Stability of an equilibrium for a hydrodynamical model of charge transport in semiconductors (Q1963383)

The following mathematical model is under consideration: \[ \begin{gathered} R_{\tau} +u R_s + Ru_s=0,\;\;\tau>0,\;s\in (0,1), \\ u_{\tau} + u u_s + \frac{\theta}{R}R_s + \theta_s +\frac{1}{R}\Sigma_s=Q- \frac{u}{\tau_p}, \\ \theta_{\tau} + u\theta_s +\frac{2}{3}\left(\theta +\frac{\Sigma}{R}\right) u_s + \frac{2}{3R}\Theta_s=\frac{2}{3}\left\{u^2\left(\frac{1}{\tau_p}-\frac{1}{2\tau_w}\right) + \frac{3(1-\theta)}{2\tau_w}\right\}, \\ \Sigma_{\tau} + u\Sigma_s + \frac{4R\theta+7\Sigma}{3} u_s + \frac{8}{15}\Theta_s = \frac{4Ru^2}{3\tau_p}- \frac{\Sigma}{\tau_{\sigma}}, \\ \begin{aligned} \Theta_{\tau} &+u \Theta_s +\frac{5}{2}(R\theta+\Sigma)\theta_s - \frac{\Sigma\theta}{R} R_s + \left(\theta - \frac{\Sigma}{R}\right) \Sigma_s + \frac{16}{5}\Theta u_s \\ &= -\frac{\Theta}{\tau_q} + \Sigma u \left(\frac{1}{\tau_\sigma}+\frac{1}{\tau_p}- \frac{1}{\tau_q}\right) + \frac{5}{2}Ru\theta\left(\frac{1}{\tau_p}- \frac{1}{\tau_q}\right) \\ &\qquad+ \frac{Ru^3}{2}\left(\frac{5}{3\tau_w}- \frac{1}{\tau_q}- \frac{3}{\tau_p}\right) - \frac{5(1-\theta)}{2\tau_w}Ru, \end{aligned} \\ \varepsilon^2\phi_{ss}=R-\rho. \end{gathered} \] Here \(R\) is the density, \(u\) is the velocity, \(\theta\) stands for the temperature, \(\Sigma\) is the voltage, \(\Theta\) denotes the heat flux, \(\phi\) is the electric potential, \(\varepsilon^2\) is a constant, \(Q=\phi_s\), \(\tau_p(E),\tau_w(E),\tau_{\sigma}(E), \tau_q(E)\) (\(E=u^2/2 + 3\theta/2\)) denote the relaxation times, and \(\rho(s)\) is a prescribed function. These equations are furnished with the boundary conditions \[ \begin{gathered} R(\tau,0)=R(\tau,1)= \theta(\tau,0)=\theta(\tau,1)=1,\;\Sigma(\tau,1)=0, \\ \phi(\tau,0)=A,\;\phi(\tau,1)=A+B, \end{gathered} \] where \(A\) and \(B>0\) are some constants. This system describes the charge transport in semiconductors. The equations are linearized on a stationary solution. The system obtained is reduced to a symmetric hyperbolic system. The main result of the article consists in obtaining a priori bounds in Sobolev spaces for solutions to this hyperbolic system. The bounds ensure stability of the equilibrium state in a linear approximation.