AI 中文总结
本文针对描述表面张力驱动薄膜对流的一维四阶非线性退化抛物型方程初边值问题，通过正则化、Galerkin近似、能量熵泛函估计与紧性引理，证明了其弱解的全局存在性与非负性。
AI 中文摘要
本文研究一维四阶非线性退化抛物型方程初边值问题弱解的全局存在性与非负性，该模型描述由表面张力驱动的薄膜对流现象。分析方法从正则化问题及对应的Galerkin近似格式的构建开始，先证明近似问题解的存在性；随后通过构造专门的能量与熵泛函，推导近似解的一致先验估计；利用Aubin-Lions紧性引理取极限，证明极限函数的非负性；最终验证该极限确为原初边值问题的全局弱解。
英文摘要
This paper investigates the global existence and non-negativity of weak solutions to an initial-boundary value problem for a one-dimensional fourth-order nonlinear degenerate parabolic equation. This model governs the convection phenomena in thin films driven by surface tension. Our analytical approach begins with the formulation of a regularized problem and a corresponding Galerkin approximating scheme. We first establish the existence of solutions to the approximate problem. Subsequently, by constructing specialized energy and entropy functionals, we derive uniform a priori estimates for the approximating solutions. Leveraging the Aubin-Lions compactness lemma, we pass to the limit and establish the non-negativity of the limit function. Finally, we demonstrate that this limit is indeed a global weak solution to the original initial-boundary value problem.
Comments17 pages
