AI 中文总结
针对薄膜流动或二元混合物相分离的四阶非线性偏微分方程，提出能量一致的双曲松弛系统，证明其弱熵解收敛至极限方程光滑解，并通过数值算例验证分析结果。
AI 中文摘要
本文针对薄膜流动建模或二元混合物相分离中出现的一类一般四阶非线性偏微分方程，提出一种新颖的双曲松弛系统。该逼近构造为耗散能量，当松弛参数趋于零时可恢复极限方程的能量（李雅普诺夫）泛函。利用相对能量框架，证明了松弛系统的弱熵解收敛至极限方程的光滑解。通过一系列薄膜方程和Cahn-Hilliard方程的数值算例验证了分析结果。
英文摘要
In this article, we propose a novel hyperbolic relaxation system for a general class of fourth-order nonlinear partial differential equations arising in the modelling of thin film flows or the phase separation of binary mixtures. The approximations are constructed to dissipate energies which recover the energy (Lyapunov) functional of the limit equation when the relaxation parameters vanish. Using the relative energy framework, we prove the convergence of weak entropy solutions of the relaxation system to smooth solutions of the limit equation. We validate our analysis with a series of numerical examples for thin film equations and Cahn-Hilliard equations.
