Spectrum of a stationary uniform-speed transport equation with reflection-type boundary conditions
The following stationary single-velocity transport equation with boundary conditions of reflection type is considered: \[ (1)\quad v^{-1}u_ t(x,\Omega,t)=-\Omega \cdot \nabla u(x,\Omega,t)-\sigma (x,\Omega,t)+(\gamma /4\pi)\int_{\Gamma}d\Omega 'u(x,\Omega ',t), \] \[ (2)\quad u(s,\Omega,t)=\alpha u(s,\Omega -2(n(s)\cdot \Omega)n(s),t),\quad s\in \partial V,\quad \Omega \cdot n(s)\geq 0. \] Here: \(x\in V\), V is a bounded, convex, open domain with sufficiently smooth boundary \(\partial V\); \(t>0\); \(\Gamma\) is the surface of the unity sphere of \(R^ 3\); \(\Omega\) is the unity vector defined on \(\Gamma\) which gives the sense of movement of the particles with constant velocity \(v>0\); n(s) is the unity vector of the interior normal at \(\partial V\); \(\sigma\), \(\gamma\) are positive constants; \(\alpha\) is a parameter, \(\alpha\in [0,1].\) Some spectral properties for the stationary problem (1), (2) are studied.
stationary single-velocity transport equation
scientific article; zbMATH DE number 4159385
scientific article