论文标题
布朗运动的非线性爱因斯坦范式和解决方案的本地化特性
Nonlinear Einstein paradigm of Brownian motion and localization property of solutions
论文作者
论文摘要
我们采用了爱因斯坦随机步行范式的概括，以扩散以非差异形式得出一类多维退化非线性抛物线方程。具体而言，在这些方程式中，扩散系数可以取决于因变量及其梯度，并且当后者中的任何一个时，它都会消失。众所周知，这种退化方程的溶液可以表现出有限的传播速度（溶液的所谓定位特性）。我们使用de giorgi-ladyzhenskaya迭代程序为非差异方程式提供了此属性的证明。然后，为一个空间维度的情况建立了映射定理的映射定理。通过有限差异方案的数值结果用于说明此特殊情况的主要数学结果。为了完整性，我们还提供了具有有限传播功能速度的一维自相似解决方案的明确结构，从kompaneets的意义上讲，它是zel'dovich-barenblatt。因此，我们展示了传播的有限速度如何定量地取决于模型的参数。
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.
