论文标题
在具有扩散边界条件的通道中，玻尔兹曼方程的传热问题
Heat transfer problem for the Boltzmann equation in a channel with diffusive boundary condition
论文作者
论文摘要
在本文中，我们研究了通道中的一维稳定玻尔兹曼流。假定该通道的墙壁具有消失的速度，并给定温度$θ_0$和$θ_1$。 Esposito等人[13,14]研究了这个问题，其中他们表明该解决方案趋向于局部麦克斯韦，其参数满足了具有无滑动边界条件的可压缩Navier-Stokes方程。但是，许多数值实验表明，流体层并不完全粘在边界上。在Knudsen数量相当小的状态下，滑移现象在边界附近很重要。因此，我们通过考虑滑动边界条件来重新审视这个问题。遵循[9]的线，我们将首先给出形式的渐近分析，以确保由玻尔兹曼方程控制的流可以通过具有温度跳跃条件的稳定CNS方程的叠加来准确地近似于温度跳跃条件和两个位于末端点的Knudsen层。然后，我们将在其余部分建立一个均匀的$ l^\ infty $估计，并为可压缩的Navier-Stokes方程得出滑动边界条件。
In this paper, we study the 1D steady Boltzmann flow in a channel. The walls of the channel are assumed to have vanishing velocity and given temperatures $θ_0$ and $θ_1$. This problem was studied by Esposito et al [13,14] where they showed that the solution tends to a local Maxwellian with parameters satisfying the compressible Navier-Stokes equation with no-slip boundary condition. However, a lot of numerical experiments reveal that the fluid layer does not entirely stick to the boundary. In the regime where the Knudsen number is reasonably small, the slip phenomenon is significant near the boundary. Thus, we revisit this problem by taking into account the slip boundary conditions. Following the lines of [9], we will first give a formal asymptotic analysis to see that the flow governed by the Boltzmann equation is accurately approximated by a superposition of a steady CNS equation with a temperature jump condition and two Knudsen layers located at end points. Then we will establish a uniform $L^\infty$ estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously.
