Stochastic analysis method for hopf's equation ut+ uux=0

Abstract

It is proved that the stochastic charactcrtstics of Burgers’ equation u t+uux=μuxxconverge in probability to the character is tcs of Hopf ’ s cqua lion ut+uux=0 as the viscosity μ →0. It follows naturally that the solution of Burgers’ equation converges to the solution of Hopf 's equation satisfying entropy condition. This is well known result due to E. Hopf in 1950. The method here is new. This paper suggests that this method may be useful for proving the validity of “vanishing viscosity method” in certain cases. Those problems are usually extremely difficult.