The existence of mild and classical solutions for time fractional Fokker–Planck equations
The existence of mild and classical solutions for time fractional Fokker–Planck equations
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DOI：
10.1007/s00605-022-01710-4
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发表时间：
2022-04
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影响因子：
--
通讯作者：
Li Peng;Yong Zhou
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文献类型：
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作者：
Li Peng;Yong Zhou
时间分数Fokker-Planck方程可以用来描述依赖于时间和空间的外部力场F(t，x)中的次扩散。在这篇论文中，我们将其转换为以下问题的形式：\Docentclass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$\Begin{Begin_tu-\kappa_\Alpha\Partial_t^{1-\Alpha}\Delta u=\nabla\cdot(F\Partial_t^{1-\Alpha}u)+f，\end{已对齐}$\end{文档}，其中。其次，在研究经典解问题时，我们分析了“工作空间”与的取值范围之间的关系。最后，通过构造一个合适的加权Hölder连续函数空间，得到了经典解的存在性，而不受以下条件的限制：\Docentclass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amsbsy}\usepackage{amsbsy}\usepackage{upgreek}\setLong{oddsidemargin}{-69pt}\Begin{Document}$\Alpha\in(\frac{1}}\usepackage{upgreek}\setlong{oddsidemargin}{-69pt}\{document}$\pha\in(\frac{1}\{2}，1\右)$$\end{文档}。
Time fractional Fokker–Planck equations can be used to describe the subdiffusion in an external time-and space-dependent force fieldF(t,x). In this paper, we convert it to the form of the following problems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\partial _tu-\kappa _\alpha \partial _t^{1-\alpha }\Delta u=\nabla \cdot (F\partial _t^{1-\alpha }u)+f,\end{aligned}$$\end{document}where. We obtain some results on existence and uniqueness of mild solutions allowing the “working space" that may have low regularity. Secondly, we analyze the relationship between “working space" and the value range ofwhen investigating the problem of classical solutions. Finally, by constructing a suitable weighted Hölder continuous function space, the existence of classical solutions is derived without the restriction on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in \left( \frac{1}{2},1\right) $$\end{document}.