arXiv 1902.025642019-02-07math.AP

具有一般强迫的时间分数阶Fokker-Planck方程解的存在唯一性和正则性

Existence, uniqueness and regularity of the solution of the time-fractional Fokker-Planck equation with general forcing

Kim-Ngan Le, William McLean, Martin Stynes

备注：: 20 pages, no figures
发表信息：: Commun. Pure Appl. Anal. 18(5), 2765-2787, 2019
DOI：: 10.3934/cpaa.2019124

摘要：

考虑了具有微分算子$u_t-\nabla\cdot(\partial_t^{1-α}κ_α\nabla u-\textbf{F}\partial_t^{1-α}u)$的时间分数阶Fokker-Planck初边值问题，其中$0<α<1$ .强迫函数$\textbf{F} = \textbf{F}(t,x)$，这是更难以分析的情况下$\textbf{F}=\textbf{F}(x)$以前研究的其他作者。空间域$Ω\subset\mathbb{R}^d$，其中$d\ge 1$具有平滑边界。在初始数据$u_0$位于$L^2(Ω)$中的假设下，证明了温和解$u$的存在性、唯一性和正则性.对于$1/2<α<1$和$u_0\in H^2(Ω)\cap H_0^1(Ω)$，它表明，$u$成为一个经典的解决方案的问题。经典解的时间导数的估计推导-这些都是已知的需要在这个问题的数值分析。

英文摘要：

A time-fractional Fokker-Planck initial-boundary value problem is considered, with differential operator $u_t-\nabla\cdot(\partial_t^{1-α}κ_α\nabla u-\textbf{F}\partial_t^{1-α}u)$, where $0<α<1$. The forcing function $\textbf{F} = \textbf{F}(t,x)$, which is more difficult to analyse than the case $\textbf{F}=\textbf{F}(x)$ investigated previously by other authors. The spatial domain $Ω\subset\mathbb{R}^d$, where $d\ge 1$, has a smooth boundary. Existence, uniqueness and regularity of a mild solution $u$ is proved under the hypothesis that the initial data $u_0$ lies in $L^2(Ω)$. For $1/2<α<1$ and $u_0\in H^2(Ω)\cap H_0^1(Ω)$, it is shown that $u$ becomes a classical solution of the problem. Estimates of time derivatives of the classical solution are derived---these are known to be needed in numerical analyses of this problem.

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摘要：

英文摘要：

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