On the asymptotic behavior of the one-dimensional motion of the polytropic ideal gas with stress-free condition. (English) Zbl 0693.76075
We consider the following initial-boundary value problem:
\[
u_ t=v_ x,\quad v_ t=((-R\theta /u)+(\mu v_ x/u))_ x,\quad c_ v\theta_ t=(-R\theta v_ x/u)+(\mu v^ 2_ x/u)+\kappa (\theta_ x/u)_ x,
\]
for \((x,t)\in [0,1]\times [0,+\infty)\) with the initial condition \((u,v,\theta)(x,0)=(u_ 0,v_ 0,\theta_ 0)(x)\), \(u_ 0>0\), \(\theta_ 0>0\), and the boundary condition (stress-free condition) \(((- R\theta /u)+(\mu v_ x/u))(0,t)=((-R\theta /u)+(\mu v_ x/u))(1,t)=0\), \(\theta_ x(0,t)=\theta_ x(1,t)=0.\)
This problem is a model of the one-dimensional motion of the polytropic ideal gas with adiabatic ends which is put into a vacuum. (u,v,\(\theta)\), unknown functions, represent the specific volume, the velocity, the absolute temperature of the gas; \((R,\mu,c_ v,\kappa)\), given positive constants, stand for the gas constant, the coefficient of viscosity, the heat capacity at constant volume, and the coefficient of heat conduction, respectively.
A. V. Kazhykhov showed the global existence of a unique solution to this problem [C. R. Acad. Sci., Paris Ser. A 284, 317-320 (1977; Zbl 0355.35071)]. He constructed the solution (u,v,\(\theta)\) in the Hölder class \[ \cap_{T>0}\{B_ T^{1+\alpha}\times H_ T^{2+\alpha}\times H_ T^{2+\alpha}\}\;(0<\alpha <1) \] provided \((u_ 0,v_ 0,\theta_ 0)\) belongs to \(H^{1+\alpha}\times H^{2+\alpha}\times H^{2+\alpha}\). We call this solution classical in this paper.
More recently M. Okada [Japan J. Appl. Math. 4, 219-235 (1987; Zbl 0636.76065)] and S. Kawashima [Large-time behavior of solutions to the free boundary problem for the equations of a viscous heat-conductive gas (preprint)] showed the asymptotic behavior of the solution. The problem has a trivial solution \(u(x,t)=\bar u(1+t)\), \(v(x,t)=\bar u(x- (1/2))\), \(\theta(x,t)={\bar \theta}\), corresponding to initial data \(u_ 0(x)=\bar u\), \(v_ 0(x)=\bar u(x-(1/2))\), \(\theta_ 0(x)={\bar \theta}\), where \(\bar u\) and \({\bar\theta}\) are positive constants satisfying the relation (*) \(\mu\bar u=R{\bar\theta}.\)
In this paper the author attempts to show the convergence of the classical solution and its rate without any restricted assumptions. We have the following result:
Let \((\bar u,\bar\theta)\) be a positive root of the simultaneous equations (*) and \[ \int^{1}_{0}((1/2)v^ 2_ 0(x)+c_ v\theta_ 0(x))dx=\int^{1}_{0}1/2\{\int^{1}_{0}v_ 0(x)dx+\bar u(x- (1/2))\}^ 2dx+c_ v{\bar \theta}. \] Then there exist positive constants \(\lambda\) and C which depend on \(R,\mu,c_ v,\kappa\), and the initial data but not on t such that the classical solution \((u,v,\theta)\) to the given initial-boundary value problem satisfies the estimate \[ \| ((u(x,t)/1+t)-\bar u,\quad v(x,t)-\int^{1}_{0}v_ 0(x)dx-\bar u(x-(1/2)),\quad \theta (x,t)-{\bar \theta})\|^ 2_{1,2}\leq C(1+t)^{-\lambda}. \] Here \(\| \cdot \|_{1,2}\) is the norm of the Sobolev space \(W^{1,2}(0,1)\).
This problem is a model of the one-dimensional motion of the polytropic ideal gas with adiabatic ends which is put into a vacuum. (u,v,\(\theta)\), unknown functions, represent the specific volume, the velocity, the absolute temperature of the gas; \((R,\mu,c_ v,\kappa)\), given positive constants, stand for the gas constant, the coefficient of viscosity, the heat capacity at constant volume, and the coefficient of heat conduction, respectively.
A. V. Kazhykhov showed the global existence of a unique solution to this problem [C. R. Acad. Sci., Paris Ser. A 284, 317-320 (1977; Zbl 0355.35071)]. He constructed the solution (u,v,\(\theta)\) in the Hölder class \[ \cap_{T>0}\{B_ T^{1+\alpha}\times H_ T^{2+\alpha}\times H_ T^{2+\alpha}\}\;(0<\alpha <1) \] provided \((u_ 0,v_ 0,\theta_ 0)\) belongs to \(H^{1+\alpha}\times H^{2+\alpha}\times H^{2+\alpha}\). We call this solution classical in this paper.
More recently M. Okada [Japan J. Appl. Math. 4, 219-235 (1987; Zbl 0636.76065)] and S. Kawashima [Large-time behavior of solutions to the free boundary problem for the equations of a viscous heat-conductive gas (preprint)] showed the asymptotic behavior of the solution. The problem has a trivial solution \(u(x,t)=\bar u(1+t)\), \(v(x,t)=\bar u(x- (1/2))\), \(\theta(x,t)={\bar \theta}\), corresponding to initial data \(u_ 0(x)=\bar u\), \(v_ 0(x)=\bar u(x-(1/2))\), \(\theta_ 0(x)={\bar \theta}\), where \(\bar u\) and \({\bar\theta}\) are positive constants satisfying the relation (*) \(\mu\bar u=R{\bar\theta}.\)
In this paper the author attempts to show the convergence of the classical solution and its rate without any restricted assumptions. We have the following result:
Let \((\bar u,\bar\theta)\) be a positive root of the simultaneous equations (*) and \[ \int^{1}_{0}((1/2)v^ 2_ 0(x)+c_ v\theta_ 0(x))dx=\int^{1}_{0}1/2\{\int^{1}_{0}v_ 0(x)dx+\bar u(x- (1/2))\}^ 2dx+c_ v{\bar \theta}. \] Then there exist positive constants \(\lambda\) and C which depend on \(R,\mu,c_ v,\kappa\), and the initial data but not on t such that the classical solution \((u,v,\theta)\) to the given initial-boundary value problem satisfies the estimate \[ \| ((u(x,t)/1+t)-\bar u,\quad v(x,t)-\int^{1}_{0}v_ 0(x)dx-\bar u(x-(1/2)),\quad \theta (x,t)-{\bar \theta})\|^ 2_{1,2}\leq C(1+t)^{-\lambda}. \] Here \(\| \cdot \|_{1,2}\) is the norm of the Sobolev space \(W^{1,2}(0,1)\).
