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Free boundary value problem of the one-dimensional model of polytropic ideal gas. (English) Zbl 0727.35145

The one-dimensional free boundary value problem of the polytropic ideal gas is studied. The equation has four unknowns; the density \(\rho\), the velocity u, the absolute temperature \(\theta\), and the free boundary X(t), and is described as \[ \rho_ t+(\rho u)_ x=0;\quad (\rho u)_ t+(\rho u^ 2+p)_ x=(\mu u_ x)_ x, \]
\[ \{\rho (e+u^ 2/2)\}_ t+\{\rho u(e+u^ 2/2)+pu\}_ x=(\kappa \theta_ x)_ x+(\mu uu_ x)_ x, \] for \(t>0\), \(X(t)<x<0\), with the initial condition \[ (\rho,u,\theta)(0,x)=(\rho_ 0,u_ 0,\theta_ 0)(x),\quad d\leq x\leq 0\quad (X(0)=d), \]
\[ \rho_ 0,\theta_ 0>0\text{ on } (d,0],\quad \rho_ 0(d)=\theta_ 0(d)=0, \] and the boundary condition \[ u(t,0)=\theta_ x(t,0)=0,\quad \rho (t,X(t))=\theta (t,X(t))=0;\quad dX(t)/dt=u(t,X(t)),\quad t>0. \] Here, \(p=R\rho \theta\) and \(e=R\theta /(\gamma -1)\), \(R>0\), \(\gamma >1\), \(\kappa >0\), and \(\mu >0\) being physical constants. Sought is the weak solution (\(\rho\),\(\theta\),u) belonging to a class of function spaces and satisfying the above equation in integral form. By the well known change of the variables (t,x) into \((t',x')\); \(t'=t\), \(x'=\int^{x}_{0}\rho (t,\xi)d\xi\), the problem is transformed into the initial-boundary value problem with the fixed boundary (after rescaling and dropping the primes): \[ \rho_ t+\rho^ 2u_ x=0;\quad u_ t+a(\rho \theta)_ x=\mu (\rho u_ x)_ x, \]
\[ \theta_ t+a\rho \theta u_ x=\mu \rho u^ 2_ x+\kappa (\rho \theta_ x)_ x,\quad t>0,\quad 0<x<1, \] with the transformed initial and the boundary conditions. Similar problems have been studied so far. The difficulty here is that \(\rho_ 0\) is not bounded away from 0. By assuming instead a nonnegative lower bound for \(\rho_ 0\), a finite difference scheme in space is proposed to construct a series of functions which approximates a weak solution with sufficiently small \(\theta_ 0\) and \(u_ 0\). An exponential decay in time for \(\theta\) and u is also obtained, the decay rate being independent of small \(\theta_ 0\) and \(u_ 0\).
Reviewer: T.Nambu (Kumamoto)

MSC:

35R35 Free boundary problems for PDEs
35M20 PDE of composite type (MSC2000)
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35D05 Existence of generalized solutions of PDE (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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References:

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