Maxwell及其相关方程的均匀化
Homogenisation for Maxwell and Friends
AI总结：
本文在Picard演化方程框架下，利用非局部H拓扑细化解算子对系数的连续依赖性，证明部分分量强收敛、其余弱收敛，并通过数值与解析例子验证理论尖锐性，适用于非局部系数及PDE类型混合振荡情形。
摘要：
我们在Picard意义下的演化方程框架中，细化了在非局部H-拓扑（即Schur拓扑）下解算子对系数的连续依赖性的理解。我们证明了解算子的某些分量强收敛。在常微分方程均匀化问题中已知的弱收敛行为，在其余解算子分量上得以恢复。这些结果由一类丰富的例子支撑，这些例子也经过了数值处理，表明理论发现具有一定的尖锐性。我们还提供了证明这种尖锐性的一个例子的解析处理。尽管所有考虑的示例都包含局部系数，但主要定理和结构见解具有算子理论性质，因此也适用于非局部系数。所考虑问题类别的主要优势在于它们包含不同类型的混合，可能在不同类型的偏微分方程之间高度振荡；一个典型例子是Maxwell方程在经典方程与相应的涡流近似之间高度振荡。
英文摘要：
We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal $H$-topology viz Schur topology in the setting of evolutionary equations in the sense of Picard. We show that certain components of the solution operators converge strongly. The weak convergence behaviour known from homogenisation problems for ordinary differential equations is recovered on the other solution operator components. The results are underpinned by a rich class of examples that, in turn, are also treated numerically, suggesting a certain sharpness of the theoretical findings. Analytic treatment of an example that proves this sharpness is provided too. Even though all the considered examples contain local coefficients, the main theorems and structural insights are of operator-theoretic nature and, thus, also applicable to nonlocal coefficients. The main advantage of the problem class considered is that they contain mixtures of type, potentially highly oscillating between different types of PDEs; a prototype can be found in Maxwell's equations highly oscillating between the classical equations and corresponding eddy current approximations.
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