布朗运动的非线性 Einstein 范式与解的局部化性质
Nonlinear Einstein paradigm of Brownian motion and localization property of solutions
AI总结：
本文推广 Einstein 随机游走范式导出多维退化非线性抛物方程，证明解的有限传播速度，并给出一维映射、自相似解及数值验证。
摘要：
我们采用 Einstein 关于扩散的随机游走范式的推广，推导出一类非散度形式的多维退化非线性抛物方程。具体而言，在这些方程中，扩散系数可以同时依赖于因变量及其梯度，并且当后两者中任一者为零时，该系数也为零。已知这类退化方程的解可以表现出有限传播速度（即所谓解的局部化性质）。我们利用针对非散度形式方程的 De Giorgi–Ladyzhenskaya 迭代程序给出了该性质的证明。随后，在一维空间情形下，建立了到控制方程散度形式版本的映射定理。通过有限差分格式得到的数值结果，用于说明这一特殊情形下的主要数学结论。为完整起见，我们还按 Kompaneets–Zel'dovich–Barenblatt 意义，显式构造了具有有限传播速度函数的一维稳自相似解。由此，我们展示了有限传播速度如何定量地依赖于模型参数。
英文摘要：
We employ a generalization of Einstein's random walk paradigm for diffusion to derive a class of multidimensional degenerate nonlinear parabolic equations in non-divergence form. Specifically, in these equations, the diffusion coefficient can depend on both the dependent variable and its gradient, and it vanishes when either one of the latter does. It is known that solutions of such degenerate equations can exhibit finite speed of propagation (so-called localization property of solutions). We give a proof of this property using a De Giorgi--Ladyzhenskaya iteration procedure for non-divergence-from equations. A mapping theorem is then established to a divergence-form version of the governing equation for the case of one spatial dimension. Numerical results via a finite-difference scheme are used to illustrate the main mathematical results for this special case. For completeness, we also provide an explicit construction of the one-dimensional self-similar solution with finite speed of propagation function, in the sense of Kompaneets--Zel'dovich--Barenblatt. We thus show how the finite speed of propagation quantitatively depends on the model's parameters.
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