Viscous limit to contact discontinuity for the 1-D compressible Navier-Stokes equations
The one-dimensional model to the ideal polytropic gas is considered. The solutions to the Euler equations \[ \begin{cases} \frac{\partial v}{\partial t}-\frac{\partial u}{\partial x}=0,\quad \frac{\partial u}{\partial t}+\frac{\partial p}{\partial x}=0,\\ \frac{\partial }{\partial t}\left(e+\frac{u^2}{2}\right)+\frac{\partial }{\partial x}(pu)=0,\quad x\in \mathbb{R},\;t>0 \end{cases} \] and to the compressible Navier-Stokes equations \[ \begin{cases} \frac{\partial v}{\partial t}-\frac{\partial u}{\partial x}=0,\quad \frac{\partial u}{\partial t}+\frac{\partial p}{\partial x}=\mu\frac{\partial }{\partial x}\left(\frac{1}{v}\frac{\partial u}{\partial x}\right),\\ \frac{\partial }{\partial t}\left(e+\frac{u^2}{2}\right)+\frac{\partial }{\partial x}(pu)= k\frac{\partial }{\partial x}\left(\frac{1}{v}\frac{\partial\theta}{\partial x}\right)+ \mu\frac{\partial }{\partial x}\left(\frac{u}{v}\frac{\partial u}{\partial x}\right) ,\quad x\in \mathbb{R},\;t>0 \end{cases} \] are compared. Here \(v\) is the specific volume, \(u\) is the velocity, \(\theta\) is the temperature, \(p\) is the pressure, \(e\) is the internal energy. Let \((V,U,\Theta)\) be a contact discontinuity (at \(x=0\)) solution to the Euler system. Then there exists a sequence of smooth solutions to the Navier-Stokes system which converges to \((V,U,\Theta)\) at a rate of \(k^{3/4}\) as \(k\) tends to zero provided that the coefficient \(\mu\) tends to zero with order not lower than \(k.\)
scientific article; zbMATH DE number 5994361
scientific article