A New Viscous Regularization of the Riemann Problem for Burgers' Equation

Abstract

This paper gives a new viscous regularization of the Riemann problem for Burgers' equation u_t + (\frac{u²}{2})_z = 0 with Riemann initial data u = u_(x ≤ 0), u = u_+(x > 0} at t = 0. The regularization is given by u_t + (\frac{u²}{2})_z = εe^tu_{zz} with appropriate initial data. The method is different from the classical method, through comparison of three viscous equations of it. Here it is also shown that the difference of the three regularizations approaches zero in appropriate integral norms depending on the data as ε → 0_+ for any given T > 0.