AI 中文总结
本文针对描述薄流体膜表面张力驱动对流的二维四阶非线性退化抛物方程，通过正则化近似、Galerkin方法等，证明了其非负弱解的整体存在性。
AI 中文摘要
本文建立了描述薄流体膜中表面张力驱动对流的二维四阶非线性退化抛物方程非负弱解的整体存在性。首先，我们构造正则化近似问题，通过Galerkin方法证明其可解性。结合能量与熵泛函以及奇异熵条件$1/h_0 \in L^1(\Omega)$，我们获得高阶空间和时间导数的一致先验界。这些界使我们能运用Aubin-Lions引理和Gagliardo-Nirenberg不等式得到强紧性和本质$L^6$可积性。此外，我们采用Alber-Zhu框架严格定义高阶局部弱导数并取极限。最后，证明极限函数非负，确认为原问题的整体弱解。
英文摘要
This paper establishes the global existence of non-negative weak solutions to a two-dimensional, fourth-order nonlinear degenerate parabolic equation modeling surface-tension-driven convection in thin fluid films. First, we construct a regularized approximate problem and prove its solvability via the Galerkin method. Utilizing energy and entropy functionals alongside a singular entropy condition $1/h_0 \in L^1(Ω)$, we secure uniform a priori bounds for higher-order spatial and time derivatives. These bounds enable the use of the Aubin-Lions lemma and Gagliardo-Nirenberg inequalities to achieve strong compactness and essential $L^6$-integrability. Furthermore, we adopt the Alber-Zhu framework to rigorously define higher-order local weak derivatives and pass to the limit. Finally, we prove the limit function is non-negative, confirming it as a global weak solution to the original problem.
Comments20 pages
