed. It is shown in particular that the tails of the signed probability density function generally decay as |ξ|-3 for large |ξ|. Further considerations give bounds on the cumulative probability density function for the velocity gradient of the solution of Burgers equation. copyright 1999 American Institute of Physics