具有反常标度的奇异非线性扩散方程的自洽展开与场论重整化群
Self-consistent expansion and field-theoretic renormalization group for a singular nonlinear diffusion equation with anomalous scaling
AI总结：
本研究将自洽展开方法应用于偏微分方程的奇异摄动问题，以Barenblatt非线性扩散方程为例验证其能改进反常维数的近似精度，还构建了场论框架并指出该方法可推广至更多奇异摄动场景。
摘要：
自洽展开方法是处理强耦合问题的有力工具，这类问题原本可能超出微扰理论的研究范围，该方法即使在低阶近似下也能提供精度惊人的结果。它最初以雏形形式应用于充分发展湍流研究，随后被成功应用于包括聚合物统计、界面动力学以及非简谐振子高阶微扰理论在内的各类问题。本文表明，自洽展开可应用于偏微分方程理论中出现的奇异摄动问题。我们将其应用于描述多孔介质渗流的Barenblatt非线性扩散方程，该方程的长时间渐近行为表现出反常维数，可通过微扰重整化群进行系统计算。我们发现，即使是一阶自洽展开，也能改进一阶微扰重整化群得到的反常维数近似结果，在强耦合区域尤为明显。我们还为确定性偏微分方程构建了场论框架，以推动自洽展开在其他动力系统中的应用，并以Barenblatt方程为例说明了其应用方式。我们关于重整化群与自洽展开结合的研究结果，适用范围仅限于长时间渐近行为由不完全相似性控制的偏微分方程。但我们的工作表明，这些方法可应用于更广泛的奇异摄动问题，例如边界层理论、多尺度分析和匹配渐近展开，目前仅使用重整化群方法就已能在这些问题中得到出色的近似结果。
英文摘要：
The method of self-consistent expansions is a powerful tool for handling strong coupling problems that might otherwise be beyond the reach of perturbation theory, providing surprisingly accurate approximations even at low order. First applied in its embryonic form to fully-developed turbulence, it has subsequently been successfully applied to a variety of problems that include polymer statistics, interface dynamics and high order perturbation theory for the anharmonic oscillator. Here we show that the self-consistent expansion can be applied to singular perturbation problems arising in the theory of partial differential equations. We demonstrate its application to Barenblatt's nonlinear diffusion equation for porous media filtration, where the long-time asymptotics exhibits anomalous dimensions that can be systematically calculated using the perturbative renormalization group. We find that even the first order self-consistent expansion improves the approximation of the anomalous dimension obtained by the first-order perturbative renormalization group, especially in the strong coupling regime. We also develop a field-theoretic framework for deterministic partial differential equations to facilitate the application of self-consistent expansions to other dynamic systems, and illustrate its application using the example of Barenblatt's equation. The scope of our results on the combination of renormalization group and self-consistent expansions is limited to partial differential equations whose long-time asymptotics is controlled by incomplete similarity. However, our work suggests that these methods could be applied to a broader suite of singular perturbation problems such as boundary layer theory, multiple scales analysis and matched asymptotic expansions, for which excellent approximations using renormalization group methods alone are already available.
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