Contact discontinuity with general perturbations for gas motions

Abstract

The contact discontinuity is one of the basic wave patterns in gas motions. The stability of contact discontinuities with general perturbations for the Navier–Stokes equations and the Boltzmann equation is a long standing open problem. General perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties. In this paper, we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the Navier–Stokes equations and the Boltzmann equation in a uniform way. One of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate ( 1 + t ) 1 4 , it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of ( 1 + t ) − 1 4 . Thus, this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the L ∞ norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern.