AI 中文总结
该研究将加权能量耗散（WED）变分方法用于构造Cahn-Hilliard型四阶退化抛物型方程的解，突破了WED仅适用于Wasserstein型二阶偏微分方程的局限，在时间维度实现椭圆正则化并获良好近似性质。
AI 中文摘要
我们应用加权能量耗散（WED）变分方法，得到Cahn-Hilliard型四阶退化抛物型方程的全局时间近似解。与存在性理论的标准方法不同，WED在时间维度而非空间维度引入椭圆正则化；该近似解已保证了相场的约束性，无需修改李雅普诺夫泛函即可得到先验估计，且近似解在时间上弱可微——在迁移率为线性时甚至二阶可微。尽管WED已广泛用于构造希尔伯特空间中的梯度流，但这似乎是其首次应用于Wasserstein型二阶偏微分方程之外的度量梯度流。
英文摘要
We apply the variational method of Weighted Energy Dissipation (WED) to obtain a global-in-time approximation of solutions to fourth order degenerate parabolic equations of Cahn-Hilliard type. Differently from the standard approach to the existence theory, WED induces an elliptic regularization in time, not in space. The confinement on the phase field is guaranteed already for the approximation, a priori estimates follow without modification of Lyapunov functionals, and the approximations are weakly differentiable in time --- even twice in the case of linear mobility. While WED has been widely used for the construction of gradient flows in Hilbert spaces, this appears to be the first application of WED to a metric gradient flow beyond second order PDEs of Wasserstein type.
Comments32 pages, no figures
