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, u(t)+uu(x)=f, is considered. An exact equation for the signed probability density function of these solutions and their gradient xi=u(x), 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 \xi\(-3) for large \xi\. Further considerations give bounds on the cumulative probability density function for the velocity gradient of the solution of Burgers equation. (C) 1999 American Institute of Physics. [S1070-6631(99)01708-0].