A theory of weak shocks

It is shown that the profile of weak shocks in the theory of steady plane shock wave in fluids is valid for a wide class of dissipative nonlinear equations involving the Navier-Stokes equation as a special case. The reductive perturbation method is applied to a hyperbolic system of conservation laws, yielding an approximate equation for a simple wave. It is shown that a weak shock is obtained as a weak solution of a certain conservation law. The method is then applied to a weakly dissipative nonlinear system which is introduced by the linear dispersion relation from which the coefficient of the shock solution of the Burgers equation is derived. It is shown that the Burgers equation is modified to give the solution to the profile equation with the exact shock velocity while the secular terms on the higher-order approximations are eliminated.