Asymptotic stability of solutions with strong discontinuities for the equations of isothermal gas dynamics

Abstract

We study the large-time behavior of weak solutions for the equations of isothermal gas dynamics:ut+(k2/v)x=vt−ux=0, when the initial data have bounded total variation. The weak stability theorem of T. P. Liu says that the solution converges to the solution to the Riemann problem whose initial data are composed of (u0(±∞),v0(±∞)). The aim of this note is to give the rate of convergence, which is to study the large-time behavior of strong shock waves interacting with weak waves. If two strong shock waves emerge, the speed and the strength of these shock waves approach those of the solution to the Riemann problem at the ratet−k(K>0). If a single shock wave emerges, the speed and the strength of this shock wave approach at the rate oft−3/2 and the total variation of the solution outside the strong shock approaches zero at the ratet−1/2.