Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases

Abstract

T h e one-dimensional m otion o f com pressible, viscous a n d heat-conductive fluids is described by th e equations in th e L a g ra n g ia n m a s s coordin ate (the suffix denotes th e partial differentiation with respect to th e v a ria b le t or x) { Pt+ p 2 u.v=0 ut+ P .= (110u .).r (e + 1.172) t (p u) x =-c D O z) + (g p u u x) , w h ere p is th e density, u is th e velocity, 0 is th e absolute temperature, g is th e viscosity coefficient and i s th e co efficien t o f heat conduction, and the pressure p a n d th e internal energy e a r e rela ted to p , 0 b y t h e equation of state of th e fluids. H e r e w e assum e that th e equation of state is one for the ideal polytropic gas: (i) p = e-R " + constant, r-w h ere R is th e ga s co n stan t an d th e constant r> i is the adiabatic exponent, an d also assum e that (ii) /c are positive constants. I f w e ta k e p a n d S = e n tro p y / R a s th e basic variables, t h e equation o f sate (i) can be w ritten in th e fo r m (c f. [1 ]) (1. 2) e-R O (0 9 / 0 7-1 exP (r-{ r-1 1) (s-s)-1) p = (Ye = R & (p /) /15) rexp (y-1) (S-S) 0= e s /R=0 (p/p) r 'e x p (r-1) (S-S),