A theory of weak shocks
Abstract
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 NavierStokes 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 higherorder approximations are eliminated.
 Publication:

Communications in Pure Applied Mathematics
 Pub Date:
 September 1983
 Bibcode:
 1983CPAM...36..679T
 Keywords:

 Burger Equation;
 Flow Theory;
 Hyperbolic Differential Equations;
 NavierStokes Equation;
 Shock Wave Profiles;
 Energy Dissipation;
 Gas Dynamics;
 Nonlinear Evolution Equations;
 Shock Discontinuity;
 Shock Wave Attenuation;
 Wave Dispersion;
 Wave Equations;
 Fluid Mechanics and Heat Transfer