弱滑移条件下一类随机薄膜方程非负能量耗散解的存在性：第一部分——正解
Existence of nonnegative energy-dissipating solutions to a class of stochastic thin-film equations under weak slippage: Part I -- positive solutions
AI总结：
本文针对 $n\in(2,3)$ 的随机薄膜方程，通过新的有限元离散非负性方法控制 Itô 修正项，证明严格正解存在并建立能量—界面势衰减估计。
摘要：
对于迁移率指数 $n\in(2,3)$，我们证明了具有奇异有效界面势和 Stratonovich 型低阶项的随机薄膜方程严格正解的存在性。着眼于在第二部分中利用这些解来构造具有紧支集初始数据的随机薄膜方程的表面张能耗散解（未来工作将证明其有限传播速度），我们建立了表面张能与有效界面势之和的衰减估计，而无需依赖涉及初始数据的其他泛函。除了推广此前在 $n=2$ 情形中使用的技术并改进离散解的振荡估计外，本文主要的分析创新是一种离散化方法：在周期边界条件及参数区间 $n\in(2,3)$ 下，该方法证明了积分 $\int_{\mathcal O}(u^{n-2}u_x)_x u_{xx}\\,dx$ 的有限元对应项具有非负性。这一非负性质用于控制表面张能衰减估计中的 Itô 修正项。由此，它成为获得表面张能与有效界面势之和所需衰减估计的关键，而这些估计使第二部分能够建立有效界面势趋于零的奇异极限。
英文摘要：
For mobility exponents $n \in (2,3)$, we prove existence of strictly positive solutions to stochastic thin-film equations with singular effective interface potential and Stratonovich-type lower-order terms. With the perspective of using these solutions in Part II to construct surface-tension-energy dissipating solutions to stochastic thin-film equations with compactly supported initial data, for which finite speed of propagation is shown in future work, we establish decay estimates on the sum of surface-tension energy and effective interface potential -- without relying on further functionals involving initial data. Besides an extension of earlier techniques used in the case $n=2$ and a refinement of oscillation estimates for discrete solutions, the main analytical novelty of this paper is a discretization method which shows nonnegativity for a finite-element counterpart of the integral $\int_{\mathcal O} (u^{n-2}u_{x})_x u_{xx} dx$ under periodic boundary conditions in the parameter regime $n \in (2,3)$. This nonnegativity property serves to control Itô-correction terms in the estimate for the decay of the surface-tension energy. This way, it is the key to obtain the desired decay estimates for the sum of surface-tension energy and effective interface potential which permit to establish the singular limit of vanishing effective interface potential in Part II.
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