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We show the existence of the global solutions in time of the equations of one-dimensional motion of barotropic gas with a free boundary and the asymptotic behaviors of the solutions as time tends to infinity.Key words: barotropic gas, solutions global in time, free boundary value problem, asymptotic behaviors of the solution w 1. lntroduction We consider afrec boundary value problem where the barotropic gas is in contact continuously with the vacuum. The equation of one-dimensional motion of barotropic gas is desr by the following in the Eulerian coordinate:(1.1)in r>0, y(r)<~<0.Here the unknown functions p and u represent the density and the velocity, respectively; p=ap ~ is the pressure where ais a positive constant and 7> l; the positive constant # is the coefficient of viscosity; 9 > 0 is the gravitation constant and when the external force is not considered, 9 =0; ~ =0 is the fixed boundary, (1.3) u(z, 0) = 0, and y(z) is the free boundary, i.e. the interface of the gas and the vacuum:This problem can be reformulated in Lagrangean mass coordinates by using the transformation:Here, considering the position of the boundary X(z)=x(r, y(r)) ~YI~) " =Jo Ptr, ~)d~ and using (1.1), (1.3), (1.4), we have dX/dz = 0, i.e. X is independent of z. Therefore we can 162 M. OKADA set X= ~£ p(0, Oda. Thr are two cases, i.e. (i) y(0) = -oo and (ii) y(0) > -oo even if X> -co. We consider y(0) > -oo, i.e., a finite total mass on a finite interval, which is most interesting asa free boundary problem. Afler rescaling the variables, the problem (1. l)-(1,4) is transformed to the following fixed boundary problem;(1.5) pt + p2ux=O , (1.6) u~+px=(#pUx)x+g, t>0, 0_0 in (0, 1], and we assume the compatibility conditions at x=0, 1, (1.9) po(0)=0, uo(1)=0.Since the density becomes zero on the free surface x=0, the known local existence theorem of the solution for (1.5) and (l.6) doesn't apply directly. Therefore the existence of a generalized solution of the problem (1.5)-(1.9) will be proved by using the line method. That is, we consider systems of 2N ordinary differential equations when N goes to infinity:(1.10). Z U2n+l--U2n-l=O " P2mt+Pzn Ax (1.11) u2,_l.t+a p~"-p~z"-21/ u2"+a-u2.-' U2n-l _~__U2n-3. )/ , Ÿ237 Ax --q p2" ~x P2,-~ for n= 1, 2, ".., N, where Ax = 1/(N+ 1q and the boundary conditions are (1.12) po(t)=uzu+l(t)=O, t>O.When n = l, the second term in the right hand side of (1.11) is regarded asIf the initial data are given as [ p2,(0)= p(0, nAx)>O, (1.14) ( (1) ) u2, l(0)=u O, nAx , n=l,2,'",N, the local existence of solutions (p2.(t), u2,_l(t) ), n = 1, 2, 9 9 N, for (1.10)-(1.14) is trivial for fixed Nas (1.10) and (1.11) satisfy the Lipschitz' condition when 7 > 1, and Free Boundary Value Problems 163 the solution satisfies the positivity of density inside: p2,( t) > O , n = l , 2...

We study the spherically symmetric motion of viscous barotropic gas surrounding a ball. The basic equation is the compressible Navier-Stokes equation. We are interested in the density distribution which contacts with vacuum at a finite radius. This is a free boundary problem. After rewriting the equation in the Lagrangean coordinate, we construct approximate solutions by discretizing the mass variable. Passing to a limit, we find a global weak solution.

We consider a free boundary problem for the equation of one-dimensional isentropic motions with density-dependent viscosity µ = bpO, where b and /3 are positive constants, when the density function connects to a vacuum continuously on the boundary. We prove that there exists a weak solution globally in time, provided that ß < -1

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