A nonstationary problem of complex heat transfer
The following initial value problem for the normalized evolution diffusion model describing radiative, conductive, and convective heat transfer in a bounded region \(\Omega \subset {\mathbb R}^{3} \) is considered: \[ {\partial \theta \mathord{\left/ {\vphantom {\partial \theta \partial t}} \right. \kern-\nulldelimiterspace} \partial t} -a\Delta \theta +\mathbf{v}\cdot \nabla \theta +b\kappa _{a} \left(\left|\theta \right|\theta ^{3} -\varphi \right)=0, \eqno (1) \] \[ {v\partial \varphi \mathord{\left/ {\vphantom {v\partial \varphi \partial t}} \right. \kern-\nulldelimiterspace} \partial t} -\alpha \Delta \varphi +\kappa _{a} \left(\varphi -\left|\theta \right|\theta ^{3} \right)=0,\; x\in \Omega ,\; t\in \left(0, T\right), \eqno (2) \] \[ a\partial _{n} \theta +\beta \left(\theta -\theta _{b} \right)=0,\; a\partial _{n} \varphi +\gamma \left(\varphi -\varphi _{b}^{4} \right)=0, \eqno (3) \] \[ \left. \theta \right|_{t=0} =\theta _{0} ,\; \left. \varphi \right|_{t=0} =\varphi _{0} , \eqno (4) \] where \(\partial _{n} \) denotes the derivative in the direction of the external normal \({\mathbf n}\). The nonnegative function \(\theta _{b} \); the functions \(\beta =\beta (x)\), and \(\gamma =\gamma (x)\), \(x\in \Gamma \), which describe the reflective properties of the boundary and the initial functions \(\theta _{0} \) and \(\varphi _{0} \) are given. It should be noted that conditions of the third kind for the temperature are usually set on a rigid wall, where \(\mathbf v . n = 0\). In the given case, the third-kind conditions impose on the whole boundary and, in particular, on the inflow section simulate heat transfer at low normal velocities, \(\theta \) is the normalized temperature, \(\varphi \) is the normalized radiation intensity averaged over all directions, \(\mathbf v\) is a given velocity field, and \(\kappa _{a} \) is the absorption coefficient. The constants \(a,\, b\), and \(\alpha \), characterize the specifics of the studied process. The main results of this work consist in obtaining a priori estimates for the initial boundary value problem (1)--(4), which are then used to prove the unique solvability of the problem. In addition, the authors show that the equilibrium state \(\theta =\theta _{s} =\mathrm{const}\), \(\varphi =\theta _{s}^{4} \) is an exponentially stable singular point of the evolution system (1)--(4).
scientific article; zbMATH DE number 6505330
scientific article