AI 中文总结
研究区间上非线性扩散方程，通过EDP收敛研究相关梯度结构，展示自由能等性质向有效动力学关系的迁移，从线性昂萨格关系出发得到指数增长的马塞兰 - 德东德动力学。
AI 中文摘要
我们考虑区间上的非线性扩散方程，其中中心附近小区域的扩散系数按比例缩放，使其近似膜处的传输条件。虽然解的极限行为已为人熟知，但我们研究由自由能和耗散势给出的相关梯度结构在能量耗散原理（EDP）意义下的收敛性。EDP收敛提供了一个唯一指定的极限梯度结构，它根据膜的有效动力学关系重新表述传输条件，该关系将化学势的跃变与通过膜的通量联系起来。我们展示了所选自由能的\AAA性质和小尺度扩散的迁移率如何迁移到有效动力学关系。一个惊人的结果是，从线性扩散方程的奥托梯度结构的线性昂萨格关系出发，可得到指数增长的动力学关系，即所谓的马塞兰 - 德东德动力学。
英文摘要
We consider nonlinear diffusion equations on an interval where the diffusion coefficient in a small region near the center is scaled such that it approximates a transmission condition at a membrane. While the limiting behavior of the solutions is well-understood, we study the convergence in the sense of the energy-dissipation principle (EDP) of associated gradient structures given in terms of a free energy and a dissipation potential. EDP-convergence provides a uniquely specified limiting gradient structure that reformulates the transmission condition in terms of an effective kinetic relation for the membrane, which relates the jump of the chemical potential and the flux through the membrane. We show how \AAA properties of the chosen free energy and the mobility of small-scale diffusion migrate to the effective kinetic relation. A surprising result is that starting from the linear Onsager relation of Otto's gradient structure for the linear diffusion equation, one obtains an exponentially growing kinetic relation, the so-called Marcelin-De Donder kinetic.
Comments47 pages, 2 figures
