A COMPARISON OF 2 VISCOUS REGULARIZATIONS OF THE RIEMANN PROBLEM FOR BURGERS EQUATION

This note compares solutions of two regularizations for Burgers' equation u(t) + (u(2)/2)(x) = 0 with Riemann initial data u = u-(x less than or equal to 0), u = u(+)(x > 0) at t = 0. The regularizations given by u(t)(epsilon) + (u(epsilon 2)/2)(x) = epsilon u(xx)(epsilon) and u(t)(epsilon) (u(epsilon 2)/2)(x) = epsilon tu(xx)(epsilon) with appropriate initial data in each case. The first regularization is more traditional while the second preserves the space-time dilational invariance of the Riemann problem for the inviscid equation. Here it is shown that the difference of the two regularizations approaches zero (in appropriate integral norms depending on the data) as epsilon --> 0(+) for 0 < t less than or equal to 1.