发表机构
Southern University of Science and Technology（南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为超线性随机反应-扩散方程建立适定性、正则性及强逼近的一般框架，首次构造驯化有限元方法并证明其最优强收敛速率。
AI 中文摘要
本文为具有超线性漂移和扩散系数的随机反应-扩散方程（SRDE）建立了适定性、正则性和强逼近的一般框架。我们首先将W. Liu和M. Röckner（J. Funct. Anal., 2010, 2902–2922）以及W. Liu（J. Differential Equations, 2013, 572–592）中的适定性结果推广到Gelfand三元组V ↪ H ↪ V*中具有超线性扩散的情形，其中V配备范数‖·‖_V，并通过建立关于‖X‖_V^p（p ≥ 2）的新Itô公式推导出矩估计。然后我们将这一抽象结果应用于SRDE，当初始数据位于同一Sobolev空间时，对任意γ ∈ [0,1]建立更高的空间正则性Ḣ^{1+γ}，并获得时间Hölder正则性。最后，我们在驯化函数的一般假设下为SRDE构造了一族驯化有限元方法（tamed-FEMs），推导了其长期无条件稳定性，并建立了最优强收敛速率。据我们所知，这是首个针对具有超线性扩散系数的SPDE的强逼近结果。
英文摘要
This paper develops a general framework for the well-posedness, regularity, and strong approximation of the stochastic reaction--diffusion equation (SRDE) with superlinear drift and diffusion coefficients. We first extend the well-posedness results in \emph{W. Liu and M. Röckner, J. Funct. Anal., 2902--2922, 2010} and \emph{W. Liu, J. Differential Equations, 572--592, 2013} to the case of superlinear diffusion in the Gelfand triple \(V \hookrightarrow H \hookrightarrow V^*\), with \(V\) equipped with the norm \(\|\cdot\|_V\), and derive a moment estimate by establishing a new Itô formula for \(\|X\|_V^p\) with general \(p \ge 2\). We then apply this abstract result to the SRDE, establish higher spatial regularity \(\dot H^{1+γ}\) for any \(γ\in [0,1]\) whenever the initial datum lies in the same Sobolev space, and obtain temporal Hölder regularity. Finally, we construct a family of tamed finite element methods (tamed-FEMs) for the SRDE under general assumptions on the tamed functions, derive their long-time unconditional stability, and establish optimal strong convergence rates. To our knowledge, this is the first strong approximation result for SPDEs with superlinear diffusion coefficients.
