热Smoluchowski流体的稳定性
Stability of a Hot Smoluchowski Fluid
摘要：
我们研究耦合的非线性抛物方程描述的流体的材料密度和温度，空间和时间的函数。在一维中，我们找到了一些固定的解决方案，对应于固定的边界上的温度，与无逃逸边界条件的材料。对于特殊的情况下，在边界上的温度是相同的两端，小扰动的线性方程的固定的解决方案，他们是受边界条件，狄利克雷的温度和无流动条件的材料。本文证明了时间演化的生成元L的谱是实的、离散的和非正的，即使L不是自伴的，它被看作是[0，1]上平方可积函数的Hilbert空间上的算子。这个结果对于定态的稳定性是必要的，但可能还不够。问题在于L不是扇形算子，它的数值范围是整个复平面。
英文摘要：
We study coupled non-linear parabolic equations for a fluid described by a material density and a temperature, both functions of space and time. In one dimension, we find some stationary solutions corresponding to fixing the temperature on the boundary, with no-escape boundary conditions for the material. For the special case, where the temperature on the boundary is the same at both ends, the linearised equations for small perturbations about a stationary solution are derived; they are subject to the boundary conditions, Dirichlet for the temperature and no-flow conditions for the material. The spectrum of the generator L of time-evolution, regarded as an operator on the Hilbert space of square-integrable functions on [0,1], is shown to be real, discrete and non-positive, even though L is not self-adjoint. This result is necessary for the stability of the stationary state, but might not be sufficient. The problem lies in the fact that L is not a sectorial operator; its numerical range is the whole of the complex plane.
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